
Go{pglit}^'°. 



qiq 



CORfRIGHT DEPOSm 



WORKS OF 

WALTER LORING WEBB 

PUBLISHED BY 

JOHN WILEY & SONS, Inc. 



Railroad Construction. — Theory and Practice. 

A Text-book for the Use of Students in Col- 
leges and Technical Schools. Sixth Edition. 
Rewritten and Enlarged. 16mo. xv + 831 
pages and 218 figures and 10 plates. Mo- 
rocco, $4.00 net. 

Technic of Surveying Instruments and 
Methods. 

16mo. Morocco. In collaboration with Prof. 
J. C. L. Fish. (In preparation.) 

The Economics of Railroad Construction. 

Small Svo. Second Edition, vii+347 pages, 
35 figures. Cloth, $2.50 net. 

The American Civil Engineers' Pocket Book. 

{Author of Section on Railroads.) 
Large 16mo. Morocco, $5.00 net. 



RAILROAD CONSTRUCTION 

THEORY AND PRACTICE 

A TEXT-BOOK FOR THE USE OF STUDENTS 
IN COLLEGES AND TECHNICAL SCHOOLS, 

AND 

A HAND-BOOK FOR THE USE OF ENGINEERS 
IN FIELD AND OFFICE, 



BY 

WALTER LORING WEBB, C.E., 

Member American Society of Civil Engineers; Member American Railway Engi- 
neering Association; Assistant Professor of Civil Engineering (Railroad 
Engineering) in the [ University of Pennsylvania, 1893-1901; 
Major^ Engineer Officers* Reserve Corps, U, S. A., etc. 



SIXTH EDITION, REVISED AND ENLARGED 
TOTAL ISSUE, THIRTEEN THOUSAND 



NEW YORK 

JOHN WILEY & SONS, Inc. 

London: CHAPMAN & HALL, Limited 

1917 



-Tf''"' 



^^^ AM 



Copyright, 1899, 1903. 1908. 1913. 1917, 

BY 

WALTER LORING WEBB. 




/y^- 



JUL -7 1917 



PRESS OF 

BRAUNWORTH & CO. 

BOOK MANUFACTURERS 

BROOKLYN, N. Y. 



CLA467775 



ri^ . / 



r^ PREFACE TO SIXTH EDITION. 

The revision of the fifth edition has been so extensive that 
^ it has almost amounted to a rewriting of the book. Compara- 
C lively few pages have been left without some revision. 

The last few years have seen a greater advance in the science 

J of railroad construction than any similar period in its previous 

history. This has been largely due to the combined work of 

the several Standing Committees of the American Railway 

I Engineering Association. The writer has received special per- 

' mission to quote from the Association's publications and has 

availed himself of the privilege, because he considers that the 

decisions of such an Association are, in general, the highest 

j authority obtainable. 

Considerable new matter has been added on the general sub- 
ject of railroad surveys, and the handHng of surveying parties. 
One feature of the additions has been the emergency medical 
and surgical treatment which the engineer-in-charge, as respon- 
sible head of the party, must sometimes supply when regular 
professional advice is absolutely unobtainable and the engineer 
must choose between seeing the victim die (or become perma- 
nently injured), or assuming the unwelcome responsibility of 
applying simple instructions plus common sense. It usually 
means choosing the lesser of two evils. The author wishes to 
acknowledge his indebtedness to his friends. Dr. G. Victor 
Janvier and Dr. Henry P. DeForest, for advice and the revision 
of these sections, which may thus be depended on to be tech- 
nically correct. 

Those familiar with the former editions of this work will note 
that the computations previously given for the unit values of 
saving one foot (or mile) of distance, one degree of curvature, or 
one foot of rise-and-fall, have now been omitted. This is due 
to the belief, as expressed by the Economics Committee of the 

iii 



IV PEEFACE TO SIXTH EDITION.- 

Am. Rwy. Eng. Assoc, that all previously published methods 
of making such calculations are unreliable since they ignore 
certain operating conditions peculiar to each road, and that the 
application of such unit figures may lead to unwarranted con- 
clusions. It may be that a method will be sometime devised 
by which some simple and satisfactory form of unit value may 
be used. At present, the most practicable method yet proposed 
is to compute the costs of operating two suggested routes on 
the basis of an assumed amount and kind of traffic and compare 
the results. 

Walter Loring Webb. 

Philadelphia, Pa.; 
Nov., 1916. 



I , » < • » 



TABLE OF CONTENTS 



CHAPTER I. 

RAILKOAD SURVEYS. 



PAGE 
Reconnoissance 1 

1. Character of a reconnoissance survey. 2. Selection of a gen- 
eral route. 3. Valley route. 4. Cross-country route. 5. Moun- 
tain route. 6. Existing maps. 7. Determination of relative 
elevations. Barometrical method. 8. Horizontal measurements, 
bearings, etc. 9. Importance of a good reconnoissance. 
Preliminary surveys 14 

10. Character of a survey. 11. Cross-section method. 12. 
Cross-sectioning. 13. Stadia method. 14. Form for stadia notes. 
15. The reduction of stadia observations. 16. Stadia method vs. 
cross-section method. 17. "First" and "second" preliminary 
surveys. 
Location surveys 24 

18. "Paper location." 19. Preparation of the notes. 20. 
Surveying methods. 21. Form of notes. 22. Number of men 
required in surveying parties. 
Maintenance of surveying parties 36 

23. Economy and efficiency. 24. Country hotels and farm 
houses. 25. Camping outfits. 26. Tent floors. 27. Tent stoves. 
28. Dining tables. 29. Cooking utensils, table 'ware, tools, etc. 
30. Drawing tables. 31. Stationery and map chest. 32. Pro- 
visions. 33. Beds. 34. Transportation. 35. Clothing. 
Medical and surgical treatment 47 

36. Responsibility of engineer-in-charge. 37. Appliances. 38. 
Antiseptics. 39. Drinking water. 40. Bleeding. 41. Ailments 
and diseases; medicines. 42. Drowning; electric shock; as- 
phyxiation. 43. Fractures. 44. Snake or insect bites. 45. 
Wounds. 

CHAPTER II. 

alinement, 
Simple curves 55 

46. Designation of curves. 47. Metric curves. 48. Length 
of a subchord. 49. Length of a curve. 50. Curve notation. 51. 
Elements of a curve. 52, Relation between T, E, and A. 53. 
Elements of a 1° curve. 54. Exercises. 55. Curve location by 
deflections. 56. Instrumental work. 57. Curve location by 



VI TABLE OF CONTENTS. 

PAGE 

two transits. 58. Curve location by tangential offsets. 59. Curve 
location by middle ordinates. 60. Curve location by offsets from 
the long chord. 61. Use and value of the above methods. 62. 
Obstacles to location. 63. Modifications of location. 64, Limita- 
tions in location. 65. Determination of the curvature of existing 
track. 66. Problems. 

Compound curves 77 

67. Nature and use. 68. Mutual relations of the parts of a 
compound curve having two branches. 69. Modifications of loca- 
tion. 70. Problems. 

Transition curves 82 

71. Superelevation of the outer rail on curves. 72. Practical 
rules for superelevation. 73. Transition from level to inclined 
track. 74. Fundamental principle of transition curves. 75. Va- 
rieties of transition curves. 76. Proper length of spiral. 77. 
Symbols. 78. Deflections. 79. Location of spirals and circular 
curve with respect to tangents. 80. Field-work. 81. To replace 
a simple curve by a curve with spirals. 82. Application of tran- 
sition curves to compound curves. 83. To replace a compound 
curve by a curve with spirals. 

Vertical curves 100 

84. Necessity for their use. 85. Required length. 86. Form 
of curve. 87. Numerical example. 

CHAPTER III. 

earthwork. 

Form of excavations and embankments 104 

88. Usual form of cross-section in cut and fill. 89. Terminal 
pyramids and wedges. 90. Slopes. 91. Compound sections. 
92. Width of roadbed. 93. Form of subgrade. 94. Ditches. 
95. Effect of sodding the slopes, etc. 

Earthwork surveys 112 

96. Relation of actual volume to the numerical results. 97. 
Prismoids. 98. Cross-sectioning. 99. Position of slope-stakes. 
100. Setting slope-stakes by means of "automatic" slope-stake 
rods. 

Computation of volume 118 

101. Simple approximations. 102. Approximate volume, level 
sections. 103. Numerical example, level sections. 104. Equiva- 
lent sections. 105. Three-level sections. 106. Computation of 
products. 107. Irregular sections. 108. Volume of an irregular 
prismoid. 109. Numerical example; approximate volume, irreg- 
ular sections. 110. Prismoidal correction. 111. Correction for 
triangular prismoid. 112. Correction for level sections. 113. 
Prismoidal correction for "equivalent" sections. 114. Prismoidal 
correction for three-level sections. 115. Prismoidal correction; 
irregular sections. 116. Magnitude of the probable error of this 
method. 117. Numerical illustration of the accuracy of the ap- 
proximate rule, 118. Cross-sectioning irregular sections. 119. 



TABLE OF CONTENTS. Vll 

PAGE 

Side-hill work. 120. Borrow-pits. 121. Correction for curvature. 
122. Eccentricity of the center of gravity. 123. Center of gravity 
of side-hill sections. 124. Example of curvature correction. 125. 
Accuracy of earthwork computations. 126. Approximate com- 
putations from profiles. 

Formation of embankments 149 

127. Shrinkage of earthwork. 128. Proper allowance for shrink- 
age. 129. Methods of forming embankments. 

Computation of haul 155 

130. Nature of subject. 131. Mass diagram. 132. Properties 
of the mass curve. 133. Area of the mass curve. 134. Value of 
the mass diagram. 135. Changing the grade line. 136. Limit of 
free haul. 

Elements of the cost of earthwork 163 

137. Analysis of the total cost into items. 138. Loosening. 
139. Loading. 140. Hauling. 141. Choice of method of haul 
dependent on distance. 142. Spreading. 143. Keeping roadways 
in order. 144. Trimming cuts to their proper cross-section. 

145. Repairs, wear, depreciation, and interest on cost of plant. 

146. Superintendence and incidentals. 147. Contractor's profit 
and contingencies. 148. Limit of profitable haul. 

Blasting. . . . ; 184 

149. Explosives. 150. Drilling. 151. Position and direction 
of drill-holes. 152. Amount of explosive. 153. Tamping. 154. 
Exploding the charge. 155. Cost. 156. Classification of ex- 
cavated material. 157. Specifications for earthwork. 

CHAPTER IV. 

trestles. 

158. Extent of use. 159. Trestles vs. embankments. 160. Two 

principal types 194 

Pile trestles 196 

161. Pile bents. 162. Methods of driving piles. 163. Pile- 
driving formulae. 164. Pile-points and pile-shoes. 165. Details 
of design. 166. Specifications for timber piles. 167. Pile-driving 
— principles of practice. 168. Cost of pile trestles. 
Framed trestles 205 

169. Typical design. 170. Joints. 171. Multiple-story con- 
struction. 172. Span. 173. Foundations. 174. Longitudanal 
bracing. 175. Lateral bracing. 176. Abutments. 
Floor systems 211 

177. Stringers. 178. Corbels. 179. Guard-rails. 180. Ties on 
trestles. 181. Superelevation of the outer rail on curves. 182. 
Protection from fire. 183. Timber. 184. Cost of framed timber 
trestles. 
Design of wooden trestles 217 

185. Common practice. 186. Required elements of strength. 
187. Strength of timber. 188. Loading. 189. Factors of safety. 
190. Design of stringers. 191. Design of posts. 192. Design 
of caps and sills. 193. Bracing. 



VIU TABLE OF CONTENTS. 

CHAPTER V. • 

TUNNELS. 

PAGE 

Surveying 227 

194. Surface surveySo 195. Surveying down a shaft. 196. 
Underground surveys, 197. Accuracy of tunnel surveying. 
Design 232 

198. Cross-sections. 199. Grade. 200. Lining. 201. Shafts. 
202. Drains. 
Construction 237 

203. Headings. 204. Enlargement, 205. Distinctive features 
of various methods of construction. 206. Ventilation during con- 
struction. 207. Excavation for the portals. 208. Tunnels vs. 
open cuts. 209. Cost of tunneling. 

CHAPTER VI. 

CULVERTS AND MINOR BRIDGES. 

210. Definition and object. 211. Elements of the design 245 

Area of the waterway 246 

212. Elements involved. 213. Methods of computation of area. 
214, Empirical formulae. 215. Value of empirical formulae. 216. 
Results based on observation. 217. Degree of accuracy required. 

Pipe culverts 250 

218. Advantages. 219. Construction. 220. Iron-pipe culverts. 
221. Tile-pipe culverts. 

Box CULVERTS , 254 

222. Wooden box culverts. 223. Stone box culverts. 224. Old 
rail culverts. 225. Reinforced concrete culverts. 

Arch culverts 258 

226. Influence of design on flow. 227. Examples of arch-cul- 
vert design. 

Minor openings 260 

228. Cattle-guards, 229. Cattle-passes. 230. Standard stringer 
and I-beam bridges. 

CHAPTER VII. 

BALLAST. 

231. Purpose and requirements. 232. Materials. 233. Cross- 
sections. 234. Classification of railroads, 235. Recommended 
sections for the several classifications. 236. Proper depth of ballast. 
237. Methods of laying ballast. 238. Cost 263 

CHAPTER VIII. 

ties and other forms of rail support. 

239. Various methods of supporting rails. 240. Economics of 

ties 276 

Wooden ties. 277 

241. Choice of wood. 242. Durability. 243. Dimensions. 
244. Spacing. 245. Specifications. 246. Regulations for laying 
and renewing ties. 247. Dating naUs. 248. Cost of ties. 



TABLE OF CONTENTS. ix 

/ 

PAGE 

Pbeservative processes for wooden ties 282 

249. General principle. 250. Creosoting. 251. Burnettizing. 
252. Kyanizing. 253. Zinc-tannin 'process. 254. Zinc-creosote 
emulsion process. 255. Two-injection zinc creosote process. 256. 
Cost of treating. 257. Economics of treated ties. 

Metal ties 290 

258. Extent of use. 259. Durability. 260. Form and dimen- 
sions of metal cross-ties. 261. Fastenings. 262. Cost. 263. 
Bowls or plates., 264. Longitudinals. 265. Reinforced concrete 
ties. 

CHAPTER IX. 

RAILS. 

266. Early forms. 26V. Present standard forms. 268. Weight 
for various kinds of traffic. 269. Effect of stiffness on traction. 
270. Length of rails. 271. Expansion of rails. 272. Rules for 
allowing ' for temperature. 273. Chemical compositipn. 274. 
Proposed standard specifications for steel rails. 275. Rail wear on 
tangents. 276. Rail wear on curves. 277. Experimental deter- 
mination of rail wear. 278. Cost of rails 296 

CHAPTER X. 

rail-fastenings. 
Rail-joints 314 

279. Theoretical requirements for a perfect joint. 280. Effi- 
ciency of the ordinary angle-bar. 281. Effect of rail-gap at joints. 
282. Supported, suspended, and bridge joints. 283. Failures of 
rail-joints. 284. Standard angle-bars. 285. Specifications for 
steel splice-bars. 

Tie-plates 320 

286. Advantages. 287. Elements of the design. 288. Methods 
of setting. 

Spikes 324 

289. Requirements. 290. Driving. 291. Screw spikes. 292. 
Wooden spikes. 

Track-bolts and nut-locks 330 

293. Essential requirements. 294. Design of track-bolts. 2 
Design of nut-locks. 

CHAPTER XI. 

switches and crossings. 

Switch construction 335 

296. Essential elements of a switch. 297. Frogs. 298. To find 
the frog number. 299. Stub switches. 300. Point switches. 301. 
Switch-stands. 302. Tie-rods. 303. Guard-rails. 

Mathematical design of switches 342 

304. Design with circular lead rails. 305. Standard design, 
using straight frog-rails and straight point-rails. 306. Design for 
a turnout from the outer side of a curved track. 307. Design 



X TABLE OF CONTENTS. 

PAGE 

for a turnout from the inner side of a curved track. 308. Con- 
necting curve from a straight track. 309. Connecting curve from 
a curA^ed track to the outside. 310. Connecting curve from a 
curved track to the inside. 311. Crossover between two parallel 
straight tracks. 312, Crossover between two parallel curved 
tracks. 313. Practical rules for switch-laying. 314. Slips. 

Crossings 361 

315. Two straight tracks. 316. One straight and one curved 
track. 317. Two curved tracks. 



CHAPTER XII. 

miscellaneous structures and buildings. 



Water stations and water supply 367 

318. Location. 319. Required qualities of water. 320. Mechan- 
ical cleaning. 321. Chemical purification. 322. Foaming and 
priming. 323. Boiler compounds. 324. Tanks. 325. Pumping. 
326. Track tanks. 327. Stand pipes. 

Buildings 377 

328. Station platforms. 329. Minor stations. Freight 
HOUSES. 330. Two types. 331. Fire risk. 332. Dimensions. 
333. Platforms. 334. Floors. 335. Doors, 336. Roofs project- 
ing over platforms. 337. Lighting. 338. Scales. 339. Ramps. 
340. Section houses. Engine houses. 341. Form. 342. Doors. 
343. Length. 344, Materials of construction. 345, Engine pits. 
346, Smoke jacks. 347. Floors, 348, Drop pits. 349, Heating. 
350. Window lighting. 351. Electric lighting. 352, Piping. 
353. Tools. 354. Hoists, 355. Turntables. Locomotive coal- 
ing STATIONS. 356. Hand shoveling. 357. Locomotive crane. 
358. Coaling trestle. 359. Coal conveyors. 360. Oil houses. 
361. Section tool houses. 362. Sand houses. 363. Ash pits. 

Snow structures 391 

364. Snow fences. 365. Snow sheds. 

Fences 393 

366. Wire fences. 367. Types. 368. Posts. 369. Braces. 370. 
Concrete posts. 371. Construction details. 

Signs , 396 

372. Highway signs. 373 Trespass signs. 374. Marker posts. 
375. Bridge warning. 

CHAPTER XIII. 

yards and terminals. 

376. Value of proper design. 377. Divisions of the subject. . . , 400 

Freight yards 401 

378. General principles. 379, Hump yards. 380 Relation of 
yard to main tracks, 381. Minor freight yards. 382. Transfer 
cranes. 383. Track scales. 

Engine yards 411 

384. General principles. 



TABLE OF CONTENTS. XI 

CHAPTER XIV. 

BLOCK SIGNALING. 

PAGE 

General principles 412 

385. Two fundamental systems. 386. Manual systems. 387. 
Development of the manual system. 388. Permissive blocking. 
389. Automatic systems. 390. Distant signals. 391. Advance 
signals. 
Mechanical details 418 

392. Signals. 393. Wires and pipes. 394. Track circuit for 
automatic signaling. 

. CHAPTER XV. 

rolling stock. 

Wheels and rails 425 

395. Effect of rigidly attaching wheels to their axles. 396. 
Effect of parallel axles. 397. Effect of coning wheels. 398. 
Effect of flanging locomotive driving wheels. 399. Action of a 
locomotive pilot-truck. 400. Types of locomotive wheel bases. 

locomotives. 

General structure 433 

401. Frame. 402. Boiler. 403. Fire box. 404. Area of grate. 
405. Superheaters. 406. Reheaters. 407. Coal consumption. 
408. Oil-burning locomotives. 409. Heating surface. 410. Loss 
of efficiency of steam pressure. 411. Tractive power. 

Running gear 444 

412. Equalizing levers. 413. Counterbalancing. 414. Mutual 
relations of the boiler power, tractive power and cylinder power 
for various types. 415. Life of locomotives. 

CARS. 

416. Capacity and size of cars. 417. Stresses to which car- 
frames are subjected. 418. The use of metal. 419. Draft gear. 
420. Gauge of wheels and form of wheel tread 455 

train-brakes. 

421. Introduction. 422. Laws of friction as applied to this 

problem 461 

Mechanism op brakes 465 

423. Hand-brakes. 424. "Straight" air brakes. 425. Auto- 
matic air brakes. 426. Tests to measure the efficiency of brakes. 
427. Brake shoes. 

CHAPTER XVI. 

train resistance. 

428. Classification of the various forms. 429. Resistances inter- 
nal to the locomotive. 430. Velocity resistances. 431. Wheel 
resistances. 432. Grade resistance. 433. Curve resistance. 434. 
Brake resistance. 435. Inertia resistance. 436. Dynamometer 



Xll TABLE OF CONTENTS. 

PAGE 

tests. 437. Gra\dty or "drop" tests. 438. Formulae for train 
resistance. 439. American Railway Engineering Association 
Formula 471 

CHAPTER XVII. 

COST OF EAILRODAS. 

440. General considerations. 441. Preliminary financiering. 
442. Surveys and engineering expenses. 443. Land and land 
damages. 444. Clearing and grubbing. 445. Earthwork. 446. 
Bridges, trestles and culverts. 447. Trackwork. 448. Buildings 
and miscellaneous structures. 449. Interest on construction. 
450. Telegraph lines. 451. Detailed estimate of the cost of a line 
of road 490 

CHAPTER XVIII. 

THE POWER OF A LOCOMOTIVE. 

452. Pounds of steam produced. 453. Numerical example. 
454, Weight of steam per stroke at full cut-off. 455. Pounds of 
steam and per cent of cut-off for multiples of M velocity. 456. 
Draw-bar pull. 457. Effect of increasing the rate of coal con- 
sumption. 458. Effect of using a better quality of coal. 459. 
Check with approximate rule. 460. Tractive force at higher 
velocities. 461. Effect of grade on tractive power. 462. Accel- 
eration-speed curves. 463. Retardation-speed curves. 464. 
Drifting. 465. Review of computed power of one locomotive. 
466. Selection of route. 467. Rating of locomotives 500 

CHAPTER XIX. 

The promotion of railroad projects. 

468. Method of formation of railroad corporations. 469. The 
two classes of financial interests, the security and profits of each. 

470. The small margin between profit and loss to the projectors. 

471. Extent to which a railroad is a monopoly. 472. Profit 
resulting from an increase in business done; loss resulting from a 
decrease. 473. Estimation of probable volume of traffic, and of 
probable growth. 474. Probable number of trains per day. In- 
crease with growth of traffic. 475. Effect on traffic of an increase 
in facilities. 476. Loss caused by inconvenient terminals and 
by stations far removed from business centers. 477. General 
principles which should govern the expenditure of money for 
railroad purposes. 478. Study of railroad economics — its nature 
and limitations. 479. Outline of the engineer's duties., 522 

CHAPTER XX. 

operating expenses. 

> 480. Distribution of gross revenue. 481. Operating expenses 

per train mile. 482. Reasons for uniformity in expenses per train 



TABLE OF CONTENTS. Xlll 

PAGE 

mile. 483. Detailed classification of expenses with ratios to the 
total expense. 484. Elements of the cost (with variations and 
tendencies) of the various items 536 

Maintenance of way and structures 539 

485. Track materials. 486. Roadway and track. 487. Main- 
tenance of track structures. 

Maintenance of equipment 543 

488. Repairs, renewals, and depreciation of steam and electric 
locomotives. 

Transportation 544 

489. Yard-engine expenses. 490. Road enginemen. 491. Fuel 
for road locomotives. 492. Road trainmen. 493. Train supplies 
and expenses. 494. Clearing wrecks, loss, damage, and injuries 
to persons and property. 495. Operating joint tracks and facil- 
ities, switching charges, etc. 

CHAPTER XXI. 

DISTANCE. C 

496. Relation of distance to rates and expenses. 497. The 
conditions other than distance that affect the cost; reasons why 

rates are usually based on distance. 550 

Effect of distance on receipts 551 

498. Classification of traffic. 499. Method of division of through 
rates between the roads run over. 500. Effect of a change in the 
length of the home road on its receipts from through competitive 
traffic. 501. The most advantageous conditions for roads forming 
part of a through competitive route, 502. Effect of the variations 
in the length of haul and the classes of the business actually done. 
503. General conclusions regarding a change in distance. 504. 
Justification of decreasing distance to save time. 505. Effect of 
change of distance on the business done. 

CHAPTER XXII. 
curvature. 

506. General objections to curvature. 507. Financial value of 
the danger of accident due to curvature. 508. Effect of curvature 

on travel. 509. Effect on operation of trains 557 

Compensation for curvature 561 

510. Reasons for compensation. 511. The proper rate of com- 
pensation. 612. The limitations of maximum curvature. ^ 



CHAPTER XXIII. 

GRADE. 

513. Two distinct effects of grade. 514. Application to the 
movement of trains of the laws of accelerated motion. 515. Con- 
struction of a virtual profile. 516. Variation in draw-bar pull. 
517. Use, value and possible misuse. 518. Undulatory grades; 
advantages, disadvantages, and safe limits 566 



XIV TABLE OF CONTENTS. 

PAGE 

Ruling grades 575 

519. Definition. 520. Choice of ruling grades. 521. Maximum 
train load on any grade. 522. Proportion of traflSc affected by 
the ruling grade. 

Pusher grades 578 

523. General principles underlying the use of pusher engines. 
624. Balance of grades for pusher service. 525. Two-pusher 
grades. 526. Operation of pusher engines. 527. Length of a 
pusher grade. 528. Cost of pusher-engine service. 

Balance op grades for unequal traffic 584 

529. Nature of the subject. 530. Computation of the theoreti- 
cal balance. 531. Computation of relative traffic. 

CHAPTER XXIV 

THE IMPROVEMENT OP OLD LINES. 

532. Classification of improvements. 533. Advantages of re- 
locations. 534. Disadvantages of re-locations 588 

Reduction op virtual grade 591 

535. Obtaining data for computations. 536. Use of the data 
obtained. 537. Reducing the starting grade at stations. 

Appendix. The adjustments op instruments 596 

Azimuth 604 

Tables. 

I. Radii of curves 612 

II. Tangents, external distances, and long chords for a 1** curve 616 
Ila. Excess length of sub-chords 619 

III. Switch leads and distances 619 

IV. Transition curves 621 

V. Logarithms of numbers 624 

VI. Logarithmic sines and tangents of small angles 644 

VII. Logarithmic sines, cosines, tangents, and cotangents 647 

VIII. Logarithmic versed sines and external secants 692 

IX. Natural sines, cosines, tangents, and cotangents 737 

X. Natural versed sines and external secants 760 

XI. Reduction of barometer reading to 32° F 783 

XII. Barometric elevations 784 

XIII. Coefficients for corrections for temperature and humidity . 784 

XIV. Useful trigonometrical formulae 785 

XV. Useful formulae and constants 787 

XVI. Squares, cubes, square roots, cube roots and reciprocals. . . 788 

XVII. Cubic yards per 100 feet of level sections 805 

XVIII. Annual charge against a tie, based on the original cost and 

assumed life of the tie 808 

XIX. Superelevation of the outer rail (in feet) for various 

velocities and degrees of curvature 83 

XX. Moduli of rupture for various timbers 220 



TABLE OF CONTENTS. XV 

Tables. page 

XXI. Working unit stresses for structural timber 221 

XXII. Number and value of cross ties, used in U. S., 1906 277 

XXIII. Angles and dimensions of standard designs for rails 299 

XXIV. Angles and dimensions of standard designs for splice bars. 319 
XXV. Rectangular coordinates of curved rail of switches 358 

XXVI. Quantity of reagents required to remove incrusting or 

/ corrosive matter from water 370 

XXVII. Capacity of cylindrical water-tanks in gallons 373 

XXVIII. Cost of fuel for various types of pumps and engines 374 

XXIX. Locomotive resistances t 473 

XXX. Number of cross ties per mile 494 

XXXI. Tons per mile (with cost) of rails of various weights 495 

XXXII. SpUce bars and bolts for various weights of rail 496 

XXXIII. Railroad spikes 497 

XXXIV. Track bolts 497 

XXXV. Number of rail joints and track-bolts per mile of track. . . . 497 

XXXVI. Average evaporation in locomotive boilers 501 

XXXVII. Weight of steam used in one foot of stroke in locomotives. 503 

XXXVIII. Maximum cut off and pounds of steam per I. H. P. hour. 504 

XXXIX. Per cent cylinder tractive power for various multiples of M 505 

XL. Locomotive rating discounts 520 

XLI. Analysis of operating expenses of railroads in U. S. in 

1912, 540, 541 

XLII. Velocity head of trains 570 

XLIII. Tractive power of various types of locomotives 577 

XLIV. Cost for each mile of pusher-engine service 583 

Index , 809 



I: . 



RAILROAD CONSTRUCTION. 



CHAPTER I. 

RAILROAD SURVEYS. 

The proper conduct of railroad surveys presupposes an 
adequate knowledge of almost the whole subject of railroad 
engineering, and particularly of some of the complicated ques- 
tions of Railroad Economics, which are not generally studied 
except at the latter part of a course in railroad engineering, if 
at all. This chapter will therefore be chiefly devoted to methods 
of instrumental work, and the problem of choosing a general 
route will be considered only as it is influenced by the topog- 
raphy or by the application of those elementary principles of 
Railroad Economics which are self-evident or which may be 
accepted by the student until he has had an opportunity of 
studying those principles in detail 

The student-engineer should be warned against the hasty and 
inadequate surveying which has resulted in so much miscon- 
struction in this country. This kind of surveying was especially 
common forty or fifty years ago, and the methods have more or 
less continued. The demand for railroad facilities was then so 
urgent that lax methods were tolerated. A general route would 
be selected which, at first sight, seemed most obvious and it 
would be immediately staked out in a manner suitable to a 
location survey. After correcting some of the most glaring 
faults, the survey was considered complete and the road was 
constructed accordingly. The cost of such a survey is compara- 
tively small, but it is almost inevitable that the line is not as 
good as could havQ been obtained with a greater amount of 



2 RAILKOAD CONSTRUCTION. § 1. 

examination and study. The cost of construction and the 
future cost of operating such a Hne is always unnecessarily high. 
The money wasted in construction, plus the capitalized value of 
the annual waste in future operating expenses, is frequently a 
hundred times the cost of the extra study and surveying which 
would have avoided these faults. This has been unquestionably 
proved by the innumerable cases of reconstruction of portions 
of old' lines which could have been constructed originally on the 
lines as revised at even less cost. The engineer is not always 
responsible for ill-advised hasty work. An impatient Board 
of Directors often insists on commencing to " throw dirt " 
before a proper survey has been made. The engineer should 
make, if necessary, the most earnest representations and even 
strenuous demands, that he be given the requisite time, oppor- 
tunity and money to conduct his survey in such a manner as 
to investigate thoroughly every possibility for improving the 
alinement. 

. A railroad survey ordinarily consists of three parts: (a) 
the reconnoissance; (h) the preliminary survey, and (c) the 
definite location. As explained later, circumstances may modify 
the relative importance of these divisions, but under ordinary 
circumstances all three are necessary. 



RECONNOISSANCE SURVEYS. 

I. Character of a reconnoissance survey. A reconnoissance 
survey is a very hasty examination of a belt of country to deter- 
mine which of all possible or suggested routes is the most prom- 
ising and best worthy of a more detailed survey. It is essentially 
very rough and rapid. It aims to discover those salient features 
which instantly stamp one route as distinctly superior to another 
and so narrow the choice to routes which are so nearly equal 
in value that a more detailed survey is necessary to decide 
between them. 

A map should be prepared, at n scale not smaller than one 
mile to the inch, which should show all general routes which are 
conceivably possible. It is particularly important that the 
mere lack of data should not exclude consideration of some 
general route which might be superior to the one or more obvious 
routes which have already been picked out. 



§ 2. RAILROAD SURVEYS. 3 

2. Selection of a general route. The general question of 
running a railroad between two towns is frequently a financial 
rather than an engineering question. Financial considerations 
usually determine that a road must pass through certain more 
or less important towns between its termini. It is also pos- 
sible that there may be certain topographical features in any 
route between two determined towns on the line, such as a low 
saddle in crossing a ridge or a difficult crossing of a large river, 
which, with the towns, may be considered as control points, and 
the problem may be narrowed down to the determination of 
the best route between these consecutive control points. But 
care should be taken that control points are not too hastily 
considered as fixed and unalterable, especially if it results in 
very xmfavorable grades and alinement between consecutive 
points. 

The reconnoissance survey should include the determination 
of the location and relative elevations of all these control points. 
These data should be obtained with sufficient accuracy to 
compute the necessary ruling grade and the general character 
of the alinement, and the map as thus amplified should be 
studied by comparing the several possible routes and elimin- 
ating all those which are unquestionably less favorable than 
others. 

The engineer should avoid, especially in a rough and wooded 
country, the influence that an existing highway, or even a path 
through the w^oods or of a clearing of the trees, may have in 
determining the choice of routes. Mere ease of travel, as long 
as it is not glaringly wrong, has caused many prepossessions in 
favor of a certain route, when a much better line could be obtained 
by plunging through the woods or over swampy or rocky ground. 
As a first trial in selecting the route, the bearing of a line joining 
two consecutive control points should be determined and then 
an effort should be made to find a general route which will have 
the least possible variation from that straight line, without sac- 
rificing the limits of ruling grade, curvature and general type 
or cost of construction which may have been fixed for the 
road. 

A difficult line between two control points should be studied 
by beginning at either end for two independent studies. The 
very obvious route, starting from A toward B, may lead into 
very difficult construction, which may be avoided by com- 



4 EAILROAD CONSTRUCTION. § 3. 

mencing at B and finally reaching A on a route which, while 
practicable, would not be considered attractive when starting 
from A . 

When a railroad runs through a thickly settled and very flat 
country, where, from a topographical standpoint, the road may 
be run by any desired route, the '' right-of-way agent " some- 
times has a greater influence in locating the road than the 
engineer. But such modifications of alinement, on account of 
business considerations, are foreign to the engineer's side of the 
subject, and it will be hereafter assumed that topography alone 
determines the location of the line. The consideration of those 
larger questions combining finance and engineering (such as 
passing by a town on account of the necessary introduction of 
heavy grades in order to reach it), will be considered in later 
chapters. ^ 

3. Valley route. This is perhaps the simplest problem. If 
two control points to be connected lie in the same valley, it is fre- 
quently only necessary to run a line which shall have a nearly 
uniform grade. The reconnoissance problem consists largely in 
determining the difference of elevation of the two termini of 
this division and the approximate horizontal distance so that the 
proper grade may be chosen. If there is a large river running 
through the valley, the road will probably remain on one side 
or the other throughout the whole distance, and both banks 
should be examined by the reconnoissance party to determine 
which is preferable. If the river may be easily bridged, both 
banks may be alternately used, especially when better alinement 
is thereby secured. A river valley has usually a steeper slope 
in the upper part than in the lower part. A uniform grade 
throughout the valley will therefore require that the road climbs 
up the side slopes in the lower part of the valley. In case the 
'' ruling grade ''* for the whole road is as great as or greater 
than the steepest natural valley slope, more freedom may be 
used in adopting that alinement which has the least cost — 
regardless of grade. The natural slope of large rivers is almost 
invariably so low that grade has no influence in determining the 
choice of location. When bridging is necessary, the river 



* The ruling grade may here be loosely defined as the maximum grade 
which is permissible. This definition is not strictly true, as may be seen later 
when studying Railroad Economics, but it may here serve the purpose. 



§ 4. RAILROAD SURVEYS. 5 

banks should be examined for suitable locationiS for abutments 
and piers. If the soil is soft and treacherous, much difficulty 
may be experienced and the choice of route may be largely 
determined by the difficulty of bridging the river except at 
certain favorable places. 

4. Cross-country route. A cross-country route always has one 
or more summits to be crossed. The problem becomes more 
complex on account of the greater number of possible solutions 
and the difficulty of properly weighing the advantages and dis- 
advantages of each. The general aim should be to choose the 
lowest summits and the highest stream crossings, provided that 
by so doing the grades between these determining points shall 
be as low as possible ^-nd shall not be greater than the ruling 
grade of the road. Nearly all railroads combine cross-country 
and valley routes to some extent. Usually the steepest natural 
slopes are to be found on the cross-country routes, and also the 
greatest difficulty in securing a low through grade. An approx- 
imate determination of the ruling grade is usually made during 
the reconnoissance. If the ruling grade has been previously 
decided on by other considerations, the leading feature of the 
reconnoissance survey will be the determination of a general 
route along which it will be possible to survey a line whose 
maximum grade shall not exceed the ruling grade. 

5. Mountain route. The streams of a mountainous region 
frequently have a slope exceeding the desired ruling grade. In 
such cases there is no possibility of securing the desired grade 
by following the streams. The penetration of such a region 
may only be accomplished by "development" — accompanied 
perhaps by tunneling. "Development" consists in deliber- 
ately increasing the length of the road between two extremes 
of elevation so that the rate of grade shall be as low as desired. 
The usual method of accomplishing this is to take advantage of 
some convenient formation of the ground to introduce some 
lateral deviation. The methods may be somewhat classified afl 
follows: ^^' 

(a) Rimning the line up a convenient lateral valley, turning 
a sharp curve and working back up the opposite slope. As 
shown in Fig. 1, the considerable rise between A and B was 
surmounted by starting off in a very different direction from 
the general direction of the road; then, when about one-half of 
the desired rise had been obtained, the line crossed the valley 



6 



RAILROAD CONSTRUCTION. 



and continued the climb along the opposite slope, (b) Switch- 
hack. On the steep side-hill BCD (Fig. 1) a very considerable 
gain in elevation was accomplished by the switchback CD, 
The gain in elevation from B to D is very great. On the other 
hand, the speed must always be slow; there are two complete 
stoppages of the train for each run; all trains must run back- 
ward from C to D. (c) Bridge spiral. When a valley is so 
narrow at some point that a bridge or viaduct of i e/ASonable 
length can span the valley at a considerable elevation above tim 




Fig. 1. 



bottom of the valley, a bridge spiral may be desirable. In Fig. 2 
the line ascends the stream valley past A, crosses the stream at 
Bj works back to the narrow place at (7, and there crosses itself, 
having gained perhaps 100 feet in elevation, (d) Tunnei 
spiral (Fig. 3). This is the reverse of the previous plan. Iv 
implies a thin steep ridge, so thin at some place that a tunnel 
through it will not be excessively long. Switchbacks and 
spirals are sometimes necessary in mountainous countries, but 
they should not be considered as normal types of construction. 
A region must be very difficult if these devices cannot be 
avoided. 



.MAP OF 

C.V.R.R. AND C.R.R. OF N.J 

FROM GLEN SUMMIT TO WILKESBARRE 

ScAle of MUes 




{To face page 6.) 



§5. 



RAILROAD SURVEYS. 



On Plate I are shown three separate ways (as actually con- 
structed) of running a railroad between two points a little over 
three miles apart and having a difference of elevation of nearly 
1100 feet. At A the Central H. R. of New Jersey runs under 
the Lehigh Valley R. R. and soon turns off to the northeast for 
about six miles^ then doubles back, reaching Z), a fall of about 
1050 feet with a track distance of about 12.7 miles. The 
L. V. R. R. at A runs to the westward for six to seven miles, 





'////iii\\\\\\y\\ 



Fig. 2. 



/ll!\\\V^> 
Fig. 3. 




then turns back until the roads are again close together at D, 
The i^rack distance is about 14 miles and the drop a Httle greater, 
since at A the L. V. R. R. crosses over the other, while at D they 
are at practically the same level. From B to C the distance is 
over eleven miles. From A directly down to D the C. R. R. of 
N..J. runs a '^ gravity" road, used exclusively for freight, on 
which cars alone are hauled by cable. The main-Hne routes 
are remarkable examples of sheer "development." Even as 
constructed the L. V. R. R. has a grade of about 95 feet per 
mile, and this grade has proved so excessive for freight work 
that the company has constructed a cut-off (not shown on the 
map) which leaves the main line at A, nearly parallels the 



8 RAILROAD CONSTRUCTION. § 6. 

C. R. R-. to C, and then running in a northeasterly direction 
again joins the main line beyond Wilkesbarre. The grade is 
thereby cut down to 65 feet per mile. 

Rack railways and cable roads, although types of mountain 
railroad construction, will not be here considered. 

6. Existing maps. The maps of the U. S. Geological Survey 
are exceedingly valuable as far as they have been completed. 
So far as topographical considerations are concerned, they 
almost dispense with the necessity for the reconnoissance and 
''first preliminary" surveys. Some of the State Survey maps 
will give practically the same information. County and town- 
ship maps can often be used for considerable information as to the 
relative horizontal position of governing points, and even some 
approximate data regarding elevations ma.y be obtained by a 
study of the streams. Of course such information will not dis- 
pense with surveys, but will assist in so planning them as to 
obtain the best information with the least work. When the 
relative horizontal positions of points are reliably indicated on a 
map, the reconnoissance may be reduced to the determina- 
tion of the relative elevations of the governing points of the 
route. 

7. Determination of relative elevations. A recent description 
X)i European methods includes spirit-leveling in the reconnois- 
sance work. This may be due to the fact that, as indicated 
above, previous topographical surveys have rendered unnecessary 
the ''exploratory" survey which is required in a new country, 
find that their reconnoissance really corresponds more nearly to 
our preliminary. 

The perfection to which barometrical methods have been 
brought has rendered it possible to determine differences of 
elevation with sufficient accuracy for reconnoissance purposes 
by the combined use of a mercurial and an aneroid barometer. 
The mercurial barometer should be kept at "headquarters," and 
readings should be taken on it at such frequent intervals that 
any fluctuation is noted, and throughout the period that observa- 
tions with the aneroid are taken in the field. At each observa- 
tion there should also be recorded the time, the reading of the 
attached thermometer, and the temperature of the external 
air. For uniformity, the mercurial readings should then be 
"reduced to 32° F." The form of notes for the mercurial 
barometer readings should be as follows: 



1^, 



§7. 



RAILROAD SURVEYS 



Time. 


Merc. 
Barom. 


Attached 
Therm. 


Reduction 
to 32° F. 


External 
Therm. 


Corrected 
reading. 


7:00 A.M. 
:15 
:30 
:45 


29.872 
.866 
.858 
.850 


72° 
73.5 
75 
76 


— .117 
.121 
.125 
.127 


73° 

75 

76 

77 


29 . 755 
.745 
.733 
.723 



The corrections in column 4 are derived from Table XI by 
interpolation. 

Before starting out, a reading of the aneroid should be taken 
at headquarters coincident with a reading of the mercurial. 
The difference is one value of the correction to the aneroid. 
As soon as the aneroid is brought back another comparison of 
readings should be made. Even though there has been con- 
siderable rise or fall of pressure in the interval, the difference 
in readings (the correction) should be substantially the same 
provided the aneroid is a good instrument. If the difference 
of elevation is excessive (as when climbing a high mountain) 
even the best aneroid will ''lag" and not recover its normal 
reading for several hours, but this does not apply to such dif- 
ferences of elevation as are met with in railroad work. The 
best aneroids read directly to y^ of an inch of mercury and 
may be estimated to ywo^ of an inch — ^which corresponds 
to about 0.9 foot difference of elevation. In the field there 
should be read, at each point whose elevation is desired, the 
aneroid, the time, and the temperature. These readings, cor- 
rected by the mean value of the correction between the aneroid 
and the mercurial, should then be combined with the reading 
of the mercurial (interpolated if necessary) for the times of 
the aneroid observations and the difference of elevation ob- 
tained. The field notes for the aneroid should be taken as 
shown in the first four columns of the tabular form. The '^ cor- 
rected aneroid" readings of column 5 are foimd by correcting 
the readings of column 3 by the mean difference between the 
mercurial and aneroid when compared at morning and night. 
Column 6 is a copy of the "corrected readings" from the office 
notes, interpolated when necessary for the proper time. Column 
7 is similarly obtained. Col. 8 is obtained, from cols. 4 and 5, 
and col. 9 from cols. 6 and 7, with the aid of Table XII. The 
correction for temperature (col. 11), which is generally small 
unless the difference of elevation is large, is obtained with the 



10 



RAILROAD CONSTRUCTION. 



(Left-hand page of Notes.) 



Time. 


Place. 


Aneroid. 


Therm. 


Corr. 
Aner. 


II ' 


7 00 


Office 
JO 
saddle-back 
river cross. 


29.628 
29.662 
29 . 374 
29.548 


73° 

72° 
63° 
70° 




29.755 


7:10 
7:30 
7:50 


29.789 
29.501 
29.675 


29.748 
29.733 
29.720 



aid of Table XIII. The elevations in Table XII are elevations 
above an assumed datum plane, where under the given atmos- 
pheric conditions the mercurial reading would be 30''. Of 
course the position of this assumed plane changes with varying 
atmospheric conditions' and so the elevations are to be con- 
sidered as relative and their difference taken. [See the author's 
*^ Problems in the Use and Adjustment of Engineering In- 
struments/' Prob. 22.] Important points should be observed 
more than once if possible. Such duplicate observations will be 
found to give surprisingly concordant results even when a 
general fluctuation of atmospheric pressure so modifies the 
tabulated readings that an agreement is not at first apparent. 
Variations of pressure produced by high winds, thunder-storms, 
etc., will generally vitiate possible accuracy by this method. 
By "headquarters" is meant any place whose elevation above 
any given datum is known and where the mercurial may be 
placed and observed while observations within a range of several 
miles are made with the aneroid. If necessary, the elevation of 
a new headquarters may be determined by the above method, 
but there should be if possible several independent observations 
whose accordance will give a fair idea of their accuracy. 

The above method should be neither slighted nor used for 
more than it is worth. When properly used, the errors are 
compensating rather than cumulative. When used, for example, 
to determine that a pass B is 260 feet higher than a determined 
bridge crossing at A which is six miles distant, and that another 
pass C is 310 feet higher than A and is ten miles distant, the 
figures, even with all necessary allowances for inaccuracy, mil 
give an engineer a good idea as to the choice of route especially 
as affected by ruling grade. There is no comparison between 
the time and labor involved in obtaining the above information 
by barometric and by spirit-leveling methods, and for recon^ 



§8. 



KAILROAD SURVEYS. 



11 



(Right-hand page of Notes.) 



Temp, at 
headqu. 


Approx. 
field read. 


Approx. 
headq. read. 


Diff. 


Corr. for 
temp. 


Diff. 
elev. 


76 

77 


192 
457 
297 


230 
244 
256 


- 38 
+ 213 
+ 41 


-(+ 2) 
+ ( + 10) 
+ (+ 2) 


— 40 
+ 223 
+ 43 



noissance purposes the added accuracy of the spirit-leveling 
method is hardly worth its cost. 

8. Horizontal measurements, bearings, etc. When reliable 
maps are unobtainable, rapid exploratory surveys become essen- 
tial. Since accuracy is sacrificed for rapidity in such surveys, 
more or less approximate methods are used. " An experienced 
saddle-horse, whose speeds at his various gaits have been learned 
accurately by previous timing," is quoted from Beahan * as 
one means of rapidly measuring distances. The percentage of 
probafele error is evidently large. A pedometer (or pace- 
measurer) is probably more accurate, but its accuracy depends 
on a knowledge of the average length of the observer's pace. 
Due allowance must be made for the fact that the length of pace 
will vary very greatly depending on whether the surface is 
smooth and level, or is plowed ground, or marshy, or slippery, or 
consists of rough boulders covered with moss, or is a wilderness 
of brambles, fallen trees, bogs, etc. It will also depend on 
whether the observer is fatigued or is in fresh physical condition. 
Under such a variety of conditions the counting of steps for long 
distances is sometimes a farce. Even when the surface is fairly 
smooth and easy, precautions must be taken that paces are not 
counted during the pauses at important points while bearings 
are being taken and other data recorded. An odometer which 
records the revolutions of a wheel of known circumference is 
far more accurate. Such a machine has been made so that it 
may be trundled like a wheelbarrow and thus go through the 
woods and over ground that would be impassable to any horse- 
drawn vehicle. The attachment of an odometer to the wheels 
of a wagon is very tempting, since it permits the engineer to 
ride, but it is probably an unreliable method for the reason men- 



* "The Field Practica of Railway Location," p. 34. 



12 RAILROAD CONSTRUCTION. § 8. 

tioned in Art. 2 — permitting the ease of travel over a road 
practicable for a horse and vehicle to deflect the engineer from 
his true course, which is perhaps over rough ground which is 
impassable for a vehicle. 

When . the country is quite open and clear of underbrush, 
very rapid work may be done by the stadia method, which is 
many times more accurate than any of the methods previ- 
ously mentioned. Some of the accuracy possible with stadia 
may be sacrificed for extreme rapidity and sights may be 
made 1200 and even 2000 feet long. By taking very few, 
if any, ^^ side-shots," the progress is very rapid and many 
miles per day may be covered, with the advantage that the 
three elements of distance, azimuth and relative elevation 
may be obtained with as great accuracy as is necessary for 
an exploratory survey. The method of using the stadia will be 
described later. 

The bearings of the various lines forming the skeleton of the 
survey, and also the bearings of the courses of streams and of side 
lines from the stations on the skeleton line, may be taken most 
easily with a prismatic compass. This instrument has a cir- 
cular card, or sometimes a metal ring, attached to the needle. 
The edge of the card is graduated into degrees and is usually 
numbered consecutively (instead of by quadrants), from 0° 
up to 360°. This is advantageous since the one number, with- 
out any qualifying letters, NE or NW, determines the quadrant 
definitely without danger of confusion or error. The observer 
sights through a narrow sUt in the desired direction and, by 
means of the prismatic reflector, can read directly the number 
of degrees, measured to the rights and usually from the magnetic 
South. The makers of prismatic compasses do not always 
number the graduations in the same manner, and, therefore, the 
engineer, who is accustomed to one particular instrument, should 
carefully study the markings of any new instrument. In any 
case it should be remembered that the prism reflects the numbers 
on that side of the movable card or ring which is toward the ob- 
server rather than on the side toward the object sighted at. The 
prismatic compass has the special advantage that, Uke a sextant, 
it can be used when supported only by hand, while an ordinary 
sight compass of equal accuracy would require a tripod, or, at 
least, a Jacob's staff. The decHnation of the needle in that 
section of the country can be readily determined with sufficient 



§9. RAILROAD STTRVEYS^ 13 

accuracy for the purposes of such a survey. Usually the decli- 
nation may be ignored. Any errors due to local attraction are 
never cumulative, but apply only to the point where those indi* 
vidual observations are taken. The angle between two linea 
radiating from any station may be obtained by subtracting one 
bearing from the other. 

Relative elevations may be obtained systematically, using a 
barometer, as already explained, but much filling in may be 
done with the use of a hand-level. Experience soon teaches an 
engineer that there are many optical illusions about the slopes 
of ground which have the practical effect of making the apparent 
slope different from the actual, and, in the case of low grade, 
may make an actual down grade appear as an up grade. For 
example, when looking along an actual but slight down grade, 
especially if there are no obstructions or natural objects which 
the eye can use as a comparative scale, the eye is apt to fore- 
shorten the distance, which has the effect of lessening the appa- 
rent down grade and perhaps of making it appear as a slight up 
grade. The hand level will immediately detect such errors and 
its frequent use by a reconnoissance engineer will not only 
enable him to avoid many errors he might otherwise make, but 
will also be an effective means of training him to guard against 
such optical illusions. Such a simple and effective instrument 
should always be at hand and it should be tested with sufficient 
frequency to know that it is always as accurate as such an 
instrument can be. The bubble should be as sensitive as is 
practicable for an instrument which is held in the hand. A 
well-made hand level has a bubble of the right sensitiveness, but 
even a super-sensitive level may be utihzed and still better 
work done by supporting it steadily on the top of a light wooden 
stick about five feet long. 

9. Importance of a good reconnoissance. The foregoing instru- 
ments and methods should be considered only as aids in exer- 
cising an educated common sense, without which a proper 
location cannot be made. The reconnoissance survey should 
command the best talent and the greatest experience available. 
If the general route is properly chosen, a comparatively low 
order of engineering skill can fill in a location which will prove 
a pajdng railroad property; but if the general route is so chosen 
that the ruling grades are high and the business obtained is small 
and subject to excessive competition, no amount of perfection in 



14 RAILROAD CONSTRUCTION. § 10. 

detailed alinement or roadbed construction can make the road a 
profitable investment. 

PRELIMINARY SHRVEYS. 

10. Character of survey. A preliminary railroad survey is 
properly a topographical survey of a belt of country which has 
been selected during the reconnoissance and within which it is 
estimated that the located line will lie. The width of this belt 
will depend on the character of the country. When a railroad 
is to follow a river having very steep banks the choice of 
location is sometimes limited at places to a very few feet of 
width and the belt to be surveyed may be correspondingly 
narrowed. 

But even in such a case, the width surveyed should be suffi- 
cient to include not only every possible location of " slope- 
stakes '' but also should indicate the contours and nature of 
any soil which might give trouble by sliding, after an excavation 
has been made at the base. It is justifiable and proper to survey 
a belt considerably wider than it is expected to use, for experi- 
ence shows that, while there is generally but little or no direct 
utilization of the extra area surveyed, it frequently becomes 
essential to know something of the character of the ground 
considerably to one side of where it was expected to run 
the line and the inclusion of this area in the original survey 
has saved an expensive trip to obtain a very small amount of 
data. 

In very flat country the desired width may be only limited by 
the ability to survey points with sufficient accuracy at a consider- 
able distance from what may be called the " backbone line " of 
the survey. 

11. Cross-section method. This is the only feasible method 
in a wooded country, and is employed by many for all kinds 
of country. The backbone line is surveyed either by observ- 
ing magnetic bearings with a compass or by carrying forward 
absolute azimuths with a transit. The compass method has 
the disadvantages of limited accuracy and the possibihty of 
considerable local error owing to local attraction. On the other 
hand there are the advantages of greater simplicity, no necessity 
for a back rodman, and the fact that the errors are purely 
local and not cumulative, and may be so limited, with care, that 



§11 



RAILROAD SURVEYS. 



15 



they will cause no vital error in the subsequent location survey. 
The transit method is essentially more accurate, but is liable 
to be more laborious and troublesome. If a large tree is en- 
countered, either it must be cut doT\Ti or a troublesome opera- 
tion of offsetting must be used. If the compass is employed 




Fig. 4. 



under these circumstances, it need only be set up on the far side 
of the tree and the former bearing produced. An error in 
reading a transit azimuth will be carried on throughout the 
survey. An error of enly five minutes of arc will cause an off- 
set of nearly eight feet in a mile. Large azimuth errors may, 
however, be avoided by immediately checking each new azimuth 



16 RAILKOAD CONSTRUCTION. § 12. 

with a needle reading. It is advisable to obtain true azimuth 
at the beginning of the survey by an observation on the sun* or 
Polaris, and to check the azimuths every few miles by azimuth 
observations. Distances along the backbone line should be 
measured with a chain or steel tape and stakes set every 100 
feet. When a course ends at a substation, as is usually the case, 
the remaining portion of the 100 feet should be measured along 
the next course. The level party should immediately obtain the 
elevations (to the nearest tenth of a foot) of all stations, and also 
of the lowest points of all streams crossed and even of dry gullies 
which would require culverts. 

12. Cross-sectioning. It is usually desirable to obtain con- 
tours at five-foot intervals This may readily be done by the 
use of a Locke level (which should be held on top of a simple 
five-foot stick), a tape, and a rod ten feet in length graduated 
to feet and tenths. The method of use may perhaps be best 
explained by an example. Let Fig. 5 represent a section per- 
pendicular to the survey line — such a section as would be made 
by the dotted lines in Fig. 4. C represents the station point. 
Its elevation as determined by the level is, say, 158.3 above 
datum. When the Locke level on its five-foot rod is placed at 
C, the level has an elevation of 163.3. Therefore when a point 
is found (as at a) where the level will read 3.3 on the rod, that 
point has an elevation of 160.0 and its distance from the center 
gives the position of the 160-foot contour. Leaving the long 
rod at that point (a), carry the level to some point (6) such that 
the level will sight at the top of the rod. h is then on the 165- 
foot contour, and the horizontal distance ah added to the hori- 
zontal distance ac gives the position of that contour from the 
center. The contours on the lower side are found similarly. 
The first rod reading will be 8.3, giving the 155-foot contour. 
Plot the results in a note-book which is ruled in quarter-inch 
squares, using a scale of 100 feet per inch in both directions. 
Plot the work up the page; then when looking ahead along the 
line, the work is properly oriented. When a contour crosses 
the survey line, the place of crossing may be similarly deter- 
mined. If the ground flattens out so that five-foot contours are 
very far apart, the absolute elevations of points at even fifty- 
foot distances from the center should be determined. The 

* The method of making such observations is given in the Appendix. 



§13. 



RAILROAD SURVEYS. 



17 



method is exceedingly rapid. Whatever error or inaccuracy 
occurs is confined in its effect to the one station where it 
occurs. The work being thus plotted in the field, unusually 
irregular topography may be plotted with greater certainty and 
no great error can occur without detection. It would even be 
possible by this method to detect a gross error that might have 
been made bv the level party, 




Fig. 5. 




Fig. 6. 



13. Stadia method. This method is best adapted to fairly 
open country where a ^'shot'^ to any desired point may be 
taken without clearing. The backbone survey Hne is the same 



18 RAILROAD CONSTRUCTION. § 13. 

as in the previous method except that each course is h* mi ted to 
the practicable length of a stadia sight. The distance between 
stations should be checked by foiesight and backsight — also the 
vertical angle. Azimuths should be checked by the needle. 
Considering the vital importance of levelmg on a railroad survey 
it might be considered desirable to run a line of levels over the 
stadia stations in order that the leveling may be as precise as 
possible ; but when it is considered that a preliminary survey is 
a somewhat hasty survey of a route that may be abandoned, and 
that the errors of leveling by the stadia method (which are con- 
pensating) may be so minimized that no proposed route would 
be abandoned on account of such small error, and that the effect 
of such an error may be usually neutralized by a slight change in 
the location, it may be seen that excessive care in the leveling 
of the preliminary survey is hardly justifiable. 

A stadia party should include a locating engineer (or chief of 
party), and perhaps an assistant, a transitman, a recorder and 
four rodmen, beside axemen. The transitman should have noth- 
ing to do but attend to his instrument. After setting up the 
transit at an advanced station, a backsight should be taken 
to the previous station. If the vertical circle is full 360°, the 
telescope should be plunged and sighted on the backsight with 
vernier A reading the same as the foresight to the station occu- 
pied. If the vertical arc is semi-circular (or less), no vertical 
angle can be taken with the telescope plunged and, therefore, 
vernier A should read 180° more (or less) than the foresight. 
The lower plate should be very firmly clamped, and then, after 
loosening the upper plate, a reference sight and reading on some 
well-defined natural object should be taken. If there is any 
reason to suspect that the instrument has been disturbed while 
occupying that station, the reference point can be sighted at 
and the instrument can be re-alined, and re-leveled, if necessary, 
without sending a rodman back to the previous station. 
When taking a backsight the rod reading for distance should 
be taken first and immediately compared with the previously 
recorded foresight. Since the distance between stations will 
always be taken with especial care so as to avoid ^^ blunders " 
of an even 10, 20 or perhaps 100 feet, the foresights and back- 
sights should agree to within the proper limits of the stadia 
method. Similarly the vertical angle should agree with the 
previous reading, hut with opposite sign. If especial care is 



§ 13. RAILROAD SURVEYS. 19 

taken in leveling the instrument immediately before taking both 
foresights and backsights, these readings should agree to within 
one minute, or even 30 seconds, with a good transit. The 
height of the telescope above the ground at the new station must 
be measured, and the middle wire sighted at that reading on the 
rod (called the H. /.), when taking any vertical angle. Theo- 
retically the rod reading for distance should be taken when the 
telescope is pointing at the proper vertical angle for that shot, 
but this will mean, in general, that both the upper and lower 
cross wires will read odd amounts and that an inconvenient 
subtraction must be made to get the difference, which is the 
*' rod reading." But it may be demonstrated that no error of 
distance, amounting to the lowest practicable unit of measure- 
ment, can result if the telescope is raised or lowered just enough 
to set it on the nearest even foot mark. The routine of observ- 
ing a shot is therefore as follows: (a) swing the instrument (the 
upper plate) horizontally until the telescope sights at the rod 
and clamp the horizontal motion — but very lightly and perhaps 
nat at all; (6) raise or lower the telescope until the middle cross 
wire is sighting at the H. L, reading on the rod; a target on the 
rod may be set at the H. I. reading for each set-up and it will 
facilitate the work; (c) read the vertical angle and report it 
to the recorder, standing at hand; {d) raise or lower the tele- 
scope just enough so thht the lower wire is on the nearest even 
foot mark and read (calhng it out to the recorder) the number of 
even feet of interval from the lower to the upper wire and the odd 
amount at the top at the reading of the upper wire; (e) dismiss 
the rodman, who is then directed to another point by the chief 
of party; (/) read the azimuth on the horizontal plate. By 
that time another rodman has been located at a point where an 
observation is required, and the routine is repeated. The work 
of the transitman is thus very strenuous, without any recording 
work, and the progress of the party depends on him. He, there- 
fore, should not be required to direct the party or even to record 
his notes, since every moment spent in that way delays the entire 
party by that amount. The recorder also has all that he can 
do to record the notes (with perhaps some sketches), as fast as 
the transitman calls them off. Usually four rodmen can be 
kept very busy, and they must be on the run between the suc- 
cessive points at which they hold their rods. One of the rod- 
men or one of the axemen, if axemen are employed, carries and 



20 



RAILROAD CONSTRUCTION, 



§14. 



drives the stakes, which are only required at the instrument 
points. One or more axemen are generally useful in lopping off 
branches or cutting down saphngs which interfere with desirable 
sights. The chief of party has plenty to do in directing the 
rodmen and axemen so that shots may be taken at points which 
will give the most significant information, and also in picking 
out the proper location for the advance station at some place 
from which a maximum of information may be observed with 
one set-up of the transit. A well-drilled organization and 
"team work" are necessary. The best work is done when every 
man is kept busy. Several hundred shots per day can be ob- 
served when it is considered advisable to obtain much detailed 
information and the average number of shots per set-up is large. 
On the other hand, when the stadia method is used for a rapid 
exploratory survey, only a few side shots (at some stations per- 
haps none at all) will be taken at each station. In such a case, 
the total number of shots taken during a day will be compara- 
tively small, but the progress will be very rapid, and the saHent 
features of several miles of a proposed route can be obtained in a 
day. 

14. Form for stadia notes. 



[Left-hand page.] 










Inst, at 


Azim. 


Rod 


Vert, angle 


^f : 1 Elev. 


Sighting at 


A24 


264° 27' 

83° 10' 

184° 23' 

5° 47' 


622 
528 
264 
218(175) 


-0° 18' 
+ 1° 16' 
-2° 18' 
+26° 20' 


1 

1 
i 

! 


A23 


HI=4:.9 .... 

M=629.2... 


A25 
bend in creek 
top of bluff 







The usual six-column note-book can be utilized by ruling an 
extra line (shown dotted in the Form of Stadia Notes), in the 
fifth colimm, since the column is wide enough for both the " dif- 
ference of elevation " and the " elevation.^' The ^' rod reading " 
(3d column) as recorded should include the (/+c), which in 
almost all American transits equals 1.0 to 1.3 feet. Since the 
wire-interval ratio is almost invariably 1 : 100, the rod interval 
in hundredths of a foot is considered as the number of feet of dis- 
tance, except that one even foot is added for the (f+c). The 
sample figures given above are typical of all that needs to be 
taken in the field. The ^' difference of elevation " and the " ele- 
vation " are computed and entered later. 



i 



§ 14. RAILROAD SURVEYS. 21 

The '' difference of elevation " may be mathematically com- 
puted from the formula 

D = kri sin 2a+(f+c) sin or, 

in which D is the difference of elevation, fc is a constant, usually 
100, r is the rod intercept and a is the angle of elevation — or 
depression. The mathematical solution of such an equation 
for every shot that is taken (except the very few shots which are 
level) is very laborious and impracticable. But the work of 
reduction can be shortened by a justifiable approximation. By 
changing the factor of {f-\-c) from sin a to i sin 2a, the formula 
may be written 

D' = [/br+(/+c)]isin2a. 

The first term (that within the bracket) is the number recorded 
under ^' Rod '' in the Form of Notes (622, 528, etc.). The 
second term (§ sin 2a) may be taken from " Stadia Tables," of 
which many are published, although the tables usually give 
these numbers merely as the factors by which the distance is to 
be multiplied in order to obtain the " Difference of Elevation,'' 
and do not mention that the factor is really ^ sin 2a. The error 
of the approximation (when (f-\-c) = 1 foot) is less than 0.01 foot 
for a vertical angle of 15° and less than 0.1 foot for the unusual 
angle of 30°. Since 0.1 foot is the usual lowest unit of measure- 
ment for stadia elevations, probably 99% of all stadia work can 
use such an approximation without appreciable error. The 
special cases with high angles can be computed separately if it 
is considered necessary. The algebraic sign of the vertical angle 
should always be recorded, even if it is plus, or upward; the sign 
+ is a positive statement that it is plus and that the sign was 
not forgotten. The difference of elevation likewise should always 
have a + or — sign. Adding the difference of elevation to the 
elevation of the station (or subtracting it), gives the elevation of 
each point. 

Theoretically the true horizontal distance for all incHned sights 
is always less than the nominal distance, as given by the rod 
reading, The formula for true distance is 

L=kr cos'^ a-i-{f^c) cos a. 



I 



22 RAILROAD CONSTRUCTION. § 15. 

As before, we may use the approximation of combining the 
(/+c) with the kr and say that 

L' = [/cr+(/+c)]cos2a, 

and that the correction^ or the reduction from the nominal reading 
to the true distance, is 

Corr. = [hr+(J+c)] sin^ a. 

The error of this approximation is usually insignificant, as illus- 
trated below. Since sin^ a. is very much less than cos^ a for the 
usual small values of a, it is easier and more accurate to compute 
the smaller quantity and mentally subtract it from the nominal 
reading. When the vertical angle and the distance are both 
small, the horizontal correction is within the lowest unit of 
measurement (one foot), and should, therefore, be ignored. The 
engineer soon learns the approximate limits at which the com- 
bination of vertical angle and distance will make a correction 
necessary. In the above notes no correction is necessary except 
in the last case, the angle being 26° 20'. The exact mathe- 
matical computation is as follows, the rod interval being 2.17 
and (/+c) = l, 

L = 217 cos2 26° 20'+l cos 26° 20' = 175.20. 

Using the approximate rule, the correction = 218 sin^ 26° 20* 
= 42.90. 

218-42.90 = 175.10. 

The above calculations have been carried to hundredths of a foot 
for the sole purpose of illustrating that the discrepancy between 
the approximate and the theoretical value is only 0.10 foot, even 
for this unusually large angle, and considering that the rod 
interval is read only to the nearest 0.01 foot, which corresponds 
to one foot of distance, this discrepancy is utterly inappreciable. 
15. The reduction of stadia observations is most easily 
accomplished by using a stadia slide rule, which has one loga- 
rithmic scale for distances and for the computed differences 
of elevation or corrections to distance, and also two other scales 
one of which gives values for ^ sin 2q:, and the other gives values 



1^ 



§ 16. KAILROAD SURVEYS. 23 

for sin^ a. Some scales give values of cos^ a. To Illustrate the 
difference, in the above case, it is evidently easier to read 43 
(two significant figures) than to read 218, which has three fig- 
ures. When the distance is over 1000 (four figures), the diffi- 
culty is even greater. The necessity for subtracting the cor- 
rection is of no appreciable importance. In this case, the cor- 
rection would be read from the shde rule as 43, and mentally 
subtracting 43 from 218, we write at once 175, which is recorded 
in parenthesis in the Rod column. The draftsman, when plot- 
ting the notes, uses this distance (175) instead of 218. Using. a 
shde rule, two men can very quickly compute the differences of 
elevation for the entire day's work in a very short time. A very 
little practice wiU enable them to run down the list, picking out 
the observations, usually less than 10% of the total number, 
where the combination of distance and vertical angle is suffi- 
ciently great to make it necessary to compute a horizontal cor- 
rection. The stadia sHde rule is so small that it may readily be 
carried into the field and used there if desired, in which respect it 
has a great advantage over diagrams, which are sometimes used 
for the same purpose. 

i6. Stadia method vs. cross-section method. There is still a 
difference of opinion among engineers as to the choice of these 
two methods. When a large part of the route is thickly wooded, 
the cross-section method is preferable. In open country the 
stadia method is more rapid and more economical. Although it 
would be inadvisable to change from one method to the other 
every mile or so, a very considerable economy is possible by 
alternating the two methods according to the character of the 
country. The locating engineer can plan such change of method 
during his reconnoissance. The real efficiency of the stadia 
method is due to the fact that the preliminary survey should be 
considered as the topographical survey of an area or belt, and 
not the survey of a line, and that in open country the stadia 
method is the most efficient method of obtaining such topo- 
graphical data. But the efficiency depends on the handhng of 
the party. When a valley widens out with easy slopes and the 
possible area in which the location may he is correspondingly 
widened, it is far easier and more accm^ate to widen the belt 
surveyed by stadia shots of 1000 feef if necessary. 

17. " First " and " Second " prelimmary surveys. Some 
engineers advocate two prehminary surveys. When this is done^ 



24 RAILROAD CONSTRUCTION. § 18. 

the first is a very rapid survey, made perhaps with a compass, 
and is only a better grade of reconnoissance. Its aim is to 
rapidly develop the facts which will decide for or against any 
proposed route, so that if a route is found to be unfavorable 
another more or less modified route may be adopted without 
having wasted considerable time in the survey of useless details. 
By this time the student should have grasped the fundamental 
idea that both the reconnoissance and preliminary surveys are 
not surveys of lines but of areaSj that their aim is to survey 
only those topographical features which would have a deter- 
mining influence on any railroad line which might be constructed 
through that particular territory, and that the value of a locating 
engineer is largely measured by his ability to recognize those 
determining influences with the least amount of work from his 
surveying corps. Frequently too Httle time is spent on the 
comparative study of preliminary lines. A line will be hastily 
decided on after very little study; it will then be surveyed with 
minute detail and estimates carefully worked up, and the claims 
of any other suggested route will then be handicapped, if not 
disregarded, owing to an unwilHngness to discredit and throw 
away a large amount of detailed surveying. The cost of two or 
three extra prehminary surveys {at critical sections and not over 
the whole fine) is utterly insignificant compared with the prob- 
able improvement in the " operating value " of a line located 
after such a comparative study of prehminary fines. 

LOCATION SURVEYS. 

i8. " Paper location." When the preliminary survey has 
been plotted to a proper scale (usually 200 feet per inch), and the 
contours drawn in, a study may be made for the location survey. 
Disregarding for the present the effect on location of transition 
curves, the alinement may be said to consist of straight lines (or 
" tangents ") and circular curves. The '^ paper location " there- 
fore, consists in plotting on the preliminary map a succession of 
straight lines which are tangent to the circular curves connecting 
them. It may be assumed that the general route of the prelim- 
inary survey has been so weU selected, as the result of the recon- 
noissance survey, that it is possible to construct a line without 
excessive earthwork between consecutive control points, and 
that the grades are within the ruling grade. If the prehminary 



§18. 



RAILROAD SURVEYS. 



25 



survey has been run by locating stations every 100 or 200 feet 
(see § 11 and Fig. 4), the profile of this Hne gives the first approx- 
imation toward the rate of grade, and from this may be deter- 
mined whether one uniform grade between the control points is 









— 1 — 


' 1 — " — 
























— t-J 


■ — '-. J 


















iA= 


1 — 
— 


1 




T 






— 










1 


N 


^: 


t 
















— 




"^ 


^ 




^ 


y 







Fig. 7, Single Gbade Between Control Points. 

practicable, or whether two or more different grades must be 
used. If the stadia method was used, the profile of a line run- 
ning through the station points will serve the same purpose. In 
Fig. 7 let AMZ represent, on a very small scale, the surface 
profile between two control points, A and Z, which are, perhaps, 




Fig. 8. Two Geades Between Control Points, 



two miles apart. The upper dotted line shows the elevations of 
the highest points in the surveyed belt at each of the several 
stations, and the lower fine the corresponding lowest points. If 
the straight line AZ does not go outside of these dotted fines, it 
indicates that the uniform grade AZ will have " supporting 
ground " for the entire distance and that such a grade is prac- 
ticable and should be tentatively selected (or at least invest^'- 



26 RAILROAD CONSTRUCTION. § 18. 

gated) for that stretch. If the straight Hne AZ passes outside 
the belt of the dotted hnes, as in Fig. 8, it impHes that there was 
some definite reason why no higher supporting ground could be 
found near M\ or the preliminary survey, if properly made, 
would have covered that ground. It then becomes necessary 
to adopt two grades, such as AM' and M'Z. Three or more 
grades might prove necessary or desirable in some cases. 

Having determined, at least tentatively and approximately, 
the rate of grade, set a pair of dividers at such a distance (to 
scale) that the distance times the rate of grade equals the con- 
tour interval. For example, with a contour interval of 5 feet 
and a 2% grade, 

distance X. 02 = 5,' 
or 

distance = 5-^.02 = 250. 

Then, with dividers set at 250 feet, put one leg where the Hne 
previously located crosses a contour and put the other leg where 
it reaches the contour next above — or below, if a down grade. 
Then step to the next contour and so on. If the desired starting 
point is not on a contour, the distance for the first step should be 
proportionately shortened. A strict application of this method 
would probably make a sidehill line run around short gullies 
where the curvature would need to be excessively sharp. To 
avoid such sharp curvature, these narrow gullies must be crossed 
by bridges, trestles or high embankments. To carry a grade 
across such a place, the length of step of the dividers should be 
doubled or trebled and the step should be to the second or third 
contour above or below. The line running through these suc- 
cessive points located on the contours will be practically a surface 
line which has nearly the desired grade. The cut and fill would 
be almost nothing — except '^ side-hill work," and the crossing of 
gullies. No accuracy need be expected on this preliminary trial 
since the distance is somewhat greater than the air-line distance 
AZ. It would, in general, be impossible to run a practicable 
combination of tangents and proper curves through these points, 
but such a line is very suggestive of a proper alinement which 
will fulfill the grade and curvature conditions and along which 
the cut and fill will be reasonably small. 

If there are long stretches where, in each case, the line joining 
a group of consecutive points is nearly straight, the tangents will 



§ 18. RAILROAD SURVEYS. 27 

predominate and should be located first and then connected by 
curves. If the Hne has numerous and long bends, it may be 
preferable to select the curves first and then connect them with 
tangents. For such work a series of curves, drawn to proper 
scale, varying by even degrees from 1° up to 15° or 20°, or what- 
ever is the maximum allowable curvature, and drawn on any 
transparent material such as tracing cloth, celluloid or glass, is 
very useful, since different curves may be tried in turn until the 
curve which best fits the ground is discovered. The contours 
and other fixed features should have been inked in and then 
the trial lines and curves may be marked in lightly with a soft 
pencil, so that trial lines may be easily erased until a satisfactory 
fine is obtained. The number of possible combinations is infinite, 
but certain conditions must be fulfilled which narrows the choice. 
(1) The connecting tangents must not be too short; 100, 200 
and even 300 feet are used as limits. (2) The curvature must be 
within the adopted limit. If two consecutive curves, which are 
connected by a very short tangent, bend in the same direction, 
it is preferable that they should be combined into one simple 
curve, or into two branches of a compoimd curve, rather than to 
make a " broken-backed " curve. If they bend in opposite 
directions (making a reverse), even 300 feet is none too long for 
the transition curves which should be used, especially if the 
curves are sharp. Actual reverse curves (changing the direction 
of curvature without any separating tangent) should never be 
used, except on switch work and track where the speed is always 
slow. It would be far preferable to sharpen the curvature 
enough to introduce a tangent at least 100 feet long. The fol- 
lowing considerations should be kept in mind. * 

'^ (1) If the location could follow the grade hne [or surface 
Hne] precisely, there would be no cuts or fills (practically speak- 
ing) on the center hne. 

" (2) Whenever the location Hes on the ] i -n 1 side of the 

[ up-hill J 

{fill 

" (3) The further the location departs from the grade contour 
the greater will be the cut or fill, as the case may be." 

* Course of Instruction on "Paper Locatipn," by Prof. J. C. L. Fish, 
Stanford University. 



28 EAILROAD CONSTRUCTION. § 19. 

After a location line has been selected which seems satisfactory 
from the standpoints of easy curvature, not too short tangents, 
a proper balance of cut and fill, and not too great cuts and fills, 
as will be approximately indicated by its distance from the 
surface line, the volume of earthwork may be estimated with 
sufficient accuracy for comparative purposes by drawing a 
profile of the surface location line and its roadbed line. Con- 
sidering the ease with which such lines may be drawn on the 
preliminary map, it is frequently advisable, after making such 
a paper location, to begin all over, draw a new line over some 
specially difficult section and compare results. Profiles of 
such lines may be readily drawn by noting their intersection 
with each contour crossed. Drawing on each profile the re- 
quired grade line will furnish an approximate idea of the com- 
parative amount of earthwork required. A comparison of 
the areas of cut and fill on the profile will show the approxi- 
mate balance in volume of cut and fill. If it is considered nec- 
essary to compute the volume with greater accuracy, it may be 
done by the use of Table XVII (see also § 126), applying the 
latter part of the table correctively to allow for side slope. 
After deciding on the paper location, the length of each 
tangent, the central angle (see § 51), and the radius of each 
curve should be measured as accurately as possible. Frequent 
tie lines and angles should be determined between the plotted 
location line and the preliminary line. When the preliminary 
line has been properly run, its '' backbone " line will lie very 
near the location line and will probably cross it at frequent 
intervals, thus rendering it easy to obtain short and numerous 
tie lines. 

19. Preparation of the notes. This and the actual transfer 
of the paper location to the ground is a problem in surveying 
which is so varied in its character that the ingenuity of the engi- 
neer is required to use the best method adapted to each partic- 
ular case, but a few principles may profitably be kept in mind. 

(1) The scale of the paper location drawing is probably 200 feet 
per inch, unless the difficulties of the problem demand a larger 
scale for a particular stretch of the road, so that the paper loca- 
tion may be more accurate. Since a variation of 1/200 inch in 
the drawing means a variation of one foot on the ground, no 
close checking of the fine on any tie-point need be expected. 

(2) Since a very small variation in alinement w^ould, if persisted 



§ 19. RAILROAD SURVEYS. 29 

in, throw the alinement very far from its desired location, it must 
be expected that there will be more or less adjustment of the 
paper location alinement (numerically) on nearly every tangent 
and curve. (3) The intersection of the preliminary line by a 
paper-location tangent (or the tangent produced) gives a pos- 
sible tie-point. The position of this tie-point on the preliminary 
line must be scaled and the angle between the lines determined 
by measuring the chord of a long arc with its center at the point 
of intersection or by scaling the sine (or tangent) produced by a 
perpendicular from one line to the other from a point whose dis- 
tance from the intersection is a convenient unit length. (4) 
When there is no intersection at some place where a tie is desired, 
a perpendicular offset from the preliminary Hne may be necessary. 
(5) When the paper location crosses the preliminary line at fre- 
quent intervals (say 500 to 1000 feet), it may be more simple to 
locate the tie-point intersections on the prehminary line and work 
from one to the other, taking up the inevitable inaccuracies by 
slight variations in the length of tangents or curves or by some 
one of the various methods detailed in § 63. When no prac- 
ticable tie can be obtained for a considerable distance (say one- 
half mile), it may be desirable to determine the ordinates (lat- 
itudes and departures) of all the points on the preliminary and 
on the paper location between two consecutive intersections. 
In such a case the precision would depend entirely on the accuracy 
of scaling the positions of the two intersections and on the accu- 
racy of the preliminary survey. While such a method requires 
considerable office computation, even that is cheaper than an 
extensive revision of a located line in the field. For a further 
development of this method, the student is referred to a 
course of instruction originally written by Prof. J. C. L. Fish, 
of Stanford University, and included in the sixth edition of 
^' Surveying Instruments," by Webb & Fish, pubhshed by 
Wiley & Sons. 

As previously stated, the above method has been developed 
as if the final located line were to be made up only of tangents 
and circular curves. But transition curves between the tan- 
gents and circular curves are essential for the easy operation of 
trains. Anticipating the more complete demonstration of the 
subject, § 41, et seq., it may be stated that the effect of the transi- 
tion curve, or " spiral," is to move the curve inward, or toward its 
center, or to move the tangent outward. The effect of this is 



i 



30 RAILROAD CONSTRUCTION. § 20. 

equivalent to offsetting the tangent outward, or offsetting the 
ciu-ve inward, and then connecting the tangent and circular 
curve by a transition curve which gradually crosses the offsetted 
distance. The amount of the offset varies with the degree of 
the central curve and the desired length of the transition curve, 
but it is seldom more than three or four feet, and is usually much 
less. No consideration need be given to these offsets when 
comparing several trial locations. It is only after the paper 
location has been settled and it is time to transfer this to the 
ground that it is necessary to compute these offsets and adjust 
the hues accordingly. Even then the offsets will seldom be so 
large that they would appreciably affect the paper location, but 
when the alinement is actually located on the ground, the proper 
offsets should be used and the aHnement laid out as described in 
detail in § 80. 

20. Surveying methods. A transit should be used for aline- 
ment, and only precise work is allowable. The transit stations 
should be centered with tacks and should be tied to witness- 
stakes, which should be located outside of the range of the earth- 
work, so that they will neither be dug up nor covered up. All 
original property lines lying within the limits of the right of way 
should be surveyed with reference to the Jocation line, so that 
the right-of-way agent may have a propei basis for settlement. 
When the property lines do not extend far outside of the re- 
quired right of way they are frequently surveyed completely. 

The leveler usually reads the target to the nearest thousandth 
of a foot on turning-points and bench-marks, but reads to the 
nearest tenth of a foot for the elevation of the ground at stations. 
Considering that j-oVo o^ ^ ^oot has an angular value of about 
one second at a distance of 200 feet, and that one division of a level- 
bubble is usually about 30 seconds, it may be seen that it is a 
useless refinement to read to thousandths unless corresponding 
care is taken in the use of the level. The leveler should also 
locate his bench-marks outside of the range of earthwork. A 
knob of rock protruding from the ground affords an excellent 
mark. A large nail, driven in the roots of a tree, which is not 
to be disturbed, is also a good mark. These marks should be 
clearly described in the note-book. The leveler should obtain 
the elevation of the ground at all station-points; also at all 
sudden breaks in the profile line, determining also the distance 
of these breaks from the previous even station. This will in- 



§ 21. RAILROAD SURVEYS. 31 

elude the position and elevation of all streams, and even dry 
ijullies, which are crossed 

Measurements should preferably be made with a steel tape, 
care being taken on steep ground to insure horizontal measure- 
ments. Stakes are set each 100 feet, and also at the beginning 
and end of all curves. Transit-points (sometimes called ^' plugs" 
or "hubs") should be driven flush with the ground, and a 
'^ witness-stake," having the "number " of the station, should 
be set three feet to the right. For example, the witness-stake 
might have on one side "137 + 69,92," and on the other side 
"PC4°R," which would signify that the transit hub is 69.92 
feet beyond station 137, or 13769.92 feet from the beginning of 
the line, and also that it is the "point of curve" of a "4° curve" 
which turns to the right. 

Alinement. The alinement is evidently a part of the loca- 
tion survey, but, on account of the magnitude and importance 
of the subject, it will be treated in a separate chapter. 

21. Form of Notes. Although the Form of Notes cannot be 
thoroughlj^ understood until after curves are studied, it is here 
introduced as being the most convenient place. The right-hand 
page should have a sketch showing all roads, streams, and 
property lines crossed with the bearings of those lines. This 
should be drawn to a scale of 100 feet per inch — the quarter- 
inch squares which are usually ruled in note-books giving con- 
venient 25-foot spaces. This sketch will always be more or less 
distorted on curves, since the center line is always shown as 
straight regardless of curves. The station points ("Sta." in 
first column, left-hand page) should be placed opposite to their 
sketched positions, which means that even stations will be 
recorded on every fourth line. This allows three intermediate 
lines for substations, which is ordinarily more than sufficient. 
The notes should read up the page, so that the sketch will be 
properly oriented when looking ahead along the line The 
other columns on the left-hand page will be self-explanatory 
when the subject of curves is understood. If the "calculated 
bearings" are based on azimuthal observations, their agreement 
(or constant difference) with the needle readings will form a 
valuable check on the curve calculations and the instrumental 
work. 

22. Number of men required in surveying parties. No fixed 
rules can be given. The general rule of economy and efficiency 



32 



RAILROAD CONSTRUCTION, 



§22, 



FORM OF NOTES. 



[Left-hand page.] 



Sta. 


Aline- 
ment. 


Vernier. 


Tangential 
Deflection. 


Calculated 
Bearing. 


Needle. 


54 












53 












+72.2 


P.T. 


9° 11' 


18° 22' 


N 54° 48' E 


N 62° 15' E 


52 


j-*^ 


7 57 








51 




6 15 








50 


o si 

> a 


4 33 








49 




CO 00 


2 51 








48 


1 1 


1 09 








0+32 


P.C. 


0° 


/ 






47 












46 




N 




N 36° 26' E 


N 44° 0' E 



should govern, and that is, that the organization should be such 
that all desired data can be obtained at a minimum of cost. 
This general rule may be subject to the modification that the 
early completion of the survey is sometimes financially so impor- 
tant as to justify the maximum speed, almost regardless of 
expense. A common violation of the general rule of economy 
is the use of too few men, with the mistaken idea that it is eco- 
nomical. This requires the high-priced efficient men to waste 
their time on work which men at one-half (or even one-third) 
their salary could do sufficiently well, thus delaying the com- 
pletion of the work or depreciating its quaUty by undue haste 



§22. 



RAILEOAD SURVEYS. 



33 



[Ri«ht-hand page.] 




53+60 
(h JAS. WILSON 




wm. brown 




JONES 



or by neglect to obtain complete data. I'he work should be so 
organized that each man is constantly busy at the kind of work 
for which he is especially qualified, and that n© men shall have 
to wait for others to complete their co-ordinate work. Even if 
100% efficiency is unobtainable, it is very uneconomical to have 
nearly the whole party idle while one or two high-priced men 
do some work which must be done before the party can proceed 
but which could have been done by some extra lower-grade men 
without delaying the party. Reconnoissance. When the ter- 
ritory of the general route has been mapped by the U. S. Geol. 
Survey, there may be uo need of instrumental work on the 



34 RAILROAD CONSTRUCTION. § 22. 

reconnoissance, since the approximate ruling grades and general 
route may perhaps be determined directly from the map, and 
the purpose of the reconnoissance is the examination of physical 
features which would affect or modify the general route. In 
such a case the engineer does his technical work alone and only 
needs a guide and cook in case camping is necessary. When the 
reconnoissance partakes more of the nature of a hasty prelim- 
inary, distances, elevations and the necessary side topography 
being determined by rapid approximate methods, more men 
should be added, keeping in mind that the work should be so 
organized that each member of the party is kept busy at his own 
co-ordinate work, and that the chief engineer is not delayed in 
his own special work by spending his valuable time on a cheaper 
grade of work which an assistant could do sufficiently well. In 
other words, it is economical to add to the party an extra assist- 
ant whenever the work that he can do will so facilitate the 
work of the party as a whole that the value of the salaries 
and expenses saved will more than offset the assistant's 
salary and expenses. Preliminary surveys. No fixed list of 
members of a party is applicable to all conditions. The fol- 
loAving list, with monthly salaries, is given by Mr. Fred La vis* 
as having been used on each of five parties in surveying the 
Choctaw, Oklahoma & Gulf R. R. The list is very full but 
justifiably so. 

Locating engineer $150 to $175 

Assistant locating engineer 115 125 

Transitman 90 100 

Levelman 80 90 

Draftsman 80 90 

Topographers, two 80 90 

Level rodman » 50 

Head chainman 50 

Rear chainman. 40 

Tapemen, two 30 

Back flagman 30 

Stake marker 30 

Axemen, three to five 25 to 30 

Cook 50 

Cook's helper 20 

Double teams and driver, furnish their own feed, 

driver boarded in camp 65 to 90 

* Methods of Railroad Location on the Choctaw, Oklahoma & Gulf 
R.R. Trans. Am. Soc. C. E., Vol. LIV, page 104. 



§ 22. RAILROAD SURVEYS. 35 

Other organizations sometimes combine the first two positions 
on this Hst and possibly call him '^ chief of party." For the 
above work, the locating engineer was relieved altogether from 
the detailed direction of the party, which was handled by the 
assistant, and spent nearly all his time in studying the country 
so as to determine how the line should advance. In nearly 
all cases, such expense is justified, perhaps many times over, (1) 
by the saving of uselessly surveying an improper route, (2) by 
an improvement in the operating value of the route selected, or 
(3) by an improvement in route which makes a decrease in 
construction cost. Sometimes those controlling the financial 
side of the project insist that the chief of party shall also run 
the transit, as a measure of " economy." Such a policy cannot 
be too strongly condemned. The work of a transitman requires 
every instant of his time and every minute that he turns from 
his transit to direct the party or study the proper route is a min- 
ute delay for the entire party. It generally means also a deteri- 
oration in the quality of his work as a leader and as a transitman, 
in his effort to hastily do at one time work which requires the 
concentrated efforts of two men. In this survey (described by 
Mr. Lavis), the skeleton or backbone line was a broken line with 
angles every few hundred feet, and the topography was taken 
by right-angled offsets every hundred feet or oftener, substan- 
tially as described in §11 and Fig. 4. These offsets were deter- 
mined by a hand level and pacing by one of the two topographers. 
The other topographer, using a transit, with the other two tape- 
men '' determined drainage areas, located property lines and 
section corners, got names of property owners, etc.'' When, as 
is usually the case, such essential work cannot be done by the 
main party without delaying their progress, there is a real econ- 
omy in adding to the party these comparatively low-priced 
assistants. It may be noted that the above party includes two 
chainmen, back flagman and stake-marker, beside three to fiVe 
axemen. The proper number of axemen manifestly depends 
on the amount of necessary cutting, but the chainmen or the 
stake-marker should not be depended on for such work. The 
steady march of the party should not be halted while a stake- 
marker or chainman stops his regular work to cut down a tree. 
One of the duties of the chief of party is to foresee the necessities 
of tree-cutting and clearing, so far in advance that, by the time 
the surveying members of the party have reached the spot, the 



36 RAILROAD CONSTRUCTION. §23. 

area is cleared. It is likewise false economy to dispense with 
the stake-marker and require the head chainman to do such work. 
A full corps of such men, properly drilled, can add 20 to 50% 
to the daily progress of the party and much more than save 
their cost» 

MAINTENANCE OF SURVEY PARTIES. 

23. Economy and efficiency. When considering the treat- 
ment and maintenance of surveying parties, it should be remem- 
bered that a false idea of economy is frequently responsible for 
making the parties too small, overworking the men, depriving 
them of physical comforts and even necessities, and that the 
result is a greater net cost and a great deterioration in the quality 
of the results. A party may cost $40 to $65 per day in salaries 
and expenses. Any policy which depreciates the net output 
of their work 20 to 50% (which is easily possible) in order to 
save a few dollars per day is manifestly poor policy. The men, 
especially those who must use their brains and who presumably 
have a finer nervous organism, have only a quite definite sum 
total of nervous energy. If a considerable part of that energy 
is spent in needlessly long tramps both morning and. evening to 
and from work, or if that nervous energy is not maintained by 
plentiful and appetizing food and by sufficient and comfortable 
rest,' there is a reduction in efficiency which is often far greater 
than any possible saving in expenses. This idea of developing 
the maximum efficiency of the party is the justification of the 
recommendations made below regarding outfit, equipment, and 
other details about managing a party. 

24. Country hotels and farm houses. In settled sections 
of the country, country hotels and even farm houses are some- 
times available where men can be provided with living facilities 
which are unobtainable in camp life and at less total expense. 
Such accommodations have the advantage that they obviate a 
considerable capital expenditure to purchase sufficient camp 
outfit. But if suitable accommodations are unobtainable over 
a considerable portion of the route and such accommodations 
as there are on the remaining distance are inconvenient and 
inadequate, it may be preferable to provide a camping outfit at 
once. Considering the fact that there is a real economy in 
making a survey with a large party and that such a party can 



§ 25. RAILROAD SURVEYS. 37 

seldom if ever be accommodated in a single farmhouse, and that 
there is a lack of efficiency if the party is separated, the farm- 
house plan is frequently impractical. But when villages are so 
located that there is always one within five miles of any point of 
the line, the house plan may be preferable, since the party 
may be taken to and from work in conveyances. The economy 
of employing conveyances may be judged by comparing the cost 
of the vehicles and the value of the time and energy saved. A 
five-mile tramp, carrying an instrument, following a full day's 
work surveying, will frequently incapacitate a man from doing 
effective work in the night-work which the higher grade men of 
the party must generally do. The day's work in the field must 
be begun later and ended earlier or else the time and strength 
spent in the morning and evening tramps are uneconomical 
drains on their total nervous energy. 

25. Camping Outfits: Tents. The Choctaw, Oklahoma & 
Gulf R.R. survey, previously referred to, provided for each 
party one office tent, with fly, 14X16 feet, three tents, evidently 
without flies, 14X16 feet, and one cook tent 16X20 feet. The 
office tent had 5-foot walls; the others 4-foot. H. M. Wilson 
('^Topographical Surveying," p. 817) recommends 9X9 foot 
tents, with 4-foot walls. These are easier to erect but have only 
36% of the floor area of the 14 X 16-foot tents and it Would 
require 15 such tents to equal the floor area of the 5 tents 
described above. For a small party the smaller tents would be 
preferable. The canvas should be mildew-proof and free from 
sizing. A " sod-flap " about 8 inches wide, should be attached 
to the bottom of the wall. When this flap is weighted down 
with stones or heavy sticks the wind and weather is kept out. 
Dirt or sod should not be used for weights, since they rot the 
canvas. It pays to use tents which conform to the U. S. Army 
specifications. Some of the specifications as to material and 
workmanship are here quoted: 

'^ Materials. — Body of tent to be made of Army standard 12A 
ounce cotton duck, 29^ inches wide and the sod cloth of Army 
standard 8-ounce cotton duck, 28^ inches wide. 

" Workmanship. — To be made by machine in a workmanlike 
manner, all seams to be stitched with two rows of stitching, not 
less than six stitches to the inch, with three-cord twelve-thread 
Sea Island cotton, white. 

" In making tents by hand, to have not less than two and one- 



38 RAILROAD CONSTRUCTION. § 26. 

half stitches of equal length to the inch, made with a double 
thread of five-fold cotton twine, drab, well waxed. 

" The seams should be not less than 1 inch in width, flat 
stitched, and no slack taken in them. 

^^Grommet holes. — Made with malleable iron rings, galvanized, 
to be worked with four-thread five-fold cotton twine, well waxed. 

'^ Sod doth, — To be 8 inches in width in the clear from the 
tabhng, into which it is inserted 1 inch and extending from door 
seam to door seam around the tent. 

" Tabling. — On foot of tent when finished to be 2i inches in 
width." (Adopted July 14, 1911.) 

A ditch should be dug outside the tent, at least on the up-hill 
side, if the ground is at all inclined. This will prevent rain- 
water from draining through the tent. Of course, the bottom of 
the ditch should have a uniform slope draining to an outfall 
amply clear of the tent. 

26. Tent floors. Dry floors are almost essential to health. 
Sectional floors, about 3X9 feet per section, made by fastening 
boards to cross cleats, provide a perfectly dry floor and often 
repay their transportation. A mere layer of canvas, cut to 
proper shape and bound on the edges, is worth providing if the 
ground is dry when the tent is erected and can be kept from 
getting rainsoaked by proper outside drainage. 

27. Tent stoves. For winter work, tents may be made quite 
comfortable with stoves. Oil stoves are convenient when the 
oil can be purchased without excessive cost for transportation. 
** Sibley " stoves, burning wood, are commonly used but they 
require smoke pipes which must pass through the canvas and 
this means that the holes must be properly protected with metal 
or asbestos. If a pipe elbow is provided, the pipe may be taken 
out through one end of the tent. This obviates a hole in the 
roof of the tent (and also the fly) ; it avoids a direct pour of rain 
on the fire or leakage into the tent around the pipe, and also 
the danger of sparks dropping on the canvas. A ^' Sibley " 
stove for mere heating is a sheet-iron frustum of a cone, about 
3 feet high; diameter at bottom 18 to 30 inches; diameter at 
top 4^ to 6 inches, or so as to fit the stovepipe which is to be 
used. It has no bottom, or in other words, the bare earth forms 
the base. A door, large enough for the insertion of such fuel 
as it is designed to use, is placed in the side. Three or four 
lengths of pipe, one of which should have a damper, and an elbow, 



§ 28. RAILROAD SURVEYS. 39 

should be provided. Draft at the bottom is obtained, and may- 
be easily controlled, by packing earth around the base, leaving 
a small opening which may be easily enlarged or diminished to 
control the draft. Cook stove. A regular 6-hole cooking range, 
perhaps made of wrought-iron or sheet-steel, is essential to cook 
meals for twenty or more hearty men. Sporting outfitters 
supply all sizes of stoves, which must always be selected with 
due regard for the facilities for transportation. Oil stoves are 
commonly used. For still smaller parties, or when no cook 
stove can be permitted in the baggage, a primitive grid may be 
made from four sticks of green timber about 6 inches in diameter 
and 2 to 4 feet long. Notch two of them, each with a pair of 
notches about 10 inches apart. Place the other two sticks 
across the notches and they will steadily support a kettle or a 
frying pan. If the sticks are sufficiently green and the fuel quite 
dry the grid will last some time. A folding grid of iron bars may 
be obtained, which is but a small addition to the weight of the 
baggage. Another method is to suspend a kettle by a chain 
or long hook either from a tripod of sticks or from a horizontal 
stick lying in two forked sticks on each side of the fire. 

28. Dining tables. These are justifiable for a large party 
when the baggage is necessarily great and camp wagons are a 
part of the equipment. Mr. Lavis, in the article previously 
referred to, describes a very good table from the standpoint of 
transportation. The table top consists of three loose planks 
If" X 12'' X 18' 0". Two similar boards are used for seats. 
During transportation these boards are placed on the bottom 
of the wagon and, of course, project from the back where they 
form a support for stoves, etc., which can be roped on. These 
boards are supported on three trestles or horses, made as shown. 
For a much smaller party, a table may be improvised by utilizing 
two " mess-boxes,'^ which carry the cooking utensils and table- 
ware. These mess-boxes are about 20 inches wide and high 
and from 24 to 30 inches long. The covers are made to open 
180° and may be fastened horizontally. An '' inside cover,'' 
which can be utilized as a bread board, covers the entire inside 
area of the box. Two such boxes, set together and with the tops 
opened out, provide a fairly even surface four times the area 
of one box. 

29, Cooking utensils, table-ware, tools, etc. The size of the 
party, the individual preferences of the person designing the 



40 



RAILROAD CONSTRUCTION. 



§29. 



outfit and the facilities for transportation, vary such lists almost 
indefinitely. Agate ware has replaced china for plates and cups. 
Aluminum ware, although expensive, is preferable from a cooking 
standpoint and has the advantage of a very material reduction 
in weight. Out of the very great number of hsts which have 
been pubhshed, the following list of articles is quoted as sug- 
gestive: Plates, cups, saucers, steel knives and forks, German- 
silver spoons, large and small, carving knives and forks, large 
cooking forks and spoons, pepper and salt boxes, tin pans about 



2 SEAT BOARDS 
1% X 10 X 

.is'o'd.d. 



FIT MORTISE AND TENONS 
AT BOTH ENDS OF POSTS 
TIGHT BUT DO NOT FASTEN, 
SO HORSE WILL KNOCK DOWN 




ia^x6x5V 



Fig. 9. — Camp Dining Table. 



6 inches diameter by 1| inches deep, utilized for serving soup, 
cereal, etc., pans and kettles of varying sizes which will ^' nest " 
and thus facilitate packing, tea kettle, coffee pot, frying pan, 
griddle, cake turner, pie plates, dripping pan, chopping bowl 
and chopper, colander, flour sieve, coffee mill, broiler, corkscrew 
and can opener, rolling pin, folding table (similar to the drawing 
table described below), wash basins, kerosene oil can, alarm 
clock, spring balance. The last two articles are important. 
The cook is the first man up in the morning — ^usually before 
daylight — and may need the alarm clock. A single delay, of 
even ten minutes of such a party, would cost more than a very 
valuable clock. A spring balance is very essential to the proper 



§29. 



RAILROAD SURVEYS, 



41 



and economical use of provisions without waste. It pays to 
have a cook who is able to compute, weigh out and use an amount 



CLEATS FASTENED 
TO TOP WITH EIGHT 
2''SCREWS 




CLEAT, LEGS AND BRACES 
MADE OF OAK OR CHESTNUT 



IfeREW 
WHICH 
WHEN 



HOLES FOR TWO SCREWS^ 
WILL FASTEN LEGS SECURELY 
FOLDED 




Fig. 10. — Folding Drafting Lable. 

'of each kind of provisions so that there will be sufficient, but 
10 waste. Besides the above, dish towels are practically essen- 



42 RAILROAD CONSTRUCTION. § 30. 

tial and, tablecloths and napkins are easily carried. A table 
oilcloth may replace the ordinary tablecloth. Wash tubs and 
wash board facilitate the washing of table linen and also under- 
wear, so essential to clean, healthy living. Illumination for 
night work must be provided. Reflecting lanterns will answer 
for all tents except the office tent, where good lamps, with 
cylindrical wick and center draft, or similar, should be provided. 
The farther the party travels from " civilization " the greater 
the necessity for providing for emergencies, breakages, etc. 
Axes are essential, apart from their use in the surveying work. 
Extra handles should be provided. A saw, brace and several 
sizes of bits, screw drivers, monkey wrench, files, pliers, hatchet, 
assorted screws and nails, pick, shovel, crowbar, whetstone, 
rope in various sizes, sailor's needles, palm and sewing twine, 
will all be useful and even invaluable in times of emergency. 
Canvas-covered canteens, for each member of the party, when 
passing through arid regions, may be essential. 

30. Drawing tables. Complete topographic drawings, made 
in the field, are absolutely essential. Suitable drawing boards 
are, therefore, required. The design shown in Fig. 10 fulfills all 
the working requirements; it also is easily handled when packed 
up and is not readily broken. By packing them together in 
pairs, face to face, the surfaces are protected during transpor- 
tation. The table consists essentially of a drawing board w^ith 
stiffening cleats. The legs are hinged to the cleats, the braces 
for each pair of legs being of just such a length that when opened 
the legs stand at the desired angle. The braces are hinged and 
fold up, jackknife fashion, so that they nowhere project beyond 
the legs. 

31. Stationery and map chest. Considering that the maps, 
drawings and notebooks may represent thousands of dollars, 
and that they are likely to be injiured, if not irreparably ruined, 
by rain, when moving camp or during a cyclonic storm, a strong, 
water-tight chest, of ample capacity for all drawings and note- 
books, should be provided. It should be required that all 
drawings and notebooks should be kept in the chest over night 
and at all other times, except sucli drawings and notebooks as are 
in actual use. The net inside length should be a little in excess 
of the longest roll or drawing, which is perhaps 36 inches. There 
should be a tray in the top with numerous compartments or 
boxes for the multitudinous small articles required by a drafts- 



§32. 



RAILROAD SURVEYS. 



43 



man. Handles should be provided for convenience and it 
should have a lock. A good " steamer " trunk of requisite size 
will answer the purpose, provided it is waterproof, and it would 
perhaps be cheaper than a chest of similar size, made to order. 
32. Provisions. A ''ration" is the estimated amount of food 
required per man per day. For men engaged in strenuous out- 
door work, the food required is far more than that eaten ordina- 
rily. Ration Hsts should average about 5 to 6 pounds of food 
per day per man. The amount that must be transported may 
be considerably less than this, in view of the fact that e.g., dried 
vegetables may be substituted for fresh vegetables in the ratio 
of 1 lb. of dried for 3 lbs. of fresh, the water used in cooking 
providing the other two pounds. For explorers, who carry their 
own provisions, and who must cut down every possible ounce of 
baggage, still further concentrations are possible. 



^ Article 

Fresh meat, including fish and poultry, (a) 

Cured meat, canned meat, or cheese (6) 

Lard 

Flour, bread or crackers 

Corn meal, cereals, macaroni, sago, or cornstarch, 

Baking powder or yeast cakes 

Sugar 

Molasses 

Coffee 

Tea, chocolate or cocoa 

Milk, condensed (c) 

Butter 

Dried fruits (d) 

Rice or beans 

Potatoes, or other fresh vegetables (e) 

Canned vegetables or fruit 

Spices 

Flavoring extracts 

Pepper or mustard 

Salt 

Pickles 

Vinegar' 



100 rations 


100 lbs. 


50 " 


15 " 


80 •• 


15 " 


5 " 


40 " 


1 gal. 


12 lbs. 


2 " 


10 cans 


10 lbs. 


20 '• 


20 " 


100 " 


30 " 


1 < ' 


4 


1 • « 






2 


4 •' 


3 qts. 


1 " 



" (a) Eggs may be substituted for fresh meat in the ratio of 8 eggs for 1 lb. 
of meat. 

" (6) Fresh meat and cured meat may be interchanged on the basis of 5 lbs. 
of fresh for 2 ll^s. of cured. [This ratio 5:2 is far higher than is usually 
allowed, 5:3 or even less is usually stated as the equivalent ratio.] 

" (c) Fresh milk may be substituted for condensed milk in the ratio of 5 
quarts of fresh for 1 can of condensed. 

" (d) Fresh fruit may be substituted for dried fruit in the ratio of 5 lbs. 
of fresh for 1 of dried. 

" (e) Dried vegetables may be substituted for fresh vegetables in the ratio 
of 3 lbs. of fresh for 1 lb. of dried." 



44 



RAILROAD CONSTRUCTION. 



§32. 



The list at bottom of p. 43 is given by H. M. Wilson (''Topo- 
graphic Surveying ") as the ration list of the U. S. Geol. Survey. 
The quantities are those required to make up 100 rations, or the 
food for 5 men for 20 days, or for 100 men for one day. They 
are considered maximum. The sum total is about 525 lbs. or 
5 J lbs. per day per man. 

Wilson states that the cost of the above list of rations should 
not average more than 45 to 55 cents per day for average con- 
ditions and with a maximum of 75 cents, but considering that 
this statement was written in 1900, some allowance may need 
to be made for higher prices since then. 

The list given below represents the provisions actually supplied 
to a mining camp in British Columbia. The list has been reduced 
to the average quantity actually consumed per man per day. 
The food supply averaged nearly 6 lbs. per day per man. 



Meat, etc.: 

Fresh beef 1 . 89 lbs. 

Bacon 076 " 

Ham 060 ** 

Codfish 007 •* 

Canned salmon 014 can 

Breads, etc.: 

Pilot bread 007 lb. 

Flour 894 ** 

Baking powder 016 * * 

Corn meal 037 ** 

Vegetables: 

Potatoes 1.421 lbs. 

Turnips 010 ** 

Carrots 047 ** 

Beets 016 ** 

Parsnips 023 ' ' 

Rice 043 " 

Cabbage 101 ** 

Dehydrated onions. . . . 0014 * ' 

rhubarb. .0029 " 

White beans 0014 ' ' 

Bayo '* 027 •* 

Lima " .013 '* 

Split peas 006 *' 

Rowan " 0014 ** 

Canned tomatoes 016 can 

• • beans 0043 * ' 

*• peas 0014 ** 

Pearl barley . 0004 lb. 

Rolled oats 117 ** 

Beverages: 

Tea 021 lb. 

Coffee 036 " 

Milk, condensed. .... . 137 can 



Fruit: 

Dried apples . 040 lb. 

'* peats 033 ** 

** peaches 029 ** 

•* prunes 020 ** 

** apricots 007 ** 

•* figs. 030 *• 

Dehydrated cranberries .004 ** 

Currants 021 ** 

Jam 001 pint 

Condiments, etc. 

Mustard 001 lb. 

Salt 036 ** 

Pepper 001 ** 

Vinegar, Klondyke ... . 0003 pint 
Worcestershire sauce . . . 0043 ' * 

Catsup 0029 gal. 

Miscellaneous: 

Sugar 594 lb. 

Lard .030 " 

Cheese 016 ** 

Cornstarch 007 ** 

Extract 049 ** 

Curry powder 0007 ** 

Cinnamon 0009 ' * 

Hops 0001 " 

Nutmeg 0009 ' ' 

Ginger , .0014 ** 

Mapleine 0011 oz. 

Candied peel 004 lb. 

Butter .014 " 

Macaroni 003 ' * 

Sago Oil ** 

Tapioca 003 ** 

Baker's chocolate 0014 ** 

Cocoanut 0003 ' * 

Pickles 003 gal. 

Supplies: candles, .03 lb.; gold dust, 
.003 lb.; soap, .024 bar. 



§33. 



RAILROAD SURVEYS. 



45 



The following list of provisions was bought to start a camp of 
20 to 25 men on the Choctaw, Oklahoma & Gulf R. R. Survey. 
(F. Lavis, Trans. Am. Soc. C. E., Vol. LIV, p. 104.) 



6 hams 



50 lbs 


. fresh beef 


1 case eggs 


25 lbs 


butter 


25 " 


lard 


, 100 " 


flour, hard wheat 


100 " 


flour, soft "wheat 


100 •• 


sugar 


5 " 


baking powder 


2 " 


tea 


50 •' 


coffee 


50 " 


navy beans 


25 " 


lima beans 


12 " 


buckwheat flour 


5 " 


macaroni 


35 " 


cornmeal 


1 che 


ese, about 15 lbs. 


12 packages oatmeal 


10 lbs 


rice 



100 cakes soap 
1 gal. molasses 
1 case condensed milk 
1 doz. tomato catsup 
^ ' ' Worcestershire sauce 
1 gal. pickles 
I doz. lemon extract 
i ' ' vanilla extract 
1 box dried prunes 
5 lbs. raisins 

4 doz. assorted canned fruits 
1 case tomatoes 

1 bushel potatoes 
1 kit salt mackerel 
20 lbs. salt 
^ * ' mustard 
1 * ' pepper 
1 qt. vinegar 

5 doz. yeast cakes 



I 



In addition to the above, there must be provided plenty of 
matches, kerosene oil and perhaps candles. As a matter of 
health conservation, and the prevention of piles, it is wise to 
provide toilet paper and to insist, if necessary, on its use. There 
is economy, when it is practicable, in making wholesale con- 
tracts for all provisions, rather than to buy haphazard from small 
local sources. 

33. Beds. When baggage wagons accompany the party, as is 
virtually necessary to transport other essential equipment, it is 
desirable that they also transport army cots. These fold up so 
as to be easily transportable. It is "a wise economy to obtain 
the regular army blankets, since they are what long experience 
has approved. Canvas covers should be provided for the bed- 
ding. This is essential to keep the bedding in even reasonably 
cleanly condition, especially while moving. The policy of 
requiring each member of the party to provide himself with cot, 
bedding and cover, and to care for them, is debatable. As a 
matter of business economy, the company should buy all cots 
and bedding wholesale. Requiring each one to purchase his 
own is virtually a reduction of salary, for, if a man leaves the 
party, he usually does not care to take his bedding with him, 
except in the hope of reaHzing something on it. But as aU this 
is considered when accepting employment, the company vir- 
tually pays for the bedding by an increase of salary over what 



46 EAILROAD CONSTRUCTION. § 34. 

they would have to pay if bedding were provided. There is the 
same reason for owning bedding as for owning dishes, etc. 
SteriHzing bedding by means of a formaldehyde candle, especially 
after a man has left the party, is a wise sanitary precaution and 
nullifies one of the strongest reasons for individual ownership. 
34. Transportation. The route of travel of a mining engineer, 
a topographical engineer or an explorer, may be over country 
with every variety of surface and slope. But, since a practicable 
railroad route is necessarily on a low grade, except as it may pass 
over a ridge or mountain to be pierced by a tunnel, the question 
of grade does not ordinarily influence the method of transporta- 
tion and wagons can ordinarily be used, provided the nature of 
the surface will permit. Strong and heavy wagons can usually 
pick their way between the camping places, even though long 
detours must be made to avoid swamps or other obstructions. 
The parties surveying the Choctaw, Oklahoma & Gulf R. R., pre- 
viously referred to, used two teams regularly, one of which stayed 
with the topographical party. They used a third team for 
hauling supplies. Two teams of horses can help each other over 
a particularly bad place in the trail or in the case of accident. 
The wagons should have canvas tops, as a protection against 
rain, especially while moving. Transportation by dogs and 
sledges is only applicable under very limited and unusual condi- 
tions. It implies winter work, which is always uneconomical 
and ineflficient compared with summer work, but in a very 
swampy country, where the transportation of any considerable 
amount of baggage is very difficult, and where it freezes during 
the winter to a comparatively smooth surface, such a method 
may be preferable in spite of short daylight hours and other 
disadvantages. " The Duluth, South Shore & Atlantic Rail- 
way employed toboggans during the construction of its road 
throughout the season of 1887.'^ The description of this work, 
and much other useful information is given in a paper by Chas. 
H. Snow, Vol. XXIX, p. 164, in the Trans. Am. Inst. Mining 
Eng'rs. A reconnoissance through a comparatively unexplored 
country, made with the object of discovering a practicable low- 
grade route through a mountainous section, might require that 
all baggage shall be reduced to what may be handled in packs 
carried by horses, mules, Indian ponies or even by men. The 
question of the necessary method of transportation must always 
be studied before beginning a survey, since the entire question 



§ 35. RAILROAD SURVEYS. 47 

of subsistence, and even many features of the method of work, 
must depend on what can be included in the baggage. 

35. Clothing. While it may seem an unwarranted inter- 
ference with personal liberty to control the clothing worn by 
members of the party, it becomes justifiable when the eflSciency 
and progress of the party is impaired by bad health or disability, 
which is plainly due to neglect of proper precautions in the way 
of clothing or personal sanitation. Sore feet are responsible for 
a large part of the disablement of men. Washing the feet 
every night, especially when they have become wet, will often 
obviate blisters. Stockings should be heavy, made of " natural 
wool " and should fit tightly enough so that wrinkles will not 
form. Shoes should have heavy soles and should be made of 
such tough leather that they will not easily tear. Rubber boots 
should not be worn; they make the feet tender. Although a 
surveying trip is usually considered as the opportunity to use up 
discarded clothing, ordinary clothing is usually very xuisuitable 
and quickly becomes un wearable. When camping conditions 
are rough and the work must last for several months, and possi- 
bly years, clothing made of specially suitable material is econom- 
ical. The material should be tough, so that it will not easily 
be torn by brambles, etc. It should be waterproof so as to 
shed rain and yet should be porous. It should be so thoroughy 
shrunken that moisture cannot appreciably shrink it further. 
"Mackinaw" is a soft, rough cloth, all wool, thoroughly shrunken, 
Hght, warm and waterproof. It is especially suitable for cold 
weather. " Pontiac " is similar. " Khaki '' is a twiUed cotton 
and is especially suitable for warm weather clothing. " Jungle 
cloth " is somewhat similar, but is particularly noted for its 
toughness and durability. 

Especial care should be taken in the choice of underclothing, 
so as to avoid sudden chills after becoming overheated. Woolen 
underclothing is almost essential. " Cholera bands," made of 
wool, should always be worn about the abdomen in tropical 
countries, 

MEDICAL AND SURGICAL TREATMENT. 

36. Responsibility of engineer-in-charge. Throughout any 
surveying trip, where camping is necessary, professional medical 
aid is usually unobtainable. There rests upon the engineer-in- 



48 RAILROAD CONSTRUCTION. § 37. 

charge, as the head of the expedition, some measure of responsi- 
bility for the health and care of the party. When some member 
of the party is seriously injured by accident, bitten by a poisonous 
snake or insect, or stricken with a sudden and violent attack of 
disease, and competent medical assistance is absolutely unob- 
tainable for several days or even weeks, the head of the party 
must choose between seeing the victim die or boldly performing 
some simple surgical operation or giving medical treatment 
which he would not dream of doing otherwise. It is the lesser of 
two evils and the engineer must not shirk his duty. Even though 
a doctor is perhaps obtainable after many days delay by 
despatching a messenger 50 miles for him, common sense first- 
aid work and the intelligent use of a few simple methods and 
remedies may save life or prevent or mitigate permanent dis- 
ablement. The outfit should include a sufficient supply of the 
medicines and medical appliances which would most probably 
be required. All bottles should be carried in cases to prevent 
breakage and the corks or stoppers secured tightly. When prac- 
ticable, the drugs should be in tablet form, rather than liquid, 
and a normal dose should be marked on each bottle or package. 
They should be doubly labeled and the labels varnished to 
prevent their coming off in a damp climate. All adhesive 
plasters, antiseptic gauze, and such appliances, should be kept 
carefully wrapped up and protected from air and moisture. 

37. Appliances. The very simplest set of instruments should 
include a pair of good scissors, which can be made antiseptically 
clean by wiping off with alcohol; a knife with two razor-sharp 
blades; a probe; a small saw with detachable handle; a pair 
of mouse-tooth forceps; silk for ligatures. No. 2 catgut, needles 
and safety pins. There should be several rolls of sterilized gauze 
and " Z. O." adhesive plaster. A two-quart fountain syringe 
should be provided, also a hj^odermic syringe and two needles. 
The engineer should thoroughly familiarize himself with the 
working and manner of use of this last instrument. Any engi- 
neer who is preparing to head an expedition into a region where 
medical attention is unobtainable should consider that he can 
very wisely spend time with some doctor friend in learning the 
elements of the use of all these appliances. 

38. Antiseptics. The engineer should warn his men of the 
danger from the infection of even slight wounds and scratches, 
especially in hot climates. The best emergency treatment for 



§ 39. RAILROAD SURVEYS. 49 

any scratch, nail gouge, or nail in the foot, is to apply pure tinc- 
ture of iodine at the base of the wound by cotton on the end of 
a small stick or probe. A few of the many effective antiseptics 
are here mentioned: Boric ointment; one part of powdered boric 
acid added to nine parts of vaseline. Carbolic ointment; one 
part of carbolic acid to nineteen parts of vaseline. Chinosol; 
a xwo solution may be used for washing fresh wounds, burns, 
etc., or as a gargle for sore throat. Iodoform powder promotes 
rapid healing of sores and wounds; one part m eight parts of 
vaselme is a good heahng ointment. Permanganate of potash; 
one grain gives a purple color to a gallon of water; if the water 
is impure, the purple color changes rapidly to brown and this is 
a rough test of organic impurity; the crystals are soluble in 20 
parts of water; it is especially useful in the treatment of snake 
bites. In a snake-infested country, it is wise for each man to 
carry permanganate of potash crystals with him, for use in 
emergency. See " Snake bites.'' 

39. Drinking water. Every chief of party should see to it 
that his party has a pure supply of drinking water and especially 
that this supply is not contaminated by excrement from the 
camp draining into it. If there is any doubt about the purity 
of the supply (especially if so indicated by the permanganate- 
of -potash test) it should be part of the duty of the camp cook to 
maintain a liberal supply of boiled and cooled water. A neglect 
of such a precaution might easily result in an epidemic of typhoid. 
In a region where all streams are contaminated, perhaps by 
decaying vegetation or other natural cause, it may be wise to 
provide canteens, which the cook should furnish each morning 
filled with sterilized water. 

40. Bleeding from an artery or vein can sometimes be stopped 
by pressing the vessel with sujQScient pressure to stop the flow 
and continuing the pressure until the blood coagulates. If the 
vein or artery is actually severed but is not too large, the bleed- 
ing may be stopped by the use of a pair of forceps; grasp and 
pinch the vessel and twist it around three or four times. In 
about ten minutes the forceps may be removed. If the vein or 
artery is larger, and especially when it is an artery, which may 
be recognized by spurts of bright red blood, it may be necessary 
to tie the vessel. This may be done with catgut ligature, which 
should previously be boiled to prevent infection. While pre- 
paring for this, bleeding should be stopped by temporary pres- 



50 RAILROAD CONSTRUCTION. § 41. 

sure. This is most easily done when the bleeding vessel may be 
pressed against a bone. A tourniquet can be improvised for 
pressing a pad (or even a stone) against the vein or artery of a 
limb by using a stick and a piece of cloth, or, perhaps, a rope and 
a small block of wood. Fasten the cloth or rope into a loose loop 
around the limb and, running the stick through the loop; then 
twist it so that the pad is pressed down as desired. The rope 
can be so disposed as to press the block, which in turn presses 
the pad against the vein or artery. 

41. Ailments and diseases; medicines. Colic or cramp. 
Essence of ginger, 5 to 20 drops, in a small amount of very hot 
water. 

Diarrhoea. Remove the bowel irritant by a castor-oil purge; 
then, if diarrhoea continues, give 20 drops of chlorodyne and 10 
drops of tincture of ginger, in two tablespoonsf ul of hot water, 
two or three times per day. 

Purgatives. Epsom salts; dose, two teaspoonsful in a small 
glass of water. Calomel; dose, two to five grains; should be 
followed by citrate of magnesia. Cascara sagrada; dose, two to 
six grains. Castor oil; dose, one to three tablespoonsful, 
which may be made more palatable by mixing with an equal 
amount of glycerine, and then putting the mixture into a glass 
of lemonade. Any tendency to constipation, which leads to 
intestinal poisoning and appendicitis may be avoided by usinfe 
a laxative, made as effective as necessary, about once a week. 

Emetics. Common salt (two tablespoonsful), or mustard (one 
tablespoonful) or Ipecacuanha (30 grains) or Zinc sulphate (30 
grains), dissolved in a glass of water. Tickling the throat with 
a feather may sometimes be effective. Strong ^' Ivory '' soap 
suds is excellent. 

Malaria. Five grains of quinine as a preventive; ten grains, 
three times a day, as an ordinary maximum dose. Larger doses 
are often given but it is dangerous unless under the care of a 
physician. 

Cold-in-head. Rhinitis tablets, given as directed on bottle, 
are effective to break up an incipient cold. " Dover's powder "; 
dose, five to ten grains. Keep patient warm, with hot- water 
bottles and hot drinks. 

42. Drowning; electric shock, asphyxiation. The trouble and 
the remedy is essentially the same in all three cases; respiration 
has been temporarily suspended and must be promptly restored 



§43. RAILROAD SURVEYS. 51 

by artificial means. Loosen the patient's clothing, especially 
about the neck. In a drowning case, lay the patient on the 
ground, face down, straddle him and raise him at the hips so 
that the water in the air passages will drain out. Remove from 
the mouth any tobacco, false teeth or anything else that might 
obstruct breathing. Draw the tongue forward with forceps or a 
handkerchief. Then lay him face down, but with the face 
turned to one side so as to facilitate breathing, and with the arms 
extended forward. Then the operator, kneehng astride the 
patient, facing his head, and with the hands pressing on the lower 
ribs, gradually presses down so as to expel the air from the 
lungs. Then he suddenly removes the pressure by swinging 
back, and thus allows air to enter the lungs. Repeat the move- 
ments every four or five seconds, until natural breathing com-* 
mences. Considering the fact that this method has successfully 
restored breathing after some hours of imsuccessful effort, 
and also that, in those cases, the patient would have died except 
for the persistency of the effort, the operator must not be dis- 
couraged because his efforts are not immediately successful. 
Promptness in beginning such treatment is so important that 
it is better to commence at once (even outdoors) rather than 
allow any material delay in order to get the patient to a house. 
The patient should be allowed plenty of air; crowding around 
him should be avoided. A blanket, extra clothing, hot bricks 
or stones, or hot-water bags, to restore heat to the body, will be 
of assistance, provided they do not interfere with the respiration 
operations. Do not attempt to make the patient swallow any- 
thing (e. g., a stimulant), until he is fully conscious; otherwise 
he will choke. 

43. Fractures. Obtain medical aid if possible, but if this is 
unobtainable, except after a delay of many days or weeks, and 
it is uncertain even then, it may be preferable to take the chances 
of common-sense treatment, even if unskilled, rather than the 
certain permanent injury due to neglect of all treatment. Frac- 
tures are (a) simple, when the skin is not broken; (b) compound, 
when the skin is so broken that the fractured bone is more or less 
exposed to the air; and (c) comminuted, when there are two or 
more breaks of the same bone; a comminuted fracture may be 
simple or compound. Great care should be used in handling 
the patient immediately after the accident so that a simple frac- 
ture does not become compound. A broken limb should be 



52 RAILROAD CONSTRUCTION. § 44. 

carefully straightened out and bound temporarily with the best 
improvised splints which are available until the patient can be 
removed to a bed. Even if amateur bone setting is decided 
to be advisable, setting should not be attempted if there is exces- 
sive swelling or tenderness. Apply ice or evaporating lotions 
to reduce any swelling. Splints should be made which are of 
proper length and are so rounded and padded with cloth that 
they cannot produce any concentrated pressure. Usually the 
dislocated bones are forced past each other, especially if the frac- 
ture is obhque rather than perpendicular, and it is always neces- 
sary to use considerable force, especially if it is a broken leg, to 
pull the bones back into position. The amateur must use his 
best common sense and knowledge of skeleton anatomy to restore 
the fragments to the same relative position they had previously, 
and then to secure them rigidly stiff with spHnts. Comparison 
with an unbroken arm or leg will be made even by a skilled 
surgeon, and such a comparison should be carefully studied by 
the amateur. While the binding should be as firm as it is safe 
to make it, it may be so tight as to produce swelling and even 
ulceration, and then the binding must be loosened. Compound 
fractures require the care of the flesh and skin wound in addition 
to the bone setting. The wound should be treated as described 
for wounds, but the spKnts and binding should be designed so 
that the wound can be properly dressed without loosening the 
splints. If the broken bone protrudes through the wound, it 
must be drawn back so that the wound can heal externally, 
even though the bone setting is beyond the skill of the amateur 
surgeon. Setting usually requires about six weeks, but, in the 
case of a hmb, the joints above and below the break should be 
very carefully moved after about three weeks, so as to avoid 
stiff joints, special care being take that there is no strain on the 
healing bone. 

44. Snake or insect bites. The majority of snake bites occur 
on the limbs. In such a case (1) tie a cord or bandage about the 
limb just above the wound as promptly as possible, so as to 
prevent the poisoned blood from getting into the system; (2) 
cut into the wound so as to induce free bleeding; (3) suck the 
wound to aid in drawing out the poisoned blood; there is little 
or no danger in this, provided the mouth is free from sores, and 
provided the mouth is immediately rinsed out, preferably with 
an antiseptic solution, such as a light purple solution of per- 



§ 45. RAILROAD SURVEYS. 53 

manganate of potash; (4) inject into the wound a strong solu- 
tion of permanganate of potash, which may be done hypoder- 
mically or, perhaps, even by rubbing into the wound crystals of 
the drug. When the case is very serious, on account of the 
known deadly character of the poison, and when no permanganate 
of potash is obtainable, heroic measures are sometimes necessary. 
Pure carbolic acid, or caustic, may be used, if available. Cauter- 
izing the wound with white-hot iron, exploding a pinch of gun- 
powder over the wound, shooting away the infected part with a 
gun, or even summary amputation with a hatchet, may some- 
times be considered the lesser of two evils. If the hmb has been 
tightly tied, it will, of course, produce great pain, discoloration 
and swelling, which must not be continued too long. A second 
ligature should be tied a few inches above the first. When the 
limb becomes very swelled and painful, loosen the first ligature 
for about ten seconds and again tighten, and then loosen the 
second ligature for ten seconds and again tighten. After fif- 
teen minutes, repeat the loosening and tightening. After about 
eight repetitions, the ligatures may be removed altogether. If 
the poison is partly sucked out, the remainder partly neutralized 
with chemicals, and does not get fully into the system for two 
hours, the danger is greatly diminished. Of course bites on the 
face or body cannot be tied up and can only be treated by suck- 
ing out the poison and by chemicals. Stimulation of the heart 
is usually essential, which may be done with one teaspoonful of 
aromatic spirits of ammonia in two tablespoonsful of water, or 
with alcoholic liquor, preferably whiskey. One l-30th grain 
strychnine tablets, dissolved in two tablespoonsful of water, is 
also a stimulant. If a hypodermic is available, one tablet may 
be dissolved in thirty drops of sterile water and inserted in the 
back or arm, well under the skin. 

45. Wounds. First, last and all the time, prevent infection. 
The marvelous success of modern surgery is due largely to anti- 
septic methods. Neglect of cleanliness almost inevitably 
induces blood poisoning. A perfectly clean cut, after being 
washed and steriHzed with iodine, may be closed with adhesive 
plaster, taking stitches, if necessary, with steriHzed catgut or silk 
or linen thread. The stitches may be removed in a week. 
But when the flesh is torn and, especially, when dirt and other 
matter, which is possibly poisonous or infectious, has been forced 
into the wound, there is great danger of blood poisoning, and 



54 RAILROAD CONSTRUCTION. § 45. 

the wound must be cleansed. First, cover the wound itself 
with a pad which has been soaked in an antiseptic solution and 
then wash the skin (shaving off all hair), all around the wound, 
using first soap and then an antiseptic solution. Then cleanse 
out all foreign matter from the wound, using antiseptics and 
pack the wound with strip gauze, soaked in the antiseptic, so as 
to extend from the deepest part of it to the outside. This will 
drain the discharges. The dressing should be renewed every 
day until the wound shows a tendency to heal. A gaping torn 
wound should not be sewed up, except to bring the edges together 
temporarily. 



CHAPTER II. 



ALINEMENT 



In this chapter the ahnement of the center line only of a pair 
of rails is considered. When a railroad is crossing a summit in 
the grade line, although the horizontal projection of the aline- 
ment may be straight, the vertical projection will consist of 
two sloping lines joined by a curve. When a curve is on a: 
grade, the center line is realh^ a spiral, a curve of double curva- 
ture, although its horizontal projection is a circle. The center 
line therefore consists of straight lines and curves of single 
and double curvature. The simplest method of treating them 
is to consider their horizontal and vertical projections separately. 
In treating simple, compound, and transition curves, only the 
horizontal projections of those curves will be considered. 



SIMPLE CURVES. 

46. Designation of curves. A curve may be designated either 
by its radius or by the angle subtended by a chord of unit 
length. Such an angle is known 
as the '' degree of curve " and is 
indicated by D. Since the curves 
that are practically used have very 
long radii, it is generally impracti- 
cable to make any use of the actual 
center, and the curve is located 
without reference to it. If AB in 
Fig. 11 represents a unit chord (C) 
of a curve of radius Rj then by the 
above definition the angle AOB 
equals D. Then 




AO sin iD = iAB = iC. 



R = 



W 
sin JD' 



(1) 



55 



56 RAILROAD CONSTRUCTION. §47. 

or, by inversion, 

''""i^-M (2) 

The unit chord is variously taken throughout the world as 
100 feet, 66 feet, and 20 meters. In the United States 100 
feet is invariably used as the unit chord length, and throughout 
this work it will be so considered. Table I has been computed 
on this basis. It gives the radius, with its logarithm, of all 
curves from a 0° 01' curve up to a 10° curve, varying by single 
minutes. The sharper curves, which are seldom used, are given 
with larger intervals. 

An approximate value of R may be readily found from the 
following simple rule, which should be memorized: 

r> 5730 

Although such values are not mathematically correct, since R 
does not strictly vary inversely as D, yet the resulting value is 
within a tenth of one per cent for all commonly used values 
of R, and is sufficiently close for many purposes, as will be 
shown later. 

47. Metric Curves. The unit chord for railroad curves on 
the metric system is 20 meters. If a curve has a 100-foot chord 
arid a central angle of 5°, the radius would, of course, be 1146.3 
feet. Since 20 meters =65.6174 feet, a 20-meter chord between 
those same radial lines would subtend an arc with a radius of 
.656174X1146.3 feet, or 752.16 feet. But this radius, measured 
in meters, would be (.656174 X 1146.3) -^ 3.28087 = 229.26 meters, 
which is 1146.3 X. 20. In other words, the radius of any metric 
curve, measured in meters, is numerically one-fifth of the radius, 
measured in feet, of the same degree curve, but in actual length 
is a little less than two-thirds. This practically -means that a 
10° curve, metric, is actually very much sharper than a 10° 
curve, using foot-measure, or that the radius is about 66% as 
much. Therefore, in selecting curves for location, an engineer, 
who is accustomed to the foot-measure system, should remember 
that a 10° curve metric, for example, has approximately the 
same radius as a 15° curve, using foot-measure. While it is 
more convenient for an engineer, who is constantly using the 
metric system for curves, to have tables computed directly on 



§48. 



ALINEMENT, 



57 




this basis, an engineer need not be dependent on such tables, 
since it is only necessary to divide the tabular quantities in the 
foot-table by 5 to obtain the correspond- 
ing quantities for the metric system. This 
appHes not only to radii, but also to 
tangents, external distances and long 
chords for a 1 ° curve. A desired logarithm 
may be obtained by subtracting 0.6989700 
from the foot-table logarithm. 

For example, anticipating the explana- 
tion in Art. 53, what is the tangent 
distance of a 6° metric curve, when the 
central]angle is 32° 40'. From Table II, we 
find that by the foot-system the tangent 
distance for a 1° curve when the central 
angle is 32° 40' is 1679.1 feet; then for 
a 6° curve it is 1679.1^6=279.85 feet; 
for a 6° metric curve it is 279.85-^-5= ^^q ^2. 

55.97 meters. The radius of the 6° metric 

curve = 955.37 -r- 5 = 191.074 meters, which is in actual length 
about 66% of 955.37 feet. 

As another illustration of the transformation from the foot- 
system to the metric system, or vice versa, the degree of a curve, 
by the foot system, may be multiplied by .66 and obtain approx- 
imately the degree of the equivalent curve by the metric system. 
For example, a 6° curve, foot system, has about the same actual 
radius as a 6 X. 66 = 3.96° metric curve, or about a 4° curve. 

48. Length of a subchord. Since it is impracticable to 
measure along a cm-ved arc, curves are always measured by 
laying off 100-foot chord lengths. This means that the actual 
arc is always a Httle longer than the chord. It also means that a 
suhchord (a chord shorter than the imit length), will be a little 
longer than the ratio of the angles subtended would call for. 
The truth of this may be seen without calculation by noting that 
two equal subchords, each subtending the angle JD, will evi- 
dently be shghtly longer than 50 feet each. If c be the length of 
a subchord subtending the angle dj then, as in Eq, 2, 



sinfd = — , 



or, by inversion, 



c= 2/2 sin hd. 



(3) 




58 RAILROAD CONSTRUCTION. § 49. 

d 
The nominal length of a subchord = 100 — . For example, 

a nominal subchord of 40 feet will subtend an angle of -^q of 
D^: its true length will be slightly more than 40 feet, and may 

be computed by Eq. 3. The difference 
between the nominal and true lengths 
is maximum when the subchord is 
about 57 feet long, but with the low 
degrees of curvature ordinarily used 
the difference may be neglected. With 
a 10° curve and a nominal chord 
length of 60 feet, the true length is 
60.049 feet. Very sharp curves should 
be laid off with 50-foot or even 25-foot 
chords (nominal length). In such 
cases especially the true lengths of 
these subchords should be computed 
and used instead of the nominal lengths. 

For example, assume that a 12° curve begins at Sta. 26+30. 
The first subchord will be noriiinally 70 feet and actually 
70.066 feet. Assume that the central angle between the 
tangents is 39° 36'. Then the nominal length of curve is 
39.6° 4- 12° =3.30 stations. 3.30 - .70 = 2.60, the nominal length 
of curve beyond the first station point on the curve. The final 
subchord is nominally 60 feet, but its actual length is 60.070 
feet. 

The values of these subchords for even degrees between 
5° and 30°, and for nominal chord lengths of 10, 20, 30, 
40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90 and 95 feet, are given 
in Table Ila. The excess values increase approximately as 
the square of the degree of curvature, but for intervals of 
1° simple interpolation will be sufficiently accurate for inter- 
mediate values. 

49. Length of a curve. The actual mean length of the two 
rails will be more than the nomihal length of the curve, as defined 
above, and even more than the sum of the full 100-foot lengths 
and the true lengths of the subchord lengths at the ends. In 
the above numerical case the mean rail length is 

39.6° X-^ XR = 39.6° X 7— X478.34 = 330.604. 
180 180 



ttlx 



§ 50. ALINEMENT. 59 

The sum of the two full-chord lengths and the two subchords is 
70.066+200+60.070 = 330.136. A large part of the excess 
(330.604-330.136 = .468) is the excess length (.183) of each 
arc of a 12° curve over the 100-foot chord. The remainder is 
the excess of the 70-foot and 60-foot arcs over the true chord 
lengths. But this excess length is of little practical importance. 
In the above case (a 12° curve) it adds about 0.2% to the length 
of rail that must be bought. The excess varies approximately 
as the square of the degree of curvature. The percentage of 
excess for the entire length of a road is utterly insignificant and 
is swallowed up by the 2% excess which is usually allowed for 
wastage in rail cutting. 

50. Curve notation. The notation adopted by the Amer. 
Rwy. Eng. Assoc, indicates any point where there is a change of 
alinement by two letters, the first of which denotes the ahnement 
on the side toward station zero and the second that away from 
station zero. Thus, the beginning of a curve, or the change from 
a tangent to a simple curve, is noted as TC] the other end of the 
curve, or the change from a simple curve to a tangent is noted as 
CT. But, since the use of two letters to indicate a point, or the 
use of four letters to indicate a fine joining the two points, is 
cumbersome in the algebraic solutions and demonstrations which 
follow (demonstrations which the A. R. E. A. do not give), the 
author has decided to retain the old notation, rather than to try 
to conform to the A. R. E. A. notation. The A. R. E. A. sys- 
tem also indicates the central angle of a curve, or the angle 
between the two tangents, by /. In the first edition of this 
work, the author, following Searles, indicated the central angle 
by A. To make even this change, for the sake of conformity, 
would require a change in all the mathematical work and 
figures involving curves throughout the book. In Fig. 14 
both notations are given, the A. R.E. A. notations being 
given in parentheses. Both notations are also shown in 
Fig. 36, which illustrates a transition curve or spiral. It 
should be noted that some of the notations coincide for some 
of the elements. 

51. Elements of a curve. Considering the line as running 
from A toward By the beginning of the curve, at A, is called 
the point of curve (PC). The other end of the curve j at By is 
called the point of tangency (PT). The intersection of the 
tangents is called the vertex (V), The angle made by the 



60 



HAILROAD CONSTRUCTION. 



§52. 



tangents at F, which equals the angle made by the radii to 
the extremities of the curve, is called the central angle (A). AV 
and BV, the two equal tangents from the vertex to the PC 
and PT, are called the tangent distances (T). The chord 
AB is called the long chord {LC) . The intercept HG from 
the middle of the long chord to the middle of the arc is called 
the middle ordinate (M). That part of the secant GV from 




Fig. 14. 

the middle of the arc to the vertex is called the external distance 
(E). From the figure it is very easy to derive ^the following 
frequently used relations: 

T=R ta^niA (4) 

LC = 2EsiniA (5) 

M^RversiA (6) 

E = R exsec |A. ....... (7) 

52. Relation between T, E, and A. Join A and G in Fig. 14. 

The angle VAG = iAj since it is measured by one half of the 
arc AG between the secant and tangent. 

AGO = 90''-iA. 




L 



§ 53. ALINEMENT. 61 

AV :VG::sm AGV : sin VAG; 
sin AGV=sm AGO =cos |A; 

T : E :: cos JA : sin lA; 

r=Ecot iA : . . (8) 

The same relation may be obtained by dividing Eq. 4 by Eq. 
7, since tan a-^exsec a = cot Ja. 

53. Elements of a 1° curve. From Eqs. 1 to Sit is seen that 
the elements of a curve vary directly sls R. It is also seen to 
be- very nearly true that R varies inversely as D. If the ele- 
ments of a 1 ° curve for various central angles are calculated and 
tabulated, the elements of a curve of D° curvature may be 
approximately found by dividing by D the corresponding ele- 
ments of a 1° curve having the same central angle. For small 
central angles and low degrees of curvature the errors involved 
by the approximation are insignificant, and even for larger 
angles the errors are so small that for many purposes they may be 
disregarded 

In Table II is given the value of the tangent distances, 
external distances, and long chords for a 1° curve for various 
central angles The student should familiarize himself with the 
degree of approximation involved by solving a large number of 
cases under various conditions by the exact and by the approxi- 
mate methods, in order that he may know when the approxi- 
mate method is sufficiently exact for the intended purpose. 
The approximate method also gives a ready check on the 
exact method. 

A closer value may be obtained by using the " Corrective Table '' 
found at the end of the main table. The correction is always 
additive and is usually very small and often even too insignificant 
for attention. A glance at the corrective table will show whether 
a correction need be made and an easily computed interpolation 
will show its amount. For example, what is the tangent dis- 
tance for a 6° curve having a central angle of 42° 15'? Inter- 
polating between 2209.0 and 2218.6, we have 2213.8 as the 
tangent distance for a 1° curve. Dividing by 6, we have 368.97 
as the approximate tangent distance. Interpolating in the cor- 
rective table, we have ,14 as the correction for a 5° curve and a 



62 



RAILROAD CONSTRUCTION. 



§54. 



central angle of 42° 15', and .28 as the correction for a 10° curve. 
Interpolating for 6° between these values of .14 and .28, we have 
.17, which added to 368.97 equals 369.14. The precise value, 
computed from Eq. 4, is 369.12. If the approximate value, even 
after correction, is not considered sufficiently accurate, Eq. 4 
should be used. The student should appreciate that the dis- 
crepancy of even .02 in the above calculation is not due to any 
real error in the main table or the corrective table, but is due to 
the fact that the tangent distances are only computed to the 
nearest tenth of a foot for values over 1000 feet, and this will 
produce such discrepancies. The table should not be used 
where precise values are required. ■ 

54. Exercises, (a) What is the tangent distance of a 4° 20' 
curve having a central angle of 18° 24'? 

(h) Given a 3° 30' curve and a central angle of 16° 20', how 
far will the curve pass from the vertex? [Use Eq, 7.] 

(c) An 18° curve is to be laid off using 25-foot (nominal) 
chord lengths. What is the true length of the subchords? 

(d) Given two tangents making a central angle of 15° 24'. 
It is desired to connect these tangents by a curve which shall 
pass 16.2 feet from their intersection. How far down the 
tangent will the curve begin and what will be its radius? (Use 
Eq. 8 and then use Eq. 4 inverted.) 

55. Curve location by deflections. The angle between a 
secant and a tangent (or between two secants intersecting on an 
arc) is measured by one half of the intercepted arc. Beginning 
at the PC (A in Fig. 15), if the 
first chord is to be a full chord 
we may deflect an angle VAa 
(=^D), and the point a, which is 
100 feet from A, is a point on the 
curve. For the next station, b, 
deflect an additional angle bAa 
(=JD) and, with one end of the 
tape at a, swing the other end 
until the 100-foot point is on the 
line Ab. The point b is then on a 
the curve. If the final chord cB \ 
is a subchord, its additional deflec- 
tion (id) is something less than JD. The last deflection (BAV) is 




§ 56. ALINEMENT. 63 

of course ^J. It is particularly important, when a curve begins 
or ends with a subchord and the deflections are odd quantities, 
that the last additional deflection should be carefully com- 
puted and added to the previous deflection, to check the mathe- 
matical work by the agreement of this last computed deflec- 
tion with JJ. 

Example. Given a 3° 24' curve having a central angle of 
18° 22' and beginning at sta. 47 + 32, to compute the deflec- 
tions. The nominal length of curve is 18° 22' ^3° 24' = 18.367 4- 
3.40=5.402 stations or 540.2 feet. The curve therefore ends 
at sta. 52 + 72.2. The deflection for sta. 48 is iVoXK3°24') 
= 0.68Xl°.7 = l°.156 = l°09' nearly. For each additional 100 
feet it is 1° 42' additional. The final additional deflection for 
the final subchord of 72.2 feet is 

72 2 

-^XK3° 24') =1°.2274 = 1° 14' nearly. 

The deflections are 

P. C . . . Sta. 47 + 32 0° 

48 0° +1°09' = 1°09' 

49 1° 09' + 1° 42' = 2° 51' 

50 2° 51' + 1° 42'=4° 33' 

51 4° 33' + 1° 42' = 6° 15' 

52 6° 15' + 1° 42' = 7° 57' 

P. T 52 + 72.2 7°57' + l°14' = 9°ll' 

As a check 9° ll' = K18° 22') =^//. (See the Form of Notes 
in §21.) 

56. Instrumental work. It is generally impracticable to 
locate more than 500 to 600 feet of a curve from one station. 
Obstructions will sometimes require that the transit be moved up 
every 200 or 300 feet. There are two methods of setting off 
the angles when the transit has been moved up from the PC. 

(a) The transit may be sighted at the previous transit station 
with a reading on the plates equal to the deflection angle from 
that station to the station occupied, but with the angle set off on 
the other side of 0°, so that when the telescope is turned to 0° it 
will sight along the tangent at the station occupied. Plunging 
the telescope, the forward stations may be set off by deflecting 
the proper deflections from the tangent at the station occupied 



64 



RAILROAD CONSTRUCTION. 



§56. 



This is a very common method and, when the degree of curva- 
ture is an even number of degrees and when the transit is only 
set at even stations, there is but little objection to it. But the 
degree of curvature is sometimes an odd quantity, and the exi- 
gencies of difficult location frequently require that substations 
be occupied as transit stations. Method (a) will then require 
the recalculation of all deflections for each new station occupied. 
The mathematical work is largely increased and the probability 
of error is very greatly increased and not so easily detected. 
Method (6) is just as simple as method (a) even for the most 
simple cases, and for the more difficult cases just referred to the 
superiority is very great. 

(b) Calculate the deflection for each station and substation 
throughout the curve as though the whole curve were to be lo- 
cated from the PC. The computations 
may thus be completed and checked (as 
above) before beginning the instrumental 
work. If it unexpectedly becomes neces- 
sary to introduce a substation at any 
point, its deflection from the PC may be 
readily interpolated. The stations actually 
set from the PC are located as usual. 
Rule. When the transit is set on any 
forward station, backsight to any previous 
station with the plates set at the deflection 
angle for the station sighted at. Plunge 
the telescope and sight at any forward 
station with the deflection angle originally 
computed for that station. When the 
plates read the deflection angle for the 
station occupied, the telescope is sighting 
along the tangent at that station — which 
is the method of getting the forward tan- 
gent when occupying the PT. Even though 
the station occupied is an unexpected sub- 
station, when the instrument is properly 
oriented at that station, the angle reading 
for any station, forward or back, is that originally computed 
for it from the PC. In difficult work, where there are ob- 
structions, a valuable check on the accuracy may be found by 
sighting backward at any visible station and noting whether 




Fig. 16. 



§56. 



ALINEMENT. 



65 



its deflection agrees with that originally computed. As a 
numerical illustration, assume a 4° curve, with 28° curvature, 
with stations 0, 2, 4, and 7 occupied. After setting stations 
1 and 2, set up the transit at sta. 2 and backsight to sta. 
with the deflection for sta. 0, which is 0°. The reading on sta. 
1 is 2°; when the reading is 4° the telescope is tangent to 
the curve, and when sighting at 3 and 4 the deflections will be 
6® and 8°. Occupy 4 ; sight to 2 with a reading of 4°. When 
the reading is 8° the telescope is tangent to the curve and, by 
plunging the telescope, 5, 6, and 7 may be located with the 
originally computed deflections of 10°, 12°, and 14°. When oc- 
cupying 7 a backsight may be taken to any visible station with 
the plates reading the deflection for that station; then when 




a 




Fig. 17. 



Fig. 18. 



the plates read 14° the telescope will point along the forward 
tangent. 

The location of curves by deflection angles is the normal 
method. A few other methods, to be described, should be con- 
sidered as exceptional. 



66 



RAILROAD CONSTRUCTION. 



§57. 



57. Curve location by two transits. A curve might be located 
more or less on a swamp where accurate chaining would be 
exceedingly difficult if not impossible. The long chord AB 
(Fig. 17) may be determined by triangulation or otherwise, 
and the elements of the curve computed, including (possibly) 
sub chords at each end. The deflection from A and B to each 
point may be computed. A rodman may then be sent (by 
whatever means) to locate long stakes at points determined 
by the simultaneous sightings of the two transits. 

58. Curve location by tangential offsets. When a curve is 
very flat and no transit is at hand the following method may be 
used (see Fig. 18) ; Produce the back tangent as far forward as 
necessary. Compute the ordinates Oa\ Ob'y Oc\ etc., and the 
abscissae a^a, b'h, c'c, etc. If Oa is a full station (100 feet), then 



Oa'=Oa' =100 cos JD, also = i^ sin D; 

06'=Oa'4-a'6' =100 cos iZ) + 100 cos fD, 

also = R sin 2D ; 
Oc' = Oa' + a'y + 6 V = 100(cos JD + cos |Z) 4- cos |D) , 

also = R sin 3D ; 



etc. 



a' a = 100 sin JD, also =R vers D ; 

Vh=a'a-Wh =100 sin JD + 100 sin |Z), 

also =i^ vers 2Z); 
c'c = Vh + c''c =100(siniD + sin|i) + sinfD), 

also =i? vers 3.0: 



(9) 



V (10) 



etc. 

The functions JZ>, |D, etc., may be more conveniently used 
without logarithms, by adding the several natural trigonometrical 
functions and pointing off two decimal places. It may also be 
noted that Oh' (for example) is one half of the long chord for 
four stations; also that Vh is the middle ordinate for four 
stations. If the engineer is provided with tables giving the long 
chords and middle ordinates for various degrees of curvature, 
these quantities may be taken (perhaps by interpolation) from 
such tables. 

If the curve begins or ends at a substation, the angles and 
terms will be correspondingly altered. The modifications may 



§ 59. ALINEMENT. 67 

bei readily deduced on the same principles as above, and should 
be worked out as an exercise by the^ student. 

In Table II are given the long chords for a 1° curve for various 
values of J. Dividing the value as given b}^ the degree of the 
curve, we have an approximate value which is amply close for 
low degrees of curvature, especially for laying out curves with- 
out a transit. For example, given a 4° 30' curve, required the 
ordinate Oc\ This is evidently one half of a chord of six stations, 
with i=27°. Dividing 2675.1 (which is the long chord of a 
1° curve with J =27°) by 4.5 we have 594.47; one half of this is 
the required ordinate, Oc' =297.23. The exact value is 297.31, 
an excess of .08, or less than .03 of 1%, The true values 
are always slightly in excess of the value as computed from 
Table II. 

Exercise. A 3° 40' curve begins at sta. 18 + 70 and runs to 
sta. 23 + 60. Required the tangential offsets and their corre- 
sponding ordinates. The first ordinate = 30 cos Htujj^^^ 40') = 
30 X. 99995 =29.9985; the offset =30 sin 0° 33' =30 X. 0096 = 
0.288. For the second full station (sta. 20) the ordinate = 
i long chord for i =2(1° 06' 4-3° 40') with i)=3°40'. Divid- 
ing 476.12, from Table II, by 3f, we have 129.85. Otherwise, 
by Eq. 9, the ordinate =30 X cos 0° 33' + 100 cos (1° 06' + 1° 50') 
= 30.00 + 99.87 = 129.87. The offset for sta. 20=30 sin 0° 33' + 
100 sin (1° 06' + 1° 50') = 0.288 + 5.12 =5.41. Workout 
similarly the ordinates and offsets for sta. 21, 22, 23, and 
23 + 60. 

59. Curve location by middle ordinates. Take first the sim- 
pler case when the curve begins at an even station. If we con- 
sider (in Fig. 14) the curve produced back to z, the chord 2a = 
2 X 100 cos JD, A'a = 100 cos JZ), and A'A=am=zn = 100 sin ^D. 
Set off A A' perpendicular to the tangent and A 'a parallel to 
the tangent. AA'=aa'=b?>'=cc', etc. = 100 sin ^D. Set off 
aa^ perpendicular to a' A. Produce Aa^ until a'6=A'a, thus 
determining b. Succeeding points of the curve may thus be 
determined indefinitely. 

Suppose the curve begins with a subchord. As before 
ra = A m'=c' cos J(i', and rA = am' =c' sin J(i'. Also sz=An^ = 
c" cos id", and sA =2/i'=c" sin Jd", in which (d' -{-d'') =D, 
The points z and a being determined on the ground, aa' may 
be computed and set off as before and the curve continued in 



68 



RAILROAD CONSTRUCTION. 



§60. 



full stations. A sub chord at the end of the curve may be 
located by a similar process. 

60. Curve location by offsets from the long chord. (Fig 21.) 
Consider at once the general case in which the curve commences 
with a subchord (curvature, d')j continues with one or more full . 




Fig. 19. 



Fig. 20. 



Fig. 21. 



chords (curvature of each, D), and ends with a subchord with 
curvature d/'. The numerical work consists in computing first 
ABy then the various abscissae and prdinates. AB=2R sin J J. 



Ah' = Aa'-\-aV ^c' cos^{A-d') + 100 cosi{A-2d' -D)\ 

ilc'=Ao' + a'6' + 6V = c'cosK^-dO + 100cosi(J-2<f'-Z)) 

+ 100 cos ^{4 -2d" -D)l 
also 

-=AJ5-Bc' ^2Rsm^A-c!'cos^{J-d''). 

a'a^a'a =c' sin ^(J— <i'); 

6'6 = a'a + m& = c'sinK^~<i') + 100smK^-2c?'-i))' 

c'c=6'6-n& = c'sini(J-dO + 100smK^-2rf'-Z)) 

-100sini(J-2(i''-£)); 
also «=:c* sin \{A — d''\ 



(11) 



(12) 



The above formulae are considerably simplified when the 



§61. 



ALINEMENT. 



69 



curve begins and ends at even stations. When the curve is 
very long a regular law becomes very apparent in the formation 
of all terms between the first and last. There are too few terms 
in the above equations to show the law. 

6i. Use and value of the above methods. The chief value 
of the above methods lies in the possibility of doing the work 
without a transit. The same principles are sometimes em- 
ployed, even when a transit is used, when obstacles prevent the 
use of the normal method (see §62, c). If the terminal tan- 
gents have already been accurately determined, these methods 
are useful to locate points of the curve when rigid accuracy is 
not essential. Track foremen frequently use such methods to 
lay out unimportant sidings, especially when the engineer and 
his transit are not at hand. Location by tangential offsets (or 
by offsets from the long chord) is to be preferred when the 
curve is flat (i.e., has a small central angle J) and there is no 
obstruction along the tangent, or long chord. Location by 
middle ordinates may be employed regardless of the length of 
the curve, and in cases when both the tangents and the long 
chord are obstructed. The above methods are but samples 
of a large number of similar methods which have been devised. 
The choice of the particular method to be adopted must be 
determined by the local conditions. 

62. Obstacles to location. In this section will be given only 
a few of the principles involved in this 
class of problems, with illustrations. The 
engineer must decide, in each case, which 
is the best method to use. It is frequently 
advisable to devise a special solution for 
some particular case. 

a. When the vertex is inaccessible. As 
shown in § 56, it is not absolutely essential 
that the vertex of a curve should be 
located on the ground. But it is very evi- 
dent that the angle between the terminal 
tangents is determined with far less prob- 
able error if it is measured by a single 
measurement at the vertex rather than as 
the result of numerous angle measurements 
Fig. 22. along the curve, involving several posi- 

tions of the transit and comparatively short sights Some- 




70 RAILROAD CONSTRUCTION. § 62. 

times the location of the tangents is already determined on 
the ground (as by bn and am, Fig. 22), and it is -required to 
join the tangents by a curve of given radius. Method. Measure 
ah and the angles Vba and haV. J is the sum of these angles. 
The distances hV and aV are computable from the above data. 
Given J and R, the tangent distances are computable, and then 
Bb and aA are found by subtracting bV and aV from the tan- 
gent distances. The curve may then be run from A, and the 
v/ork may be checked by noting whether the curve as run ends 
at B — previously located from b. 

Example. Assume at =546 82; angle a = 15° 18'; angle 
h = 18° 22' ; D = 3° 40' ; required aA and bB. 
J = 15°18' + 18°22'=33°40' 

Eq. (4) R (3° 400 3.19392 

tan iJ =tan 16° 50' 9.4808 

r=472.85 2.6747a 

sin 18° 22' ah 2.73784 

^^ sin 33° 40' log sin 18° 22' 9.49844 

co-log sin 33° 40' 0.25621 

a7=310.81 2.49250 

AF=472.85 

0^=162.04 



sin 15° 18' ab 2.73784 

^ sin 33° 40' log sin 15° 18' 9.42139 

co-log sin 33° 40' 0.25621 

67=260.29 2.41545 

^7=472.85 

feJ5=212.56 



b. When the point of curve (or point of tangency) is inacces- 
sible. At some distance (As, Fig. 23) an unobstructed line pn 
may be run parallel with AV. nv==pnf==As=R vers a. 

'. vers a = As -7-/2. 

ns=/)s=i2 sin 0!. 



§63, 



ALINEMENT.. 



71 



At 2/, which is at a distance ps back from the computed posi- 
tion of A, make an offset sA 
to p. Run p?! parallel to the 
tangent. A tangent to the 
curve at n makes an angle of a 
with np. From n the curve is 
run in as usual 

If the point of tangency is 
obstructed, a similar process, 
somewhat reversed, may be 
used. ^ is that portion of J still 
to be laid off when m is reached. 
tm=tl=R sin ^. mz=tB=lx=R 
vers ^. 

c. When the central part of 

the curve is obstructed, a is the 

central angle between two points 

of the curve between which 

a may equal any angle, but it is prefer- 




Fig. 



a chord may be run 
able that a should be a multiple 
of D, the degree of curve, and that 
the points m and n should be on 
even stations, mn =2R sin ^a. A 
point s may be located by an offset 
ks from the chord mn by a similar 
method to that outlined in § 60. 

The device of introducing the 
dotted curve mn having the same 
radius of curvature as the other, 
although neither necessary nor 
advisable in the case shown in 
Fig. 24, is sometimes the best 
method of surveying around an 
obstacle. The offset from any point on the dotted curve to 
the corresponding point on the true curve is twice the " ordinate 
to the long cord/^ as computed in § 60. 

63, Modifications of location. The following methods may 
be used in allowing for the discrepancies between the " paper 
location '^ based on a more or less rough prehminary survey and 
the more accurate instrumental location, (See § 18.) They are 




72 



RAILROAD CONSTRUCTION. 



§63. 



also frequently used in locating new parallel tracks and modify- 
ing old tracks. 

a. To move the forward tangent parallel to itself a distance «, 
the point of curve (a.) remaining fixed. (Fig. 25.) 



FF'=-T 






sin/iFy'~sin J* ' ' * 
AV'=AV + VV\ 
The triangle BmB' is isosceles and Bm=B'm. 

B'r x' 



B'-R=0'0=mB = 



vers B'mB vers A' 



(13) 



/. R'=R + 



X' 



vers A 



(14) 



The solution is very similar in case the tangent is moved in- 
ward to V"B'\ Note that this method necessarily changes the 




Fig. 25. 




Fig. 26. 



radius. If the radius is not to be changed, the point of curve 
must be altered as follows : 

b. To move the forward tangent parallel to itself a distance x^ 
the radius being unchanged. (Fig. 26.) In this case the whole 



§64. 



ALINEMENT. 



73 



curve is moved bodily a distance 00' =AA' = VV' =BB', and 
moved parallel to the first tangent A V 

B'n X 



BB' 



AA^ 



(15) 



sin nBB' sin J 

c. To change the direction of the forward tangent at the point 
of tangency. (Fig. 27.) This problem involves a change (a) in 

the central angle and also requires a 
new radius. An error in the deter- 
mination of the central angle fur- 
nishes an occasion for its use. 

R, A, a J AV, and BV are known. 




Bs^R vers J. 



\ R'=R- 



Bs==R' vers Jr 
vers J 



(16) 



vers {A — a) 

Fig. 27. As^R sin J. A's=R' sin J' . 

.'. AA'^.A's-As^R' sin J' -R sin J. . . (17) 

The above solutions are given to illustrate a large class of 
problems which are constantly arising. All of the ordinary 
problems can be solved by the application of elementary geome- 
try and trigonometry. 

64. Limitations in location. It may be required to run a 
curve that shall join two given tangents and also pass through a 
given point The point (P, Fig. 
28) is assumed to be deter- 
mined by its distance (VP) 
from the vertex and by the 
angle AFP=/?. 

It is required to determine 
the radius (R) and the tangent- 
distance (AV), J is known. 

PF(? = K180°-J)-/? 
=90°-(ii+/?). 
PP'=2yPsinPFG 
=2yPcos(JJ+/?). 




... SP^VP$^, 
sm ^J 



Fig. 28. 



74 RAILROAD CONSTRUCTION. § 65. 

AS = VSP X SP' = VSP(SP + PP^ 

^JvP^[vP^ + 2VPcos(iJ+^8)l 
\ sm JJ L sm iJ ^" ' ^ J 



VP I sin^/^ 2sin^cos(4J+./?) 
\sinn^ sm§J ' 

sin JJ 

AV=AS+SV 

yp . 

= -^[sin(ii+/?)+\/sin2/? + 2sin/?smJJcos(ii+^)]. (18) 

2^=A7cot}J. 

In the special case in which P is on the median line OY , 
fi = 90''— i J, and (ii+/?)=90^ Eq. 18 then reduces to 

VP 

AF = -r^.(l +COS Ji) =VP cot iJ, 

sm JJ^ 

as might have been immediately derived from Eq. 8. 

In case the point P is given by the offset PK and by the 
distance VK^ the triangle PKV may be readily solved, giving the 
distance VP and the angle /?, and the remainder of the solution 
will be as above. 

65. Determination of the curvature of existing track, (a) Using 
a transit. Set up the transit at any point in the center of the 
track. Measure in eech direction 100 feet to points also in the 
center of the track. Sight on one point with the plates at 0°. 
Plunge the telescope and sight at the other point. The angle 
between the chords equals the degree of curvature. 

(b) Using a tape and string. Stretch a string (say 50 feet 
long) between two points on the inside of the head of the outer 
rail. Measure the ordinate (x) between the middle of the string 
and the head of the rail. Then 

^=^-|^( very nearly) (19) 

For, in Fig. 29, since the triangles AOE and ADC are similar, 



§66. 



ALINEMENT. 



75 




AO : AE :: AD : DC or R = ^AD''-^x. When, as is usual, 
the arc is very short compared with the 
radius, AD = iAB, very nearly. Making 
this substitution we have Eq. 19. With a 
chord of 50 feet and a 10° curve, the result- 
ing difference in x is .0025 of an inch — far 
within the possible accuracy of such a 
method. The above method gives the 
radius of the y inner head of the outer rail. 
It should be diminished by ^g for the radius 
of the center of the track. With easy curvature, however, this 
will not affect the result by more than one or two tenths of one 
per cent. 

The inversion of this formula gives the required middle or- 
dinate for a rail on a given curve. For example, the middle 
ordinate of a 30-foot rail, bent for a 6° curve, is 

a: = 900 -- (8 X 955) =.118 foot = 1.4 inches. 



Fig. 29. 



Another much used rule is to require the foreman to have a 
string, knotted at the center, of such length that the middle 
ordinate, measured in inches, equals the degree of curve. To 
find that length, substitute (in Eq. 19) 5730 --2) for R and 
D^12 for X. Solving for chord, we obtain chord = 6l.S feet. 
The rule is not theoretically exact, but, considering the uncertain 
stretching of the string, the error is insignificant. In fact, the 
distance usually given is 62 feet, which is close enough for all 
purposes for which such a method should be used. 

66. Problems. A systematic method of setting down the 
solution of a problem simplifies the w^ork. Logarithms should 
always be used, and all the work should be so set down that a 
revision of the work to find a supposed error may be readily 
done. The value of such systematic work will become more 
apparent as the problems become more complicated. The two 
solutions given below will illustrate such work. 

a. Given a 3° curve beginning at Sta. 27 + 60 and running 
to Sta. 32 + 45. Compute the ordinates and offsets used in 
locating the curve by tangential offsets. 

b. With the same data as above, compute the distances to 
locate the curve by offsets from the long chord. 

c. Assume that in Fig. 22 ah is measured as 217.6 feet, the 



76 RAILROAD CONSTRUCTION. § 66. 

angle abV = 17° 42', and the angle 6a7=21°14'. Join the 
tangents by a 4° 30' curve. Determine bB and aA, 

d. Assume that in a case similar to Fig. 23 it was noted 
that a distance (As) equal to 12 feet would clear the building. 
Assume that J =38° 20' and that D=4°40'. Required the 
value of a and the position of n. Solution: 

YQTsa=As-i-R ^s = 12 Iog = l. 07918 

R (for 4° 40' curve) log = 3.08923 

a = 8°01' log vers a = 7.9899i 

ns=R sin a log sin a = 9 . 14445 

log 7^= 3.08923 
ng = 171.27 log = 2 . 23369 

c. Assume that the forward tangent of a 3° 20' curve having 
a central angle of 16° 50' must be moved 3.62 feet inward, with- 
out altering the P.C. Required the change in radius. 

/. Given two tangents making an angle of 36° 18'. It is 
required to pass a curve through a point 93.2 feet from the 
vertex, the line from the vertex to the point making an angle 
of 42"^ 21' with the tangent. Required the radius and tangent 
distance. Solviion: Applying Eq. 18, we have 

2 log= 0.30103 

/3=42°21' logsin= 9.82844 

ii = 18°09' log sin = 9.49346 

( J J + ^) = 60° 30' log cos = » 9.69234 

.20667 9.31527 

log sin^ /?=9.65688. . . . .45382 

^|9. 81987 .66049 ' 

9.90993 81271 

^ nat. sin 60° 30'. . . . .8703 

1 .6830 log= 0.22610 

FP=93.2 log = 1.96941 

2.19551 
log sin } J = 9.49346 

Tang, dist. Ay = 503.36 log= 2.70205 

log cot j^ J = 10.48437 

E = 1536.1 log= 3.18642 

2) =3° 44' 



§67. 



ALINEMENT, 



77 



COMPOUND CURVES. 

67. Nature and use. Compound curves are formed by a 
succession of two or more simple curves of different curvature. 
The curves must have a common tangent at the point of com- 
pound curvature (P.C.C.), In mountainous regions there is 
frequently a necessity for compound curves having several 
changes of curvature. Such curves may be located separately 
as a succession of simple curves, but a combination of two 
simple curves has special properties w^hich are worth investigat- 
ing and utilizing. In the following demonstrations R2 always 
represents the longer radius and Ri the shorter, no matter which 
succeeds the other. T^ is the tangent adjacent to the curve of 
shorter radius (Ri), and is invariably the shorter tangent. J^ is 
the central angle of the curve of radius R^, but it may be greater 
or less than Jj 

68. Mutual relations of the parts of a compound curve having 
two branches. In Fig. 30, AC and CB are the two branches of 




1- 



Fig. 30. 



the compound curve having radii of Ri and R2 and central angles 
of J I and J 2' Produce the arc AC to n so that AO{n = J. The 
chord Cn produced must intersect B. The line ns, parallel to 
CO2, will intersect BO2 so that Bs=sn=020i=R2—Ri, Draw 
Am perpendicular to Oiti. It will be parallel to hk. 



78 RAILROAD CONSTRUCTION. § 68. 

Br = sn vers Bsn = (R2 —Ri) vers J 2 > 
inn=AOi veis AO{n —R^ vers J; 
^A;=^FsmAF/c- =TisinJ; 
ylit = Am =mn + n/i =mn + 5r. 
.-. Ti sin J =Ei vers A + (Eg -^1) vers d^. . . (20) 
Similarly it may be shown that 

T2 sin A =i?2 vers J - {R^ -R^) vers Jj. . . (21) 

The mutual relations of the elements of compound curves 
may be solved by these two equations. For example, assume 
the tangents as fixed {A therefore known) and that a curve of 
given radius R^ shall start from a given point at a distance T^ 
from the vertex, and that the curve shall continue through a 
given angle A^. Required the other parts of the curve. From 
Eq. 20 we have 

Tj sin A — R^ vers A 



R2 — Ri- 



vers A2 



... R^=R^ + Tl^I^A^fi^ (22) 

vers(J — Ji) ^ ^ 

T2 may then be obtained from Eq. 21. 

As another problem, given the location of the two tangents, 
with the two tangent distances (thereby locating the PC and 
PT), and the central angle of each curve; required the two 
radii. Solving Eq. 20 for R^, we have 



R,= 



Ti sin A—R2 vers A 2 
vers A — vers A 2 



Similarly from Eq. 21 we may derive * 

T2 sin J— 7^2 (vers i— vers Jj) 
vers Ji 

Equating these, reducing, and solving for R2, we have 

Tj sin J vers ij — 7^2 sin J (vers J — vers J2) 

^~~ vers A 2 vers Jj — (vers J— vers Ji)(vers J— vers ^2) * 

Although the various elements may be chosen as above with 

considerable freedom, there are limitations. For example, in 

Eq. 22, since R2 is always greater than 7?j, the term to be 

added to R^ must be essentially positive — i.e., T^ sin A must be 

vers A • 

greater than R^ vers A. This means that Ti>Ri— — j-, or that 



§69. 



ALINEMENT. 



79 



7^1 > /?i tan J J, or that T^ is greater than the corresponding 
tangent on a simple curve. Similarly it may be shown that T2 
is less than R2 tan JJ or less than the corresponding tangent 
on a simple curve. Nevertheless Tj is always greater than T^. 
In the limiting case when R2=Ri, T2 = T^, and ^2 = ^1- 

69. Modifications of location. Some of these modifications 
may be solved b}^ the methods used for simple curves. For 
example : 

a. It is desired to move the tangent VB, Fig. 20, parallel to 
itseK to VB'. Run a new curve from the P.C.C. which shall 
reach the new tangent at B^, where the chord of the old curve 





Fig. 31. Fig. 32. 

intersects the new tangent. The solution is almost identical 
with that in § 63, a. 

b. Assume that it is desired to change the forward tangent 
(as above) but to retain the same radius. In Fig. 32 

(i?2~-^i) cos J2 =^2^ J 

(Ri-Ri) cos J2' =0/n\ 

X = 02n — Og'n' = (R2 — Ri) (cos J 2 ~ cos J3O . 



cos i/ = cos J, 



R2—R1 



(24) 



The P.C.C. is moved backward along the sharper curve an 
angular distance of A2 —^2 = ^^ — 4/, 

In case the tangent is moved inward rather than outward, 
the solution will apply by transposing A2 and A2. Then we 
shall have 

cos i2' = cos ^2+0 5- (25) 

K2 — Kx 



80 



RAILROAD CONSTRUCTION, 



§69. 



The P.C.C. is then moved forward. 

c. Assume the same case as (b) except that the larger radius 
comes first and that the tangent adjacent to the smaller radius 
is moved. In Fig. 33 

(R2—R1) cos J, =0{n; 
(R2-R1) cosi/=OiV. 

= (R2—Ri)(cos J/— cos Ji). 



cos J I = cos J I 



R2—R1 



(26) 




The P.C.C. is moved forward 

along the easier curve an angular 

distance of i/ — il = i2~'^2'• 
In case the tangent is moved inward , transpose as before and 

we have 



Fig. 33. 



cos J/ = cos J I 



X 



(27) 



R2—R1 
The P.C.C. is moved backward 

d. Assume that the radius of one curve is to be altered with- 
out changing either tangent. Assume conditions as in Fig. 34. 

For the diagrammatic solution 
assume that R2 is to be increased 
by O2S. Then, since /?/ must 
pass through 0^ and extend be- 
yond Oj a distance OiS, the 
locus of the new center must lie 
on the arc drawn about 0^ as 
center and* with OS as radius. 
The locus of O2 is also given 
by a line 02'p parallel to BV 
and at a distance of i?2' (equal 
to S ,,. P.C.C.) from it. The 
new center is therefore at the 
intersection O2'. An arc with ra- 
dius 7^2' will therefore be tangent 
at B^ and tangent to the old 
Draw OjTi^ perpendicular to O2B, 




Fig. 34. 



curve produced at new P.C.C. 



§ 70. ALINEMENT. 81 

With O2 as center draw the arc 0{m, and with O2' as center draw 
the arc O^m', mB=Yn'B' =R^. 

.\ mn= m^n' = {R2 — Ri) vers J-/ = (R2 — Ri) vers Jg- 

.-. versJ/ = ^^^|^versJ2 (28) 

OiU = (R2 — Ri) sin J 2 ) ' 

Oy = (^2'--Ki) sin J2'. 
BB' = 0{n' - 0{n = (R/ - R,) sin J 2' - (^2 - ^1) sin J 2- (29> 

This problem may be further modified by assuming that the 
radius of the curve is decreased rather than increased, or that 
the smaller radius follows the larger. The solution is similar 
and is suggested as a profitable exercise. 

It might also be assumed that, instead of making a given 
change in the radius R2, sl given change BB^ is to be made. Jj' 
and J?2' ^re required. Eliminate i^z' from Eqs. 28 and 29 
and solve the resulting equation for ^2'- Then determine 7^2' 
by a suitable inversion of either Eq. 28 or 29. 

As in §§ 62 and 63, the above^ problems are but a few, although 
perhaps the most common, of the problems the engineer may 
meet with in compound curves. All of the ordinary problems 
may be solved by these and similar methods. 

70. Problems, a. Assume that tjie two tangents of a com- 
pound curve are to be 348 feet and 624 feet, and that ii=22° 16' 
and ^2 = 28° 20'. Required the radii. 

[Ans. Ei= 326.92; R2 = 1574.85,] 

h, A line crosses a valley by a compound curve which is first 
a 6° curve for 46° 30' and then a 9° 30' curve for 84° 16'. It is 
afterward decided that the last tangent should be 6 feet farther 
up the hill. What are the required changes? [Note. The 
second tangent is evidently moved outward. The solution cor- 
responds to that in the first part of § 69, c. The P.C.C. is 
moved forward 16.39 feet. If it is desired to know how far the 
(P.!^. is moved in the direction of the tangent (i.e., the projection 
of BB\ Fig. 33, on V'B'), it may be found by observing that it 
lis equal to nn' = (7?2— ^i)(sin Jj— sin i/). In this case it equals 
0.65 foot, which is very small because Jj is nearly 90°. The 
value of J 2 (46° 30') is not used, since the solution is independent 
of the value of J2. The student should learn to recognize 



82 



RAILROAD CONSTRUCTION. 



§71. 



which quantities are mutually related and therefore essential 
to a solution, and which are independent and non-essential. J 



TRANSITION CURVES. 

71. Superelevation of the outer rail on curves. When a mass 
is moved in a circular path it requires a centripetal force to keep 
it moving in that path. By the principles of mechanics we 
know that this force equals Gv^-^gR, in which G is the weight, 
V the velocity in feet per second, g the acceleration of gravity in 
feet per second in a second, and R the radius of curvature. 
If the two rails of a curved track were laid on a level (trans- 
versely), this centripetal force could only be furnished by the 
pressure of the wheel-flanges against the rails. As this is very 
objectionable, the outer rail is elevated so that the reaction of 
the rails against the wheels shall 
contain a horizontal component 
equal to the required centripetal 
force. In Fig. 35, if oh represents 
the reaction, oc will represent the 
weight G, and ao will represent the 
required centripetal force. From 
similar triangles we may write 
sn : sm :: ao : oc. Call g = 32.17. 
Call 72 = 5730 --D, which is suffi- 
ciently accurate for this purpose (see 
§ 48) . Call V = 5280F - 3600, in which V is the velocity in miles 
per hour. m7i is the distance between rail centers, which, for 
an 80-lb. rail and standard gauge, is 4.916 feet sm is slightly 
less than this. As an average value we may call it 4.900, which 
is its exact value when the superelevation is 4f inches. Calling 
sn=e, measured in feet, we have 

4.9X52S0W^D 




Fig. 35. 



ao . .Gv^ 1 

e=sm~ =4.9 ~=r — 

oc gR G 



32.17X3600^X5730' 



6 = . 0000572 y^D (30) 

It should be noticed that, according to this formula, the re- 
quired superelevation varies as the square of the velocity, which 
means that a change of velocity of only 10% would call for a 
change of superelevation of 21%. Since the velocities of trains 
over any road are extremely variable, it is impossible to adopt 



§72. 



ALINEMENT. 



83 



any superelevation which will fit all velocities even approx- 
imately. The above fact also shows why any over-iefinement 
in the calculations is useless and why the above approximations, 
which are really small, are amply justifiable. For exanTple, the 
above formula contains the approximation that J? = 5730 -f- D . 
In the extreme case of a 10° curve the error involved would be 
about 1%. A change of about 4 of 1% in the velocity, or say 
from 40 to 40.2 miles per hour, would mean as much. The error 
.in e due to the assumed constant value of sm is never more than 
a very small fraction of 1%. The rail-laying is not done closer 
than this. Table XIX is based on Eq. (30) : 

Table XIX. superelevation of the outer rail (in feet) 

FOR various velocities AND DEGREES OF CURVATURE. 



Velocity in 
Miles per 


Degree of Curve. 


Hour. 


1° 


2° 


3° 


4^ 


5° ' 


6° 


7° 


8° 


9° 


10° 


30 


.05 
.09 
.14 
.20 


.10 
.18 
.29 
.41 


.15 
.27 
.43 


.20 
.37 


.26 
.46 


.31 


.36 


.41 


.46 


1.51 


40 


.86 


.64 


"tT 


"sT 




50 


|.57 

.82 


.71 




60 


1 .62 





72. Practical rules for superelevation. A much used rule for 
superelevation is to ^'elevate one half an inch for each degree of 
curvature." The rule is rational in that e in Eq. 30 varies 
directly as T>. The above rule therefore agrees with Eq. 30 
when Y is about 27 miles per hour. However applicable the 
rule may have been in the days of low velocities, the elevation 
thus computed is too small now. The rule to elevate one inch 
for each degree of curvature is also used and is precisely similar 
in its nature to the above rule. It agrees with Eq. 30 when 
the velocity is about 38 miles per hour, which is more nearly 
the average speed of trains. 

Another (and better) rule is to '' elevate for the speed of the 
fastest trains." This rule is further justified by the fact that a 
four-wheeled truck, having two parallel axles, will always tend 
to run to the outer rail and will require considerable flange pres- 
sure to guide it along the ciirve. The effect of an excess of super- 
elevation on the slower trains will only be to relieve this flange 
pressure somewhat. This rule is coupled with the limitation 



84 



RAILROAD CONSTRUCTION. 



§72. 



that the elevation should never exceed a limit of six inches — 
sometimes eight inches. This limitation implies that locomo- 
tive engineers must reduce the speed of fast trains around sharp 
curves until the speed does not exceed that for which the actual 
superelevation used is suitable. The heavy line in Table XIX 
shows the six-inch limitation. 

Some roads furnish their track foremen with a list of the super- 
elevations to be used on each curve in their sections. This 
method has the advantage that each location may be separately 
studied, and the proper velocity, as affected by local conditions 
(e.g., proximity to a stopping-place for all trains), may be de- 
termined and applied. 

Another method is to allow the foremen to determine the 
superelevation for each curve by a simple measurement taken 
at the curve. The rule is developed as follows: By an inversion 
of Eq. 19 we have 

x=chord^^8R (31) 

Putting X equal to e in Eq. 30 and solving for " chord, '^ we 
have 

chord 2 = .0000572 y2i)si^ 
=2.62172. 

chord = 1S2V. (32) 

To apply the rule, assume that 50 miles per hour is fixed as 
the velocity from which the superelevation is to be computed. 
Then 1.627 = 1.62X50 = 81 feet, which is the distance given to 
the trackmen. Stretch a tape (or even a string) with a length 
of 81 feet between two points on the concave side of the head of 
cither the inner or the outer rail. The ordinate at the middle 
point then equals the superelevation. The values of this chord 
length for varying velocities are given in the accompanying 
tabular form. 



Velocity in miles per hour. 
Chord length in feet 



20 
32.4 



25 
40.5 



30 
48.6 



35 
56.7 



40 

64.8 



45 
72.9 



50 
81.0 



55 

89.1 



60 
97.2 



The following tabular form shows the standard (at one time) 
on the N. Y., N. H. & H. R. R. It should be noted that the 
elevations do not increase proportionately with the radius, and 
that tb.ey axe higher for descending grades than for level or 



§73. 



TNEMENT. 



85 



ascending grades. This is on the basis that the velocity on curves 
and on ascending grades will be less than on descending grades. 
For example, the superelevation for a 0° 30' curve on a de- 
scending grade corresponds to a velocity of about 54 miles per 
hour, while for a 4° curve on a level or ascending grade the super- 
elevation corresponds to a velocity of only about 38 miles per 
hour. 



TABLE OF THE SUPERELEVATION OF THE OUTER RAIL ON CURVES, 

N. Y., N. H. & H. R. R. 



Degree of 


Level or as- 


Descending 


curve. 


cending grade. 


grade. 




inches. 


inches. 


0° 30' 


Oi 


1 


1 00 


U 


U 


1 15 


U 


2 


1 30 


2 


2i 


1 45 


2i 


21 


2 00 


2i 


2f 


2 15 


2i 

2i 


3 


2 30 


3i 


2 45 


3 


31 


3 00 


H 


3f 


3 15 


31 


H 


3 30 


3i 


4 


3 45 


3i 


4i 


4 00 


4 


4i 



73. Transition from level to inclined track. On curves the 
track is inclined transversely ; on tangents it is level. The tran- 
sition from one condition to the other must be made gradually. 
If there is no transition curve, there must be either inclined 
track on the tangent or insufficiently inchned track on the curve 
or both. Sometimes the full superelevation is continued through 
the total length of the curve and the "lun-off" (having a length 
of 100 to 400 feet) is located entirely on the tangents at each 
end. In other practice it is located partly on the tangent and 
partly on the curve. Whatever the method, the superelevation 
is correct at only one point of the run-off. At all other points 
it is too great or too small. This (and other causes) produces 
objectionable lurches and resistances when entering and leav- 
ing curves. The object of transition curves is to obviate these 
resistances. 

On the lichigh Valley R. R, the run-off is made in the form 
of a reversed vertical curve, as shown in the accompanying 
figure. According to this system the length of run-off varies 



86 



RAILROAD CONSTRUCTION. 



§74. 



from 120 feet, for a superelevation of one inch, to 450 feet, 
for a superelevation of ten inches. Such a superelevation 
as ten inches is very unusual practice, but is successfully 
operated on that road. The curve is concave upward for two- 
thirds of its length and then reverses so that it is convex upward. 

TABLE FOR RUN-OFF OF ELEVATION OF OUTER RAIL OF CURVES. 
Drop in inches for each 30-foot rail commencing at theoretical point of curve. 



£.2 


V 


i" 


¥ 


¥ 


¥' 


¥' 


i" 


1" 


IV 


ir 


IF 


V 


¥' 


r 


¥ 


¥ 


¥ 


¥ 


1 
il" ¥ 


N' 


"5 
o 


W^ 












































H 


1" 




30 


30 






























30 




30 




120 


r 




30 








30 




















30 


30 






30 




150 


w 




30 








30 














30 




30 




30 






30 




180 


A" 




30 




30 






30 


. 










30 




30 


30 




30 




30 




240 


W 




30 




30 








30 






. . . 


30 




30 


30 


30 




30 




30 




?70 


6" 




30 




30 






30 




30 






30 




30 


30 


30 




30 




30 




300 


7" 




30 




30 






30 




30 




30 


30 




30 


30 




30 


30 




30 




330 


W 




30 




30 




30 






30 


30 


30 


30 




30 


30 




30 




30 




30 


360 


9'' 


30 






30 




30 




30 


30 




30 


30 


30 


30 


30 


30 


30 


_ 


30 




30 


420 


10'' 


30 


■ ■ 


30 




30 






30 


30 


30 


30 


30 


30 


30 


30 


30 


30 


•• 


30 




30 


450 




The figure (and also the lower line of the tabulated form) 
shows the drop for each thirty-foot rail length. For shorter 
lengths of run-off, the drop for each 30 feet is shown by the cor- 
responding lines in the tabular form. Note in each horizontal 
line that the sum of the drops, under which 30 is found, equals 
the total superelevation as found in the first column. For 
example, for 4 inches superelevation, length of curve 240 feet, 
the successive drops are i'', i", Y', i''] f'', J'', \", and \" 
whose sum is 4 inches. Possibly the more convenient form 
would be to indicate for each 30-foot point the actual super- 
elevation of the outer rail, which would be for the above case 
(running from the tangent to the curve) J", f", J", IJ", 2|", 
3i", 3r', 4". 

74. Fundamental principle of transition curves. If a curve 



il 



§ 75. ALINEMENT. 87 

has variable curvature, beginning at the tangent with a curve 
of infinite radius, and the curvature gradually sharpens until it 
equals the curvature of the required simple curve and there 
becomes tangent to it, the superelevation of such a transition 
curve may begin at zero at the tangent, gradually increase to 
the required superelevation for the simple curve, and yet have 
at every point the superelevation required by the curvature at 
that point. Since in Eq. (30) e is directly proportional to Z), 
the required curve must be one in which the degree of curve 
increases directly as the distance along the curve. 

75. Varieties of Transition Curves. A theoretically exact 
transition curve is very complicated and its mathematical 
solution very difficult. A committee of the Amer. Rwy. Eng. 
Assoc, investigated the many systems which have been proposed 
and reported that all of them seemed to be objectionable for 
one or more of the following reasons: '^(1) If simple approximate 
formulas were used, they were not sufficiently accurate. (2) 
Accurate formulas were too complex. (3) The curve could not 
be expressed by formulas. (4) Formulas were of the endless 
series class. (5) Complex field methods were required to make 
the field-work agree with formulas with spirals of large angles." 
The committee then developed a method which gives results 
whose accuracy is beyond that of the most careful field-work and 
yet which is sufficiently simple for practical use. The mathe- 
matical development is so elaborate that it will not be detailed 
here, but the working formulas and a condensation of the table 
together with an explanation of their practical use and applica- 
tion, will be given, with numerical examples. 

The general form of these curves, whatever their precise 
mathematical character, is shown in Fig. 36. AYB are two 
tangents, joined by the simple circular curve AMB, having the 
center 0. Assume that the entire curve is moved in the direc- 
tion MO a distance 00' =MM' =BB' =AA\ At some point TS 
on the tangent, the spiral begins and joins the circular curve 
tangentially at SC. The other spiral runs from CS to ST, The 
significance of these symbols may be readily remembered from 
the letters; 7", S, and C signify tangent, spiral and circular curve; 
TS is the point of change from tangent to spiral, SCj the point 
of change from spiral to curve, etc. At the other end of the 
circular curve the letters are in reverse order, the station numbers 
increasing from A to B. The meaning of the various symbols is 



88 



RAILROAD CONSTRUCTION. 



§76. 



indicated in Fig. 36. The student should appreciate the fact of 
the necessary distortion of the figure in order to make it plain. 
Based on the figures of the following numerical problem, the 
distance MM^ is about fourteen times its proper amount. Another 
effect of the distortion is that the dimension U, instead of being 




Fig. 36. 



nearly twice V, which is usual, as given in Table IV, Part B, is 
only a little longer than V. 

76. Proper length of spiral. This can only be computed on 
the basis of certain assumptions as to the desired rate of tipping 
the car, so as to avoid discomfort to passengers, and, of course, 
this depends on the expected velocity. There is also a maximum 
limitation, since the sum of the two spiral angles cannot exceed 
the total central angle of the curve. The minimum lengths 
recommended are as follows: 



§ 77. ALINEMENT. 89 

On curves which limit the speed: 

6° and over, 240 feet; 

Less than 6% SJXspeed in m.p.h. for elevation of 8 inches. 
On curves which do not limit the speed: 

30 times elevation in inches, or 

f Xultimate speed in m.p.h. X elevation in inches. 

For example. (1) 5° curve which Hmits speed; speed limit 
48 m.p.h. by interpolation in table, § 41; 48X5| = 256 feet 
minimum length. (2) 3° curve; maximum operating speed 60 
m.p.h.; superelevation, .62 feet = 7.44 inches; 30X7.44 = 223.2 
feet; or, |X60X7.44 = 297.6 feet. Of course the higher value 
should be used, or say 300 feet as the minimum length. 

While it is generally true that the longer transition curves 
give easier riding, the spiral must not reach the center point of 
the curve. Since it is approximately true that the spiral extends 
for equal distances on each side of the original point of curve, it is 
nearly true that two spirals, each having the same length as the 
original curve, would just meet at the center. The length of a 
spiral should in general be very much less than the length of the 
original curve. 

77. Symbols. Beside the symbols whose significance is 
clearly indicated in Fig. 36, the following are defined: 

a The angle between the tangent at the TS and the chord from 

the TS to any point on the spiral; ai is the angle to the 

first chord point. 
A The angle between the tangent at the TS and the chord 

from the TS to the SC, 
D The degree of the central circular curve. 
A The central angle of the original circular curve, or the angle 

between the tangents. 
The total central angle of the spiral. 
k The increase in degree of curve per station on the spiral. 
L The length of the spiral in feet from the TS to the SC. 
S The length of the spiral in stations from the TS to the SC. 
s The length of the spiral in stations from the TS to any given 

point. ^ 

78. Deflections. The field formulas for deflections are based 
on the following two equations: 

a = 10 ks^ minutes, 
^ = 10 kS^ minutes. 



90 RAILROAD CONSTRUCTION. § 78. 

The first deflection ai — lOksi'^ minutes. But k is the increase in 
degree of curve per station, and since the degree of curve in- 
creases as the length, k = D-^S, S being expressed in stations. 

/ D\ 
For point 1, since S = 10s, ai = 10l — - ]si'^ = Dsi, which may be 



lOsi^ 

expressed as the degree of the curves times the length of the chord 
in stations. For example, if the spiral is 400 feet long (which 
means that L = 400 and /S = 4) and runs on to a 5° curve (then 
Z) = 5), one chord is 40 feet long and s = A station. Then ai = 5 
X0.4 = 2 minutes of arc for the deflection for the first chord point. 
And since the deflections are as the square of the number of sta- 
tions, the deflections from TS to succeeding stations will be 4, 9, 
16, 25, 36, 49, 64, 81, and 100 times 2 minutes, these factors being 
those given in the second vertical column of Part A of Table 
IV. The last deflection = A = 100X2' = 200' = 3° 20' = | (10°) 
= i</>, (f> being the total central angle of the spiral. Although 
it is always nearly true that A = i4>, and the error is inappreciable 
for small angles, the error amounts to 30 seconds of arc when 
<^ = 21° 30', an unusually large angle. 

The deflection from any other point of the spiral to any other 
point, either forward or backward, may be found by multiplying 
the value of ai (in this case 2'), by the coefficients in the proper 
vertical column of that table. 

The spiral angle 



Also, 



kS^_ kL'' PL 5X400 
*^~ 2 "20000 ~200~ 200 ~^ ' 



kS^ DS 5X4 
(p = — = — — — iU 



The values of the ratios U-^L and V-r-L for even degrees, and 
for A, C-r-L, X-^L, and Y-^L for half degrees are given in Parts 
B and C of Table IV. When it is desired to temporarily omit 
locating the intermediate points of the spiral, the jump from the 
TS to the SC may be made by measuring the distance U from the 
TS along the tangent. At that point a deflection <f> and a 
measured distance V will give not only the position of SC but 
also the direction of the tangent at the beginning of the circular 
curve. Another method of locating the SC without locating 
the intermediate points is to make the deflection A at the TS 



§ 79 ALINEMENT. 91 

and measure the long chord C. In the above numerical problem 
this equals 400 X. 998664 = 399.47, a little over 6 inches short of 
the full 400 feet. By setting up the transit at the SCy back- 
sighting at the TS, and turning off the angle (0— A), which in 
the above case is 10°-3° 19' 57'' = 6° 20' 03", the direction of 
the tangent at the SC is obtained. In this case, the three sec- 
onds variation from the approximate value is utterly neghgible. 
The other dimensions are easily determined from the tables if 
desired; 

X = . 996975 X 400 = 398 . 79, 

F=. 058053X400= 23.22, 

U = . 667742 X 400 = 267 . 10 

. y=. 334313X400 = 133. 73. 

For greater convenience of notation, the points TS, SC, CS, 
and ST, in Fig. 36 are also indicated by the letters Q, Z, Z' and 
Q' respectively. The same letters are used for the corresponding 
points in Figs. 37 and 38. 

79. Location of spirals and circular curve with respect to 
tangents. See Fig. 36. Let AV and BF be the tangents to be 
connected by a D° curve, having a suitable spiral at each end. 
If no spirals were to be used, the problem would be solved as in 
simple curves giving the curve AMB, Introducing the spiral has 
the effect of throwing the curve away from the vertex a distance 
MM' and reducing the central angle of the D° curve by 2<f>. 
Continuing the curve beyond Z and Z' to A' and B', we will 
have AA'=BB'=MM'. ZK= the Y ordinate and is therefore 
known. Call MM' = m. A'N = Y-R vers <f>. Then 

,.,., . ., A'^' F-i^ verse/) , , 

m = MM' = AA' = — = (33) 

cos JA cos |A 

NA =AA' sin iA = {Y-R vers <t>) tan |A. 
VQ=QK-KN+NA+AV 

=X — R sin (t)-{-{Y — R vers 4>) tan |A+ R tan |A 
=X-/2 sin + F tan jA+1^ cos </> tan |A. . . . (34) 

When A'N has already been computed, it may be more con- 
venient to write 

VQ=X+R (tan JA- sin <t>)-\-A'N tan JA (35) 



92 



RAILROAD CONSTRUCTION. 



VM' = VM+MM' 
= R exsec |A-(- 



R vers </> 



cos JA 



cos iA 



§79. 



(36) 



AQ=VQ-AV 

=X-R sin + (F-E vers 0) tan JA (37) 

Example. To join two tangents making an angle of 34° 20' 
by a 5° 40' curve and suitable spirals. Assume that the spiral 
is 300 feet long. Then 

D^^5,67X_3 „3^, 

^22 
Since, from Table IV, Part A, F^L = . 049374 for <^ = 8° 30', 
F = 14.812; similarly, we find Z = 299.344 and (7 = 299.71. 



[Eq. 33] 



R 

vers (j) 



3.00497 
8.04076 



[Eq. 36] 



[Eq. 35] 



[Eq. 37] 



312.471 
AQ = 150.971 



AV 2.49481 







y= 


11.110 
14.812 




1.04573 






■ A'N = 


3.702 


cos JA 


0.56843 
9.98021 




m=MM' = 


--AA' = 


3.875 


R 

exsec iA 


0.58822 




3.00497 
8.66863 






yM= 

m = 


47.164 
3.875 




1.67360 




=299.344 


FM' = 

nat 
nat 


51.039 


= .30891 
= .14781 

.16110 
R 


9.20709 
3.00497 


X- 


tan iA = 
.sin = 




162.954 


[See above] 


A'-N 
tan JA 


2.21206 




0.56843 
9.48984 




1.144 
= 463.442 






AN 

R 

tan |A 


0.05827 


VQ 


3.00497 
9.48984 



§ 80. ALINEMENT. 93 

It should be noted that AQ is within a foot of equaling one-half 
the length of the spiral, which illustrates the general fact that a 
spiral begins at approximately one-half its length from the 
P.C. of the simple curve. All approximate systems of spirals 
assume this to be exactly true. 

80. Field-work. When the spiral is designed during the 
original location, the tangent distance VQ should be computed 
and the point Q located. It is hardly necessary to locate all of 
the points of the spiral until the track is to be laid. The extrem- 
ities should be located, and as there will usually be two or more 
full station points on the spiral, these should also be located. 
Z may be located by setting off QK = X and KZ = Y, or else by 
the tabular deflection for Z from Q and the distance ZQ, which 
is the long chord c. Setting up the instrument at Z and sighting 
back at Q with the proper deflection, the tangent at Z may be 
found and the circular curve located as usual, its central angle 
being A—2<t>. A similar operation will locate Q' from Z'. 

To locate points on the spiral. Set up at Q, with the plates 
reading 0° when the telescope sights along VQ. Set off from 
Q the deflections computed from Table IV for the instrument at 
Qf using a chord length of L-^10, the process being like the 
method for simple curves except that the deflections are variable. 
If a full station-point occurs within the spiral, interpolate 
between the deflections for the adjacent spiral-points. For exam^ 
pie, a 400-foot spiral running on to a 3° 31' curve begins at Sta. 
56+15. The spiral points are 40 feet apart. Sta. 57 comes 
5 feet beyond the second spiral point. The first deflection ai 
= Ds = 3.5X.4 = 1.4 min. The deflection to point 2 is 4X1.4 
= 5.6 min. and that to point 3 is 9X1.4 = 12.6 min. Then the 
deflection to Sta. 57 is ^X (12.6 -5.6) +5.6 = 6.47 min. 

This method is not theoretically accurate, but the error is small. 
Arriving at Z, the forward alinement may be obtained by sight- 
ing back at Q (or at any other point) with the proper deflection 
for that point from the station occupied. Then when the plates 
read 0° the telescope will be tangent to the spiral and to the 
succeeding curve. All rear points should be checked from Z. 
If it is necessary to occupy an intermediate station, use the de- 
flections given for that station, orienting as just explained for Z, 
checking the back points and locating all forward points up to Z 
if possible. 

After the center curve has been located and Z' is reached, the 



94 



RAILROAD CONSTRUCTION. 



§81. 



other spiral must be located but in reverse order, i.e., the sharp 
curvature of the spiral is at Z' and the curvature decreases 
toward \Q\ 

8 1. To replace a simple curve by a curve with spirals. This 
may be done by the method of § 79, but it involves shifting the 
whole track a distance m, which in the given example equals 
3,87 feet. Besides this the track is appreciably shortened, 




Fig. 37. 



which would require rail-cutting. But the track may be kept 
at practically the same length and the lateral deviation from the 
old track may be made very small by slightly sharpening the 
curvature of the old track, moving the new curve so that it is 
wholly or partially outside of the old curve, the remainder of it 
with the spirals being inside of the old curve. It is found by 
experience that a decrease in radius of from 5% to 10% will 
answer the purpose. The larger the central angle the less the 
change. The solution is as indicated in Fig. 37. 

0'Ar=i2' cos <^+F. 
0'7 = 0W sec |A 

= R' cos <l> sec |A + y sec JA. 



§ 81. ALINEMENT. 95 

m = MM' = MV-MT 
=^R exsec^A- {O'V-R') 
= R exsec |A - J?' cos sec |A - F sec iA+R'. . . (38) 

AQ = QK-KN-j-NV-VA 

= X-R' sin (t>-\-{R' cos + F) tan iA-E tan JA 
=X-E'sin0+ie'cos0taniA-(E-7) tanjA. . . (39) 

A 
The length of the old curve from Q to Q' =2AQ+100 -. 

A-20 
The length of the new curve from Q to Q' =2L+100 , 

in which L is the length of each spiral. 

Example. Suppose the old curve is a 7° 30' curve with a 
central angle of 38° 40'. As a trial, compute the relative length 
of a new 8° 20' curve with spirals 240 feet long. iA = 19° 20'; 
R (for the 7° 30' curve) =76.4.49; R' (for the 8° 20' curve) = 
688.16; = 10° 0'; 7 = 13.933; X = 239.274. 

[Eq. 38] R 2.88337 

exsec |A 8.77642 

45.687 1.65979 

/e^ = 688. 16 _ 

733.847 R' , 2.83768 

cos <^ 9.99335 

sec§A 0.02521 

718.200 ...... 2.85624 

Y 1 . 14405 

sec ^A 0.02.521 

14.766 .... 1.16926 

732.966 732.966 ~ 

m= 0.881 

^=^ R' 2.83768 

[Eq. 39] Z =239.274 ' sin 9.23967 

119.497 ...... 2.07735 

R' 2.83768 

cos </» 9.99335 

tan |A 9.54512 

237.770 . ..... 2.37615 

i? =764.49 
F= 13.93 

750.56 2.87538 
tan |A 9.54512 

JJT:^ 263.333 2.42050 

382.830 382.830 
AQ= 94.214 



96 RAILROAD CONSTRUCTION. § 82. 
The length of the old curve from Q to Q' is 

100| = 100?|fl= 515.556 

2AQ = 2X94.214= 188.428 



703.984 
2L =2X240 =480.000 



New curve: 100^ = 100 ^«-««^-f""° =224.000 



704.000 704.000 
Difference in length = 0.016 

Considering that this difference may be divided among 21 
joints (using 33-foot rails) no rail-cutting would be necessary. 
If the difference is too large, a slight variation in the value of 
the new radius R' will reduce the difference as much as neces- 
sary. A truer comparison of the lengths would be found by 
comparing the lengths of the arcs. 

82. Application of transition curves to compound curves. 
Since compound curves are only employed when the location is 
limited by local conditions, the elements of the compound curve 
should be determined (as in §§ 68 and 69) regardless of the 
transition curves, depending on the fact that the lateral shifting 
of the curve when transition curves are introduced is very small. 
If the limitations are very close, an estimated allowance may be 
made for them. 

Methods have been devised for inserting transition curves 
between the braniies of a compound curve, but the device is 
complicated and usually needless, since when the train is once 
on a curve the wheels press against the outer rail steadily and 
a change in curvature will not produce a serious jar even though 
the superelevation is temporarily a little more or less than it 
should be. 

If the easier curve of the compound curve is less than 3° or 
4°, there may be no need for a transition curve off from that 
branch. This problem then has two cases according as transition 
curves are used at both ends or at one end only. 

.a. With transition curves at both ends. Adopting the method 
of § 79, calling A^ = \Aj we may compute m^=MM^'. Similarly, 
calling ^2 = ^^, we may compute m2=MM2, But ilf/ and M/ 
must be made to coincide. This may be done by moving the 
curve Z'My and its transition curve parallel to Q'V a distance 
ilf/ikfg, and the other curve parallel to QF a distance ikTz^Mg. 



82, 



ALINEMENT. 



97 



In the triangle ikf/ilfgMz', the angle at ikf/=90° — i„ the angle 
at ilf2'=90'' — iz/and the angle at M^^J. 



Then M^'M^^M.'M^ 



„^,sm(90°-i2) 



sin J 



Similarly M^'M^ ^M^'M^ 



,,^, sin (90° -JO 



, .cos Jo 

COS ii 



sin J 



= (m^—m2j 



sin J ' 



K40) 




Fig. 38. 

b. TTu/i a transition curve on the sharper curve only. Com- 
pute mi=MM/ as before; then move the curve Z^M^' parallel 
to Q'F a distance of 

cos ^2 



ilfi'ilf4=mi 



sin d 



(41) 



98 



RAILBOAD CONSTRUCTION. 



§83. 



The simple curve MA is moved parallel to VA a distance of 

T,,,-. cos J I 

MM^=m^— ^ 



sm J 



(42) 



If Ji and ^2 a^6 both small, M/M^ and MM^ may be more 
than mj, but the lateral deviation of the new curve from the old 
will always be less than m^. 

83. To replace a compound curve by a curve with spirals. 
The numerical illustration given below employs another method. 
We first solve for m^ for the sharper branch of the ciu-ve, plac- 
ing ii = |i in Eq. 38. A value for R^ may be found whose 
corresponding value of m^ will equal m^. Solving Eq. 38 for i^', 
we obtain 



R vers JA— m cos i 

R =— — 

cos ^ — cos JA 



iA-F 



(43) 



Substituting in this equation the known value of mi (=m2) 
and calling R'=R2, R = R2, and A2 = iA, solve for i^2'. Obtain 
the value of AQ for each branch of the curve separately by Eq. 
39, and compare the lengths of the old and new lines. 

Example. Assume a compound curve with Di = S°, Z>2=4°, 
Ai = 36°, and A2 = 32°. Use 240-foot spirals at each end. Assume 
that the sharper curve is sharpened from 8° 0' to 8° 15'. 



E<a. 381 





# 


Ri 

exsec 36° 


2.85538 
9.37303 


169.21 


. 2.22842 


695.09 
864.30 


^ 8.25X240 
01= 2 

=9.°9 =9°54' 

Fi =240 X. 05747 
= 13.79 


Ri' (8° 150 

cos 01 

sec Ai 
846.39 . . . , 

sec Ai 
17.05 .. . . 




2.84204 
9.99348 
0.09204 




2.92757 




1.13969 
0.09204 




. 1.23173 


863.44 


863.44 





mi 



0.86 



§ 83. ALINEMENT. 99 



FFn 431 4.05X2.4 R2 3.156I5 

tEq;43] <^2- 2. vers 32° 9.18170 

=4°.86=4°51'.6 

217.700 2.33785 

72= .02826X240. mi =0.86 9.93450 

= 6.782 cos 32° 9.92842 

0.729 9.86292 

F2= 6.782 
7.511 7.511 

210.189 . 2.32261 

nat. cos 02 = . 99640 
nat. cos A2 = . 84805 

.14835 9.17129 

i22' =1416.84 [4° 2' 41"] 3.15132 

Eq. 39] Xi= 239.286 Xi = .997024 X240 == 

= 239.286 Ri' 2.84204 

sin 01 9.23535 

.119.505 2.07739 

Ri' 2.84204 

cos 01 9.99348 

tan iA[Ai =36°] 9.86126 

497.489 c 2.69678 

i2i =716.78 
Fi= 13.70 

703.08 2.84700 

tan ^A 9.86126 

736.775 

630.325 510.820 ..... 2.70826 

^Qi =106.450 630.325 

[Eq. 39] R2' 3.15132 

X2 = . 999284 X240 sin 02 8.92799 

=239.828 120.035 2.07931 

Ri' 3.15132 

cos 02 9 . 99843 

tan |A(A2 =32°) 9.79579 

882.145 ....!:• 2.94554 

i22 =1432,7 

F2= 6.8 

1425.9 3.15403 

tan AA 9.79579 

891.00 ..."..... 2.94988 

1121.973 1011.03 
1011.03, 

AQi^ 110.94 



100 RAILROAD CONSTRUCTION. § 84, 

For the length of the old track we have: 

10o|i = 100^= 450. 

100^^ = 100^= 800. 

AQi= 106.45 
AQ2= 110.94 

= 1467.39 

For the length of the new track we have: 

100^=100p!2i = 316.36 

100^^ = 100f^= 671.11 

Spiral on 8° 15' curve = 240.00 

Spiral on 4° 02* 41 ' curve = 240.00 

Length of new track =1467.47 
Length of old track =1467.39 



Excess in length of new track = . 08 feet. 

Since the new track is slightly longer than the old, it shows 
that the new track runs too far outside the old track at the 
P.C.C. On the other hand the offset m is only 0.86. The 
maximum amount by which the new track comes inside of the 
old track at two points, presumably not far from Z' and Z, is 
very difficult to determine exactly. Since it is desirable that 
the maximum offsets (inside and outside) should be made as 
nearly equal as possible, this feature should not be sacrificed to 
an effort to make the two fines of precisely equal length so that 
the rails need not be cut. Therefore, if it is found that the offsets 
inside the old track are nearly equal to m (0.86), the above 
figures should stand. Otherwise m may be diminished (and the 
above excess in length of track diminished) by increasing Ri 
very slightly and making the necessary consequent changes. 

VERTICAL CURVES 

84. Necessity for their use. Whenever there is a change in 
the rate of grade, it is necessary to eliminate the angle that 
would be formed at the point of change and to connect the two 
grades by a curve. This is especially necessary at a sag between 
two grades, since the shock caused by abruptly forcing an up- 
ward motion to a rapidly moving heavy train is very severe both 
to the track and to the rolling stock. The necessity for vertical 
curves was even greater in the days when link couplers were in 
universal use and the ''slack" in a long train was very great. 



§ 85. ALINEMENT. 101 

Under such circumstances, when a train was moving down a 
heavy grade the cars would crowd ahead against the engine. 
Reaching the sag, the engine would begin to pull out, rapidly 
taking out the slack. Six inches of slack on each car would 
amount to several feet on a long train, and the resulting jerk on 
the couplers, especially those near the rear of the train, has fre- 
quently resulted in broken couplers or even derailments. A 
vertical curve will practically eliminate this danger if the curve 
is made long enough. 

85. Required length. Theoretically the length should de- 
pend on the change in the rate of grade and on the length of the 
longest train on the road. A sharp change in the rate of grade 
requires a long curve; a long train requires a long curve; but 
since the longest trains are found on roads with light grades and 
small changes of grade, the required length is thus somewhat 
equalized. The A.R.E.A. rule is: ^'On class A roads (see § 198) 
rates of change of 0.1 per cent per station on summits and 0.05 per 
cent per station in sags should not be exceeded. On minor roads 
0.2 per cent per station on summits and 0.1 per cent per station 
in sags may be used.'^ When changing from a down grade to an 
up grade (or vice versa) the change of grade equals the numerical 
sum of the two rates of grade. For example, if a 0.5 per cent 
down grade is followed by a 0.7 per cent up grade, the road being 
a "minor" road, then, by the above rule the length of the curve 
should be at least [0.5 -(-0.7)] -=-0.1 =12 stations or 1200 feet. 
Added length increases the amount of earthwork required both 
in cuts and fills, but the resulting saving in operating expenses 
will always justify a considerable increase. 
\ 86. Form of curve. In Fig. 39 assume that A and C, equi- 



m 



d 



t 

LEVEL LINE 



.1 



Fig. 39. 

distant from B, are the extremities of the vertical curve. Bisect 
AC at e; draw Be and bisect it at h. Bisect AB and BC at k 
and I. The line kl will pass through h. A parabola may be 
drawn with its vertex at h which will be tangent to AB and BC 
at A and C. It may readily be shown * from the properties of 
a parabola that if an ordinate be drawn at any point (as at n) 
we will have 

* See note at end of this chapter. 



102 RAILROAD CONSTRUCTION. § 87. 

a parabola that if an ordinate be drawn at any point (as at n) 
we will have 

sn : eh (or KB) : : An? : Ae^ 

or sn=eh^-- (44) 

Ae^ 

In Fig. 39 the grades are necessarily exaggerated enormously. 
With the proportions found in practice we may assume that 
ordinates (such as mt, eB, etc.) are perpendicular to either 
grade, as may suit our convenience, without any appreciable 
error. In the numerical case given below, the variation of 
these ordinates from the vertical is 0° 07', while the effect of 
this variation on the calculations in this case (as in the most 
extreme "cases) is absolutely inappreciable. It may easily be 
shown that the angle (7A5=half the algebraic difference of the 
rates of grade. Call the difference, expressed in per cent of 
grade, r; then CAB = ^r. Let Z=length (in ^'stations" of 100 
feet) of the line AC, which is practically equal to the horizontal 
measurement. Since the angle CAB is one-half the total change 
of grade at B, it follows that Be = ilX ir Therefore 

Bh = \lr. . (45) 

Since Bh (or eh) and Ae are constant for any one curve, the cor- 
rection sn at any point (see Eq. 44) equals a constant times Am^. 
87. Niimerical example. Assume that B is located at Sta. 
16+20; that the grade of AB is -0.5%, and of BC +0.7%; 
also that the elevation of B above the datum plane is 162.6. 
Then the algebraic difference of the grades, r, =0.7 — (—0.5) = 
1.2; Z = 12. m = iZr = jXl2X1.2 = 1.8. A is at Sta. 10+20 
and its elevation is 162.6 + (6X0.5) = 165.6; C is at Sta. 22+20 
and its elevation is 162.6 + (6X0.7) = 166.8. The elevation of 
Sta. 11 is found by adding sn to the elevation of s on the 
straight 'grade Hne. The constant](e/i-^Ae^) equals in this case 
1.8-i-6002 = 2ooVoo- Therefore the curve elevations are 

A, Sta. 10+20, 162.6+(6.00X0.5) =165.60 

11 165.6-(0.80X0.5) + Wo^xF 802=165.23 

12 165.6-(1.80X0.5) + 2ocfVoo 1802=164.86 

13 165.6-(2.80X0.5) + Woo^2802=164.59 

14 165. 6 - (3. 80X0. 5) +5uoW 3802 =164. 42 

15 165.6 -(4.80X0.5) + 200V0TT 4802 =164.35 

16 165.6-(5.80X0.5) + W^u 5802 =164.38 



§ 87. ALINEMENT. 103 

B, 





ALINEMENT. 


16+20, 


162.6+ 1.80 =164.40 


17 


166.8-(5.20X0.7)+ 200^000 5202=164.51 


18 


166 . 8 - (4 . 20 X . 7) +, 2 oc^o^o 4202 = 164 . 74 


19 


166.8 -(3.20X0.7) + Wooo 3202 =165.07 


20 


166 . 8 - (2 . 20 XO . 7) + ^ooW 2202 = 165 . 50 


21 


166.8-(1.20X0.7)+ ^5^^1202=166.03 


22 


166. 8-(0. 20X0. 7) + 55i^ 202=166.66 


22+20, 


>162.6+(6.00X0.7) =166.80 

• 



c. 



DEMONSTRATION OF EQ. 44. 

The general equation of a parabolai passing through the point n (Fig. 36) 
may be written 

2/2 + 2/^2 = 2p(x + x^h 

y2 2/„2 

from which x^, =-71 h -z-- — X, 

^ 2p 2p 



When 3; == a;^ V '^ Va ^^^ ^^ have 

The general equation of a tangent passing through the point A may be 
written 

Wa " P(^ + ^^)' 

vva 



from which 








X = 






When X = Xg, 


y 


-Vsl- 


= yn] 


and 


we have 

VuVa 

p ^^. 








an = 


^n - 


= 

2p = 

sn = 


2p 

(yA-y'v)^ 

2p 
yA^ Ae^ 
^A eh 

~Am^ 

eh -„. 

Ae^ 


'^y-nyj 

Am 
2p' 



Tiiis proves the general proposition that if secants are drawn parallel to 
the axis of re, intersecting a parabola and a tangent to it, the intercepts be- 
tween the tangent and the parabola are pioportional to the square of the 
distances (measured parallel to y) from the tangent point. 



J 



CHAPTER in. 

EARTHWORK. 
FORM OF EXCAVATIONS AND EMBANKMENTS. 

88. Usual form of cross-section in cut or fill. The normal 
form of cross-section in cut is as shown in Fig. 40, in which 
C . . . gr represents the natural surface of the groiuid, no matter 



e ^s 




how irregular ; ctb represents the position and width of the re- 
quired roadbed; ac and hd represent the "side slopes" which 
begin at a and h and which intersect the natural surface at such 




\ 



Fig. 41. 



points {c and d) as will be determined by the required slope 
angle (/?). 

The normal section in fill is as shown in Fig. 41. The points 
c and d are likewise determined by the intersection of the re- 

104 



§89. 



EARTHWORK, 



105 



quired side slopes with the natural surface. In case the required 
roadbed (ab in Fig. 42) intersects the natural surface, both cut 




Fig. 42. 

and fill are required, and the points c and d are determined as 
before. Note that /? and /?' are not necessarily equal. Their 
proper values will be discussed later. 

89. Terminal pyramids and wedges. Fig. 43 illustrates the 
general form of cross-sections when there is a transition from 
cut to fill. a..,g represents the grade line of the road which 




Fig. 43. 

passes from cut to fill at d. sdt represents the surface profile. 
A cross-section taken at the point where either side of the road- 
bed first cuts the surface (the point m in this case) will usually 
be triangular if the groimd is regular. A similar cross-section 
should be taken at o, where the other side of the roadbed cuts 
the surface. In general the earthwork of cut and fill terminates 



106 RAILROAD CONSTRUCTION. § 90. 

in two pyramids. In Fig. 43 the pyramid vertices are at n 
and k, and the bases are Ihm and o'pq. The roadbed is generally 
wider in cut than in fill, and therefore the section Ihm and the 
altitude In are generally greater than the section opq and the 
altitude ph. When the line of intersection of the roadbed and 
natural surface (nodkm) becomes perpendicular to the axis of 
the roadbed (ag) the pyramids become wedges whose bases are 
the nearest convenient cross-sections. 

90. Slopes, a. Cuttings. The required slopes for cuttings 
vary from perpendicular cuts, which may be used in hard rock 
which will not disintegrate by exposure, to a slope of perhaps 
4 horizontal to 1 A^ertical in a soft material like quicksand or in 
a clayey soil which flows easily when saturated. For earthy 
materials a slope of 1 : 1 is the maximjim allowable, and even 
this should only be used for firm material not easily affected by 
saturation. A slope of IJ horizontal to 1 vertical is a safer 
slope for average earthwork. It is a frequent blunder that 
slopes in cuts are made too steep, and it results in excessive work 
in clearing out fr6m the ditches the material that slides down, 
at a much higher cost per yard than it would have cost to take 
it out at first, to say nothing of the danger of accidents from 
possible landslides. 

b. Embankments. The slopes of an embankment A^ary from 
1 : 1 to 1.5 : 1. A rock fill will stand at 1 : 1, and if some care 
is taken to form the larger pieces on the outside into a rough 
dry wall, a much steeper slope can be allowed. This method is 
sometimes a necessity in steep side-hill work. Earthwork em- 
bankments generally require a slope of IJ to 1. If made 
steeper at first, it generally results in the edges giving way, re- 
quiring repairs until the ultimate slope is nearly or quite IJ : 1. 
The difficulty of incorporating the added material v/ith the old 
embankment and preventing its sliding off frequently makes 
these repairs disproportionately costly. 

91. Compound sections. When the cut consists partly of 
earth and partly of rock, a compound cross-section must be 
made. If borings have been made so that the contour of the 
rock surface is accurately known, then the true cross-section may 
be determined. The rock and earth should be calculated sepa- 
rately, and this will require an accurate knowledge of where the 
rock ''runs out"— a difficult matter when it must be deter- 



§ 92. EARTHWORK. 107 

mined by boring. During construction the center part of the 
earth cut would be taken out first and the cut widened until a 
sufficient width of rock surface had been exposed so that the 
rock cut would have its proper width and side slopes. Then the 
earth slopes could be cut down at the proper angle. A^'berm" 
of about three feet should be left on the edges of the rock cut as 




Fig. 44. 

a margin of safety against a possible sliding of the earth slopes. 
After the work is done, the amount of excavation that has been 
made is readily computable, but accurate prehminary estimates 
are difficult. The area of the cross-section of earth in the figure 
must be determined by a method similar to that developed for 
borrow-pits (see § 120). 

Q2. Width of roadbed. Owing to the large and often dis- 
proportionate addition to volume of cut or fill caused by the 
addition of even one foot to the width of roadbed, there is a 
natural tendency to reduce the width until embankments become 
unsafe and cuts are too narrow for proper drainage. The cost 
of maintenance of roadbed is so largely dependent on the drain- 
age of the roadbed that there is true economy in making an 
ample allowance for it. The practice of some of the leading 
railroads of the country in this respect is given in the following 
table, in which are also given some data belonging more properly 
to the subject of superstructure. 

It may be noted from the table that the average width 
for an earthwork cut, single track, is about 24.7 feet, with a 
minimum of 19 feet 2 inches. The widths of fills, single track, 
average over 18 feet, with numerous minimums of 16 feet. 
The widths for double track may be found by adding the distance 
between track centers, which is usually 13 feet. 



108 



KAILEOAD CONSTRUCTION. 



§92. 

























- 








i 


© d i 








S S* o S 






^ 


J ^ c3-g 




rt^oocb CCCOCO COOICC cc 


^ 


"qq-*^> C3 




fH tH 1-H 1— 1 I— 1 1— 1 1— 1 1— 1 i-H 1—1 


CM 


5^^<3 






1— ( 








i 




'^ 


1-t 






tHtH 


iO»OiOiOiOiOiCuDiOiOiO 


.fj 






,—1 ,— 1 ,_^ tH I— 1 1— 1 1—1 T— 1 I— i 1— i r-H 1— I 


1— t r-i 


03 
















'■- 


"^ 










9^ 






tH 1-1 T-l 1-1 ,-1 tH iHi-ItH 


1-! 


o 






i-li-l tH _i-I r-*T- 


* ,, 1-H 


s 






•• "lOiCiOiOiO "lO "lO^O " * 


■ lo •• 






1— (i-t* 1— < •!— ( • • • H*Hf< • 1— 1 1 








1-1 tH tH T-H rH tH i-| ,H r-i 


1—1 










.' t^ 


^ 




- 








1=3 




. CO 

•o ceo 


OOOCM 


coc^^ • 


^ 








fiH 




•CO ^00 


^ coco 


cccc^ 


1— H 




o* 








* CO 

'. CO 


CO 
CO 


^ '. 


CO 




c3 


















H 


























^^ 




__^ 




3 








■lOzO^ 






X 




S 








:xxx 


^^t^ 


CM 




o 
Q 




-(J 

;3 




.(MC^C^ 


ooXx 


Xc "^ ^ 

C1 C0 5: 


+ 








O 




;oor-co 

.(NCOCO 


CAW CM 




Tf - 

1—1 
CO 












! ^ :t 














^' 


O 


•CO ooooQOco 


o 




'^ : 


^^- 






S 


CM 


• r-i -t- (M T-H ^ 1-1 






?^ : 


cr.a>-^ 














1—1 


r-i 1—1 


M 
















n 






























o 








:S^ea^ ^ 


lo 




5: 


g- to 

^ >-.c<^ 

rJ3 Ce -^ 


1 




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CM 




CM 

1-1 CO 
CM 1-i 


fccScM 








>-..-'-^'-^CM r^ 


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•Ji 

> 
















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r ^j:2 :'-::2'C d) c3 o-- • o 








< 


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Kh:i 


h-1 


H^l 


s 


z 


IZi 




Ph 


p 


1 



cd 





o 




(h 




o 








■+3 




(U 




(D 




«*-i 




CO 


>> 


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03 


-*i 


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a 


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05 




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73 






el 


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a; 


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TJ 


T) 






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+ 


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r/j 




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a 


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^ 


bll 


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(1 


^ 


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CJ 


+j o 


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73 
O 


=<^ 


^ 


73 








03 ro 


"a 


|2 


bfl 


"^O 


ou 


o m 




O 02 


»o 


03 C8 
m 1— > 


X^O 


CM 






MO 


* 


fl^ 




Wli^ 




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73 




sg 




<:r„ 




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03 




O 




»4 



§ 93. EARTHWORK. 109 

93. Form of subgrade. Specifications (or the cross-section 
drawings) formerly required that the subgrade should have a 
curved form, convex upward, or that it should slope outward 
from a slight ridge in the center, with the evident purpose of 
draining to the sides all water which might percolate through the 
ballast. If the subsoil were hard and impenetrable by the ballast, 
the method might answer, but experience has shown that, with 
ordinary subsoils, the ballast immediately under each rail is 
forced a Httle deeper into, the subsoil by the passage of each train. 
Periodical retamping of ballast under the ends of the ties, and 
Httle or no tamping under the center, only adds to the accumula- 
tion under each rail. A cross-section of a very old roadbed will 
frequently show twice as much depth of ballast under the rails 
as there is under the center. This method of tamping quickly 
obliterates the original line of demarcation between ballast and 
subsoil and any expected improvement in drainage due to sloping 
subsoil is not reaUzed. Therefore the A.R.E.A. specifications 
call ioT flat subgrades. 

94« Ditches. '^ The stability of the track depends upon the 
strength and permanence of the roadbed and structures upon 
which it rests; whatever will protect them from damage or pre- 
vent premature decay should be carefully observed. The worst 
enemy is water, and the further it can be kept away from the 
track, or the sooner it can be diverted from it, the better the 
track will be protected. Cold is damaging only by reason of 
the water which it freezes; therefore the first and most impor- 
tant provision for good track is drainage." (Rules of the Road 
Department, Illinois Central R. R.) 

The form of ditch generally prescribed has a flat bottom 12" 
to 24" wide and with sides having a minimum slope, except in 
rock-work, of 1 : 1, more generally 1.5 : 1 and sometimes 2:1. 
Sometimes the ditches are made V-shaped, which is objection- 
able unless the slopes are low The best form is evidently that 
which will cause the greatest flow 'for a given slope, and this 
will evidently be the form in which the 
ratio of area to wetted perimeter is the 
^ulD/////////r" largest. The semicircle fulfills this con- 

dition better than any other form, but the 
Fig. 45. , • 1 . , , , , ,.^ , 

nearly vertical sides would be difficult to 

maintain. (See Fig. 45.) A ditch, with a flat bottom and such 




110 RAILROAD CONSTRUCTION. ^ §95. 

slopes as the soil requires, which approximates to the circular 
form will therefore be the best. 

When the flow will probably be large and at times rapid it 
will be advisable to pave the ditches Avith stone, especially if the 
soil is easily washed away. Six-inch tile drains, placed 2' under 
the ditches, are prescribed on some roads. (See Fig, 46.) No 
better method could be devised to insure a dry subsoil. The 
ditches through cuts should be led off at the end of the cut so 
that the adjacent embankment will not be injured. 

Wherever there is danger that the drainage from the land 
above a cut will drain down into the cut, a ditch should be made 
near the edge of the cut to intercept this drainage, and this 
ditch should be continued, and paved if necessary, to a point 
where the outflow will be harmless. Neglect of these simple 
and inexpensive precautions frequently causes the soil to be 
loosened on the shoulders of the slopes during the progress of a 
heavy rain, and results in a landslide which will cost more to 
repair than the ditches which would have prevented it for all 
time. 

Ditches should be formed along the bases of embankments; 
they facilitate the drainage of water from the embankment, 
and may prevent a costly slip and disintegration of the em- 
bankment. 

95* Effect of sodding the slopes, etc. Engineers are unani- 
mously in favor of rounding off the shoulders and toes of em- 
bankments and slopes, sodding the slopes, paving the ditches, 
and providing tile drains for subsurface drainage, all to be put 
in during original construction. (See Fig. 46.) Some of the 
highest grade specifications call for the removal of the top layer 
of vegetable soil from cuts and from under proposed fills to 
some convenient place, from which it may be afterwards spread 
on the slopes, thus facilitating the formation of sod from grass- 
seed. But while engineer^ favor these measures and their 
economic value may be readily demonstrated, it is generally 
impossible to obtain the authorization of such specifications 
from railroad directors and promoters. The addition to the 
original cost of the roadbed is considerable, but is by no means 
as great as the capitalized value of the extra cost of mainte- 
nance resulting from the usual practice. Fig. 46 is a copy of 



^ 



§95. 



EARTHWORK. 



Ill 



designs * presented at a convention of the Ameiican Society of 
Civil Engineers by Mr. D. J. Whittemore, Past President of 
the Society and Chief Engineer of the Chi., Mil. & St. Paul 




'^-f^ PROPOSED SECTION OF ROADBED IN EXCAVATION. 




PROPOSED SECTION OF ROADBED ON EMBANKMENT. 




Fig. 46. — " Whittemore on Railway Excavation and Embankments" 
Traos. Am. See. C. E., Sept. 1894. 

R. R. The " customary sections " represent what is, with some 
variations of detail, the practice of many railroads. The '^ pro- 



* Trans. Am. Soc. Civil Eng., Sept. 1894. 



112 RAILROAD CONSTRUCTION. §96. 

posed sections'* elicited unanimous approval. They should be 
adopted when not prohibited by financial considerations. 

EARTHWORK SURVEYS. 

96. Relation of actual volume to the numerical result. It 

should be realized at the outset that the accuracy of the result 
of computations of the volume of any given mass of earthwork 
has but little relation to the accuracy of the mere numerical 
work. The process of obtaining the volume consists of two 
distinct parts. In the first place it is assumed that the volume 
of the earthwork may be represented b}^ a more or less com- 
plicated geometrical form, and then, secondly, the volume of 
such a geometrical form is computed. A desire for simplicity 
(or a frank willingness to accept approximate results) will often 
cause the cross-section men to assume that the volume may be 
represented by a very simple geometrical form which is really 
only a very rough approximation to the true volume. In such 
a case, it is only a waste of time to compute the volume with 
minute numerical accuracy. One of the first lessons to be 
learned is that economy of time and effort requires that the 
accuracy of the numerical work should be kept proportional to 
the accuracy of the cross-sectioning work, and also that the 
accuracy of both should be proportional to the use to be made 
of the results. The subject is discussed further in § 125. 

97. Prismoids. To compute the volume of earthwork, it is 
necessary to assume that it has some geometric form whose vol- 
ume is readily determinable. The general method is to consider 
the volume as consisting of a series of prismoids, which are 
solids having parallel plane ends and bounded by surfaces which 
may be formed by lines moving continuously along the edges of 
the bases These surfaces may also be considered as the sur- 
faces generated by lines moving along the edges joining the cor- 
responding points of the bases, these edges being the directrices, 
and the lines being always parallel to either base, which is a 
plane director. The surfaces thus developed may or may not 
be planes. The volume of such a prismoid is readily determin- 
able (as explained in § 110 et seq.), while its definition is so very 
general that it may be applied to very rough ground. The 
'^ two plane ends" are sections perpendicular to the axis of the 
road. The roadbed and side slopes (also plane) form three of 



§98. 



EARTHWORK. 



113 



the side surfaces. The only approximation lies in the degree of 
accuracy with which the plane (or warped) surfaces coincide with 
the actual surface of the ground between these two sections. 
This accuracy will depend (a) on the number of points which 
are taken in each cross-section and the accuracy with which the 
lines joining these points coincide with the actual cross-sections; 
(h) on the skill shown in selectiDg places for the cross -sections so 
that the warped surfaces shall coincide as nearly as possible with 
the surface of the ground. In fairly smooth country, cross- 
sections every 100 feet, placed at the even stations, are suf- 
ficiently accurate, and such a method simplifies the computations 
greatly; but in rough countr}^ cross-sections must be inter- 
polated as the surface demands. As will be explained later, 
carelessness or lack of judgment in cross-sectioning will introduce 
errors of such magnitude that all refinements in the computa- 
tions are utterly wasted. 

98. Cross-sectioning. The process of cross-sectioning con- 
sists in determining at an}^ place the intersection by a vertical 
plane of the prism of earth lying between the roadbed, the side 
slopes, and the natural surface. The intersection with the road- 




FiG. 47. 



bed and side slopes gives three straight lines. The intersection 
with the natural surface is in general an irregular line. On 
smooth regular ground or when approximate results are accept- 
able this line is assumed to be straight. According to the jrreg- 



114 RAILROAD CONSTRUCTION. § 99. 

ularity of the ground and the accuracy desired more and more 

^'intermediate points'' are taken. 

The distance (d in Fig. 47) of the roadbed below (or above) 
the natural surface at the center is known or determined from 
the profile or by the computed establishment of the grade line. 
The distances out from the center of all ^' breaks " are deter- 
mined with a tape. To determine the elevations for a cut, set 
up a level at any convenient point so that the line of sight is 
higher than any point of the cross-section, and take a rod read- 
ing on the center point. This rod reading added to d gives the 
height of the instrument (H. I.) above the roadbed. Sub- 
tracting from H. I. the rod reading at any "break" gives the 
height of that point above the roadbed (hi, ki, hrj etc.). This 
is true for all cases in excavation. For fill, the rod reading at 
center minus d equals the H. L, which may be positive or nega- 
tive. When negative, add to the "H. I." the rod readings of 
the intermediate points to get their depths below "grade"; 
when positive, subtract the "H. I." from the rod readings. 

The heights or depths of these intermediate points above or 
below grade need only be taken to the nearest tenth of a foot, 
and the distances out from the center will frequently be suffi- 
ciently exact when taken to the nearest foot. The roughness of 
the surface of farming land or woodland generally renders use- 
less any attempt to compute the volume with any greater accu- 
racy than these figures would imply unless the form of the ridges 
and hollows is especially well defined. The position of the slope- 
stake points is considered in the next section. Additional dis- 
cussion regarding cross-sectioning is found in § 107. 

99. Position of slope-stakes. The slope-stakes are set at the 
intersection of the required side slopes with the natural surface,; 
which depends on the center cut or fill (J). The distance of 
the slope-stake from the center for the lower side is x = ^h 
■\-s{d-{-y)\ for the up-hill side it is x' = \h-\-s{d—y'). s is the 
"slope ratio" for the side slopes, the ratio of horizontal to ver 
tical. In the above equation both x and y are unknown. There- 
fore some position must be found by trial which w^ill satisfy the 
equation. As a preliminary, the value of x for the point a = \h 
■\-sdj which is the value of x for level cross-sections. In the 
case of fills on sloping ground the value of x on the down-hill 
side is ^freaier than this; on the up-/i/ZZ side it is Z^ss. The differ- 
ence in distance is s times the difference of elevation. Take # 



§ 99. EARTHWORK. 115 

numerical case corresponding with Fig. 48. The rod reading 
on c is 2.9; (i=4.2: therefoie the telescope is 4.2—2.9 = 1.3 
helow grade. 6- = 1.5 : 1, 6 = 16. Hence for the point a (or for 
level ground) 0^ = ^X16 + 1.5X4.2 = 14.3. At a distance out 
of 14.3 the ground is seen to be about 3 feet lower, which will 
not only require 1.5X3=4.5 more, but enough additional dis- 
tance so that the added distance shall be 1.5 times the additional 
drop. As a first trial the rod ma}^ be held at 24 feet out and a 
reading of, say, 8.3 is obtained. 8.3 + 1.3=9.6, the depth of 
the point below grade. The point on the slope line (n) which 
has this depth below grade is at a distance from the center 



Fig. 48. 

a; =8 + 1.5X9.6 =22.4. The point on the surface (s) having 
that depth is 24 feet out. Therefore the true point im) is 
nearer the center. A second trial at 20.5 feet out gives a rod 
reading of, say, 7.1 or a depth of 8.4 below grade. This corre- 
sponds to a distance out of 20.6. Since the natural soil (espe- 
cially in farming lands or woods) is generally so rough that a 
difference of elevation of a tenth or so may be readily found hy 
slightly varying the location of the rod (even though the dis- 
tance from the center is the same), it is useless to attempt too 
much refinement, and so in a case like the above the combina- 
tion of 8.4 below grade and 20.6 out from center may be taken 
to indicate the proper position of the slope-stake. This is 
usually indicated in the form of a fraction, the distance out being 
the denominator and the height above (or below) grade being 
the numerator; the fact of cut or iill may be indicated by C or F* 
Ordinarily a second trial will be sufficient to determine with 
sufficient accuracy the true position of the slope-stake. Ex- 
perienced men will frequently estimate the required distance 



116 RAILROAD CONSTRUCTION. § 100. 

out to within a few tenths at the first trial. The left-hand pages 
of the note-book should have the station number, surface eleva- 
tion, grade elevation, center cut or fill, and rate of grade. The 
right-hand pages should be divided in the center and show the 
distances out and heights above grade of all points, as is illus- 
trated in § 84. The notes should read up the page, so that when 
looking ahead along the line the figures are in their proper 
relative position. The ^^ fractions" farthest from the center 
line represent the slope-stake points. 

100. Setting slope-stakes by means of " automatic " slope- 
stake rods. The equipment consists of a specially graduated tape 
and a specially constructed rod. The tape may readily be prepared 
by marking on the back side of an ordinary 50-foot tape which is 
graduated to feet and tenths. Mark "0" at " J6 '' from the tape- 
ring. Then graduate from the zero backward, at true scale, to 
the ring. Mark off *^feet" and ** tenths'^ on a scale propor- 
tionate to the slope ratio. For example, with the usual slope 
ratio of 1.5:1 each ''foot" would measure 18 inches and each 
''tenth'' in proportion. 

The rod, 10 feet long, is shod at each end and has an endless 
tape passing within the shoes at each end and over pullej^s — to 
reduce friction. The tape should be graduated in feet and 
tenths, from to 20 feet — the and 20 coinciding. By moving 
the tape so that is at the bottom of the rod — or (practically) 
so that the 1-foot mark on the tape is one foot above the bottom 
of the shoe, an index mark may be placed on the back of the 
rod (say at 15 — on the tape) and this readily indicates when the 
tape is "set at zero." 

The method of use may best be explained from the figure and 
from the explicit rules as stated. The proof is given for two 
assumed positions of the level. 

(1) Set up the level so that it is higher than the "center" 
and (if possible) higher than both slope-stakes, but not more 
than a rod-length higher. On very steep ground this may be 
impossible and each slope-stake must be set by separate positions 
of the level. 

(2) Set the rod-tape at zero (i.e., so that the 15-foot mark 
on the hack is at the index mark). 

(3) Hold the rod at the center-stake (B) and note the read- 
ing (ni or n2). Consider n to be always plus; consider d to be 
plus for cut and minus for fill. 



§100. 



EARTHWORK. 



117 



(4) Raise the tape on the face side of the rod (n + d). Apphed 
literally (and algebraically), when the level is helow the roadbed 
(only possible for fill), (n + c?) = (riz + {—df))=n2 —dp This being 
numerically negative, the tape is lowered {df—n^. With level 
at (1), for fill, {n + d)={n^ + {—df))^{n^—df)', this being positive, 
the tape is raised. With level at (1), for cut, the tape is raised 
(ni + dc). In every case the effect is the same as if the telescope 
were set at the elevation of the roadbed. 




Fig. 49. 



(5) With the special distance-tape, so held that its zero is J6 
from the center, carry the rod out until the rod reading equals 
the reading indicated by the tape. Since in cut the tape is 
raised {n-\-d), the zero of the rod-tape is always higher than the 
level (unless the rod is held at or below the elevation of the road- 
bed — which is only possible on side-hill work), and the reading 
at either slope-stake is necessarily negative. The reading for 
slope-stakes in fill is always positive. 

(6) Record the rod-tape reading as the numerator of a frac- 
tion and the actual distance out (read directly from the other 
side of the distance-tape) as the denominator of the fraction. 

Proof. Fill. Level at (i). Tape is raised (ji^—df). When 
rod is held at C/, the rod reading is -\-Xj which —Tfi — in^—df), 
But the reading on the back side of the distance-tape is also x. 

Fill. Level at (2). Tape is raised (ng— c?/), i.e., it is lowered 
{df—n^. When rod is held at C/, the rod reading is -{-x, which 
similarly = rf^ — iji-i—df) = r/2 + {df—n^. Distance-tape as be- 
f-^re. 



118 RAILROAD CONSTRUCTION. § 101. 

Cut Level at (i). Tape is raised (ni + ^c)- When rod is 
held at Cc the rod reading is— 2, which = rci — (n^ + dc), i.e., 
2J = (?ii + d^ — Tci. The distance-tape will read z. 

Side-hill work. It is easily demonstrated that the method, 
when followed literally, may be applied to side-hill work, al- 
though there is considerable chance for confusion and error, 
when, as is usual, J6 and the slope ratio are different for cut and 
forfni. 

The method appears complicated at first, but it becomes 
mechanical and a time-saver when thoroughly learned. The 
advantages are especially great when the ground is fairly level 
transversely, but decrease when the difference of elevation 
of the center and the slope-stake is more than the rod length. 
By setting the rod-tape ' ' at zero,'' the rod may always be used 
as an ordinary level rod and the regular method adopted, as in 
§ 99. Many engineers who have thoroughly tested these rods 
are enthusiastic in their praise as a time-saver. 

COMPUTATION OF VOLUME 

§ 1 01. Simple approximations. The principles developed in 
§§96 and 97 show that, except where the ground is abnormally 
smooth and level, the earthwork to be excavated has a geometrical 
form whose volume cannot be accurately computed by any simple 
rule. The usual method is to consider that the volume is approx- 
imately measured by the product of the mean of the areas of 
two consecutive sections and the distance between those sec- 
tions. When the ground is so regular that the error of such 
an approximation may be tolerated, or when only a rough approx- 
imation is necessary, such a computation may be accepted 
without correction. In any case, the " volume by averaging 
end areas " is computed as a first approximation and then 
correction is computed if desired. It should, therefore, be 
remembered that this approximate method, which is so common 
that it is often accepted without correction as the true volume, 
is never mathematically correct except under conditions which 
practically never exist. Whether a correction should be com- 
puted depends on the percentage of accuracy required, on the 
irregularity of the ground, and on the differences in the depth 
of adjacent center cuts — or fills. Experience gives the engineer 
such an idea of the probable amount of this correction under 



§ 102. EARTHWORK. 119 

any given conditions that he may judge when it is necessary to 
compute the correction in order to obtain the true volume with 
any desired degree of accuracy. The methods of computing 
this correction will be given later. 

102. Approximate voltime, level sections. When the coun- 
try is very level or when only approximate preliminary results 




Fig. 5o. 

are required, it is sometimes assumed that the cross-sections are 
level. The area of the cross-section may be written 

{a-hdys-j (46) 

in which a, b and d are dimensions as indicated by the figure and 
s is the " slope ratio '' or the ratio of the horizontal projection 
of the slope to the vertical. A table is very readily formed 
giving the area in square feet of a section of given depth and for 
any given width of roadbed and ratio of side slopes. Usually 
these tables give a number which equals that area times 100 and 
divided by 27, which is the volume in cubic yards of a prism 100 
feet long and with that cross-sectional area. Table XVII is 
such a table. 

The volume may also be readily determined (as illustrated in 
the following example), without the use of such a table; a table 
of squares will facilitate the work. Assuming the cross-sections 
at equal distances ( = I) apart, the total approximate volume for 
any distance will be 

-[Ao+2(Ai+A2+...A^_i)+AJ. .... (47) 

103. Numerical example : level sections. Given the following 
center heights for the same number of consecutive stations 100 
feet apart; width of roadbed 18 feet; slope IJ to 1. 



120 



RAILROAD CONSTRUCTION. 



§104, 



The products in the fifth column may be obtained very 
readily and with sufficient accuracy by the use of the slide-rule 
described in § 106. The products should be considered as 

(a-\-d){a-\-d) -^—. In this problem s = lj, — = .6667. To apply 
s s 

the rule to the first case above, place 6667 on scale B over 89 

on scale A, then opposite 89 on scale B will be found 118.8 on 

scale A . The position of the decimal point will be evident from 

an approximate mental solution of the problem. 



Sta. 


Center 
Height. 


a-^d 


(a+rf)2 


(a+dys 


Areas. 


17 
18 
19 
20 
21 
22 


2.9 
4.7 
6.8 
11.7 
4.2 
1.6 


8.9 
10.7 
12.8 
17.7 
10.2 

7.6 


79.21 
114.49 
163.84 
313.29 
104 . 04 

57.76 


118.81 
171.741 
245.76 1 
469.93 f 
156.06 J 
86.64 


118.81 

(343.48 

vo_ J 491.52 

^^-1939.86 

1312.12 

86.64 



ab 6X18 
2 ~ 2 



54 



10X54 



1752.43X100 
2X27 



= 3245 cub. yards 



2292.43 
: 540 

1752.43 
approx. vol. 



104. Equivalent sections. When sections are very irregular 
the following method may be used, especially if great accuracy 




Fig. 51. — Equivalent Section. 



is not required. The sections are plotted to scale and then a 
uniform slope line is obtained by stretching a thread so that the 
undulations are averaged and an equivalent section is obtained. 
Measure the distances {xi and Xr) from the center. The area 



§ 105. EARTHWORK. 121 

may then be obtained independent of the center depth as follows: 

h 
Let s = the slope ratio of the side slopes = cot /3 = — . (See Fig. 

50.) Then the 

l/xi-{-Xr\ . , , XrXr XiXi, db 



XiXr ah 



(48) 



These approximate methods are particularly useful for rapidly 
making up monthly estimates, realizing that the inaccuracies, 
plus and minus, will be wiped out when the final computation 
is made by a more accurate method. 

105. Three-level sections. The next method of cross-section- 
ing in the order of complexity, and therefore in the order of 



Fig. 52. 

accuracy, is the method of three-level sections. The area of 

ah 
the section is h{(J'-\-d){Wr-\-wi) , which may be written 

ah , , 

^(a-]-d)w , in which w = Wj.-\-wi. If the volume is com- 

Zi 

puted by averaging end areas, it will equal 

\a-\-d')w'-ah+{a+d'')w''-ah], . . • . (49) 



122 



RAILROAD CONSTRUCTION. 



§105, 



o 

a 
ft 



Oc3 I 






.2 t: 



I I 



3 



-^ r-l 

CO lO 

1—1 r-i 

I I 





iO 


GO 


<M 


05 




a> 


^ 


O 


■^ 




»o 


^ 


o 


Tf 


o 


t^ 


Tfi 


CI 


o 


r-l 


o 


00 


o 


l:^ 


<N 


^ 


1> 


CO 





tH 00 



^ 


cq 


fc. 


r-l 


fc. 


C^ 


^ 


1—1 


ft. 


oc; 


00 


'^ 


(M 


00 


Tt^ 


T-i 


o 


CM 


c. 


CO 


r-l 


rt* 


1— J 


(M 


1— ^ 


o 





Cc 


■35 


fe.jt^ 


fe, 


CO 


fe. 


o 


&H 


t^ 


t+-i 


CC 




00 • 


o\ 




o 




00 




01 




0^ 


■lO 




r^ 




or 




»o 


l-H 


o 


CM 


lO ICO 

»— 1 1 


c 




^ 


(M 


lO 


tH 



rH 


t^ 


I^ 


73 


05 


Tt< 


tH 


O 


o 




O 


^ 


(N 




CM 


-M 




It 


II 


^ 



cri 






CI 












O) 






73 


. . 






i> 




^ . 


■^ 




;t^ ^ 


II 




tH O 


31« 






CO 


^ 


I 03 


II 


o o 


-^|(N 




e 



§ 106. EARTHWORK. 123 

If we divide by 27 to reduce to cubic yards, we have, when 
Z = 100 

Vol (, . . . ..) =f|.(a+d')t(;'-ffa6+|f (a+d'0t^"-f|-a6. 

For the next section 

Vol (// . . /.,) =^{a+d'')w'' -^ah^^{a+d''')w''' -^ah. 

For a partial station length compute as usual and multiply 

, ' length in feet 

result by . 

^ 100 

The following example is given to illustrate the method of 

three-level sections. 

In the first column of yards 

210 = ff(a+(i)i/; = ffX7.3X31.1; 

507, 734, etc., are found similarly; 

595=210-61+507-61; 

448 = 3^«o(507-61+734-61); 

602 = 1^(734-61+392-61); 

449=392-61+179-61. 

The " F " in the columns of center heights, as well as the 

columns of " right " and ^^ left " are inserted to indicate fill for 

all those points. Cut would be indicated by ^' C" 

25 
io6. Computation of products. The quantities —{a-[-d)w 

^J 

25 
and — ob represent in each case the product of two variable 

terms and a constant. These products are sometimes obtained 
from tables which are calculated for all ordinary ranges of the 
variable terms as arguments. A similar table computed for 

25 

— {d' —d"){w" —w') will assist similarly in computing the 

ol 

prismoidal correction, see § 114. Prof. Charles L. Crandall, of 
Cornell University, is believed to be the first to prepare such a set 
of tables, which were first published in 1886 '' Tables for the 
Computation of Railway and Other Earthwork." Another 
easy method^ of obtaining these products is by the use of a shde- 
rule. Any slide-rule, from which may be read directly three 
significant figures and from which the fourth may be read by 
estimation, can be utihzed for this purpose. The Thacher or 



124 RAILROAD CONSTRUCTION. § 107. 

the Stanley cylindrical rules are still more accurate. - To illus- 
trate its use, suppose (a+c?) =28.2, and w = Q2A; then 

25, . ,^ 28.2X62.4 

— (a-\-d)w = . 

27 1.08 

Set 108 (which, being a constant of frequent use, may be specially 
marked) on the sliding scale (B) opposite 282 on the other scale 
(A), and then opposite 624 on scale B will be found 1629 on 
scale A, the 162 being read directly and the 9 read by estima- 
tion. Although strict rules may be followed for pointing off 
the final result, it only requires a very simple mental calculation 
to know that the result must be 1629 rather than 162.9 or 
16290. For products less than 1000 cubic yards the result 
may be read directly from the scale; for products between 1000 
and 5000 the result may be read directly to the nearest 10 
yards, and the tenths of a division estimated. Between 5000 and 
10,000 yards the result may be read directly to the nearest 20 
yards, and the fraction estimated; but prisms of such volume 
will never be found as simple triangular prisms — at least, an 
assumption that any mass of ground was as regular as this would 
probably involve more error than would occur from faulty esti- 
mation of fractional parts. Facihties for reading as high as 
10^000 cubic yards would not have been put on the scale except 
for the necessity of finding such products as ^{9.1X^-5), for 
example. This product would be read off from the same part 
of the rule as |-f-(9lX95). In the first case the product (80.0) 
could be read directly to the nearest .2 of a cubic yard, which 
is imnecessarily accurate. In the other case, the product 
(8004) could only be obtained by estimating -^ of a division. 

The computation for the prismoidal correction (see § 114), 
may be made similarly except that the divisor is 3.24 instead of 

5 5X11 7 
1.08. For example, ff-(5.5Xll.7) =-^ '- Set the 324 on 

scale B (also specially marked like 108) opposite 55 on scale A, 
and proceed as before. 

107. Approximate volume. Irregular sections. In cross- 
sectioning irregular sections, the distance from the center and 
the elevation above " grade ^' of every ^' break " in the cross- 
section must be observed. The area of the irregular section 
may be obtained by computing the area of the trapezoids {fivCf 
in Fig. 53) and subtracting the two external triangles. For Fig. 
53 the area would be 



'\ 



§107, 



EARTHWORK, 



125 



hi + ki. . , ki + d , d + jr , jr + JCr , ^ V 

_, kr + hr , .hi/ h\ hr[ h\ 




Fig. 53. 



Expanding this and collecting terms, of which many will 
cancel, we obtain 



Area 



4[ 



xiki+yiid—hi) +Xrkr+yr(jr—hr) 



■\-Zr(d'-kr)-\' (hl + hr) 



] 



(50), 



An examination of this formula will show a perfect regu- 
larity in its formation which will enable one to write out a 
similar formula for any section, no matter how irregular or how 
many points there are, without any of the preliminary work. 
The formula may be expressed in words as follows: ^ 

Area equals one-half the sum of products obtained as follows: 

the distance to each slope-stake times the height above grade of 
the point next inside the slope-stake; 

the distance to each intermediate point in turn times the height of 
the point just inside minus the height of the point just outside; 

finally^ one-half the width of the roadbed times the sum of the 
slope-stake heights, 



126 RAILROAD CONSTRUCTION. § 108. 

If one of the sides is perfectly regular from center to slope- 
stake, it is easy to show that the rule holds literally good. The 
" point next inside the slope-stake " in this case is the center; 
the intermediate terms for that side vanish. The last term 
must always be used. The rule holds good for three-level sec- 
tions, in which case there are three terms, which may be reduced 
to two. Since these two terms are both variable quantities for 
each cross-section, the special method, given in § 105, in which 
one term (Jab) is a constant for all sections, is preferable for 
three-level sections. In the general method, each intermediate 
" break " adds another term. 

io8. Volume of an irregular prismoid. This is obtained by 
computing first the approximate volume by " averaging end 
areas '' or by multiplying the length by the half sum of the end 
areas, as computed from Eq. (50). In other words, the Approx. 

volume = X— (area' + area''). But since each area equals 

^7 ^ 

one-half the sum of products of width times height (see Eq. (50)) 

we may say that 

25 
Approx. volume = — (summation of width times height) . (51) 

z I 

the terms of width times height being hke those found within 
the bracket of Eq. (50) . 

As before, for partial station lengths, multiply the result by 
(length in feet -i- 100). There will be no constant subtractive 

25 

term, — ah, as m § 105. 

Zt 

109. Numerical example; approximate volume; irregular 

sections. Assume the earthwork notes as given below where 

the roadbed is 18 feet wide in cut and the slope is 1 J to 1. Note 

that the stations read up the page and that when the surveyor 

is looking ahead- along the line the several combinations of heights 

and distances out have approximately the same relative position 

on the notebook as they have on the ground. For example, 

. . 8.9c 
beginning at the bottom line (Sta. 16), the combmation — - 

means that the extreme left-hand point of that section (the 
'^ slope-stake ") is 22.4 feet horizontally from the center and that 
it is 8.9 feet above the required roadbed. The cut (c) would he 
8.9 feet to reach the roadbed, but of course the actual cutting is 



Vv. 



§ 109. 



EARTHWORK. 



127 



zero at the slope stake. The next point is 12.0 feet horizontally 
from the center and 7.6 feet above the road})ed. The cut at 
the center is 6.8 feet. The combinations of dimensions on the 
right-hand side are to be interpreted similarly. 



Sta. 

19 

18 

17 

+ 42 

16 



( cut 

Centers or 

fill. 



0.6c 
2.3c 
7.6c 
10.2c 
6.8c 



3.6c 



27.3 

8.9c 
22.4 



Left. 



14.4 






4.2c 


6.8c 


3.2c 


15.3 


8.4 


5.2 


8.2c 


10.2c 


8.0c 


21.3 


17.4 


6.1 


12. 2c 




12 6c 



8.2 

7.6c 
12.0 



Right. 



0.1c 
4.2 



6.2c 



0.4c 
9.6 

1.2c 

10.8 

4.2c 
15.3 




12.9 



The numerical computation is greatly facilitated by a sys- 
tematic form as given below. For Sta. 16, the first term is 
''the distance to the left slope stake'' (22.4) times 'Hhe height 
above grade of the point next inside" (the height being 7.6), 
and we place this pair of figures in the columns of ''width" 
and "height." The "distance to the point next inside" is 
12.0 and the "height of the point just inside (6.8) minus the 
height of the point just outside" (8.9) equals ( — 2.1) and these 

25 



are the next pair of widths and heights. 



Taking ^ of the 



product of each pair of numbers we have the numbers in the 

first column of "yards.''' The sum of all these numbers in the 

42 
first and second groups multiplied by ---: (that section being 

only 42 feet long) equals 378 cubic yards, the volume by averag- 
ing end areas. The determination of center heights and total 
widths and the application of Eq. (54), to obtain the approxi- 
mate prismoidal correction (see § 114), is self-evident. 

no. Prismoidal correction. The foregoing methods of cal- 
culation have been called approximate, although under many 



128 



EAILROAD CONSTRUCTION. 



§110, 



VOLUME OF IRREGULAR PRISMOID, WITH APPROXIMATE PRISMOIDAL 

CORRECTION. 



Sta. 


W'th 


H'ght 


Yards. 


Cen. 
Height. 


Total 
width 


d'-d" 


w''-w' 


Approx. 
pris.corr. 




22.4 


7.6 


158 




+ 6.8 


35.3 










12.0 


-2.1 


-23 














16 


12.9 
4.1 
9.0 


3.2 

4.2 

11.5 


40 
16 
96 
















27.3 

8.2 


12.6 
-2.0 


319 
-15 




+ 10.2 


48.9 


-3.4 


+ 13.6 


-14 


+42 


21.6 
7.5 


6.2 

1.8 


124 
13 
















9.0 


20.6 


172 


378 








, 


(-6) 




21.3 


10.2 


201 




+ 7.6 


36.6 


+ 2.6 


-12.3 


-10 




17.4 


-0.2 


- 3 














17 


6.1 
15.3 


-2.6 
7.6 


-14 
107 
















9.0 


12.4 


103 


584 










(-6) 




15.3 


6.8 


95 




+ 2.3 


26.1 


+ 5.3 


-10.5 


-17 




8.4 


-1.0 


- 7 














18 


5.2 
10.8 


-4.5 
2.3 


-22 
23 
















9.0 


5.4 


45 


528 










(-17) 




14.4 


0.6 


8 




+ 0.6 


24.0 


+ 1.7 


-2.1 


-1 


19 


9.6 


0.1 


1 














4.2 


0.2 


1 
















9 


4.0 


33 


177 










(-1) 



Approx. volume =1667 
Approx. pris. corr. = —30 



30 



Corrected volume = 1637 cubic yards 



conditions such results are considered to be sufficiently accurate 
to serve as final. In any case the approximate result is first 
computed and then the " prismoidal correction " is computed 
if necessary. The mathematical necessity for a correction may 
be at once appreciated from the consideration that the volume 
of a prismoid having dissimilar and unequal ends is NOT equal 
to the length times the average of the end areas but is usually 
somewhat less. In an extreme case the correction is one-third 
of the approximate volume, or one-half of the true volume. The 
amount of the prismoidal correction for a triangular prism will 
be first determined and from that the correction for any kind of 
prism may be deduced. 

Let Fig. 54 represent a triangular prismoid. The two tri- 
angles forming the ends lie in parallel planes, but since the angles 
of one triangle are not equal to the corresponding angles of the 



§110. 



EARTHWORK. 



129 



other triangle, at least two of the surfaces must be warped. If 
a section, parallel to the bases, is made at any point at a dis- 




•——67 -* 

Fig. 54. 

tance x from one end, the area of the section will evidently be 

The volume of a section of infinitesimal length will be Axdx, 
and the total volume of the prismoid will be * 



< 



bihix+{lh-bi)hi^j^+bi{lh-hi)^ 



+ (62 -61) (A; 



-^'4][ 



1 . 

— 2l 



&iW+[(62-6i)/ii+&i(/i2-W]|+(62-6i)(fe-w|} 



=^[ibihi+ibih2+ih2hi+ib2h2] 



= -]Ai+iAm+A,], 
D 



(52) 



^ * Students unfamiliar with the Integral Calculus may take for granted the 

fundamental formulae that | dx=x, that | xdx = ^x\ and that | x^dx=^x^; 

also that in integrating between the limits of I and (zero), the value 
of the integral may be found by simply substituting I for x after 
integration. 



130 RAILROAD CONSTRUCTION. §111. 

in which Ai, A2, and A^i are the areas respectively of the two 
bases and of the middle section. Note that A,» is not the mean 
of Ai and A 2, although it does not necessarily differ very greatly 
from it. 

The above proof is absolutely independent of the values, ab- 
solute or relative, of h^, 62; ^1? or h2. For example, hi may be 
zero and the second base reduces to a line and the prismoid be- 
comes wedge-shaped; or 62 and hi may both vanish, the second 
base becoming a, point and the prismoid reduces to a pyramid. 
Since every prismoid (as defined in § 97) may be reduced to a 
combination of triangular prismoids, wedges, and pyramids, and 
since the formula is true for any one of them individually, it is 
true for all collectively ; therefore it may be stated that * 

The volume of a prismoid equals one sixth of the perpendicular 
distance between the bases multiplied by the sum of the areas of 
the two bases plus four times the area of the middle section. 

While it i^ always possible to compute the volume of any 
prismoid by the above method, it becomes an extremely compli- 
cated and tedious operation to compute the true value of the 
middle section if the end sections are complicated in form. It 
therefore becomes a simpler operation to compute volumes by 
approximate formulae and apply, if necessary, a correction. 
The most common methods are as follows : 

III. Correction for triangular prismoid. The volume of the 

triangular prismoid (Fig. 54), computed by averaging end areas, is 

I . , 

—[2^1/^1 + 2^2/12]. Subtracting this from the true volume (as 

given in the equation above Eq. 52), we obtain the correction 

-[(6i-62)(/i2-Ai)]. . .... . (53) 

This shows that if either the /i^s or 6's are equal, the correc- 
tion vanishes ; it also shows that if the bases are roughly similar 
and b varies roughly with h (which usually occiu-s, as will be 
seen later), the correction will be negative, which means that the 
method of averaging end areas usually gives too large results. 

If the " base " at one end vanishes to a point, making a trian- 

* The student should note that the derivation of equation (52) does not 
complete the proof, but that the statements in the following paragraph 
are logically necessary for a general proof. 



§ 112. EARTHWORK. 131 

gular pyramid, then 61 and hi each equal zero and the correction 
reduces to 

-[(-6.)(W] = -— . 

But the volume of a triangular prismoid is one-third of the alti- 
tude times the area of the base or JKi&2/i2) = 4^62/^2- The approx- 
imate volume, by averaging end areas, applying the rule strictly, 
is il(i^2h2-{-0) =ilb2h2. The correction is therefore one-third of 
the approximate volume, or one-haK of the true volume, in this 
extreme case. Therefore, when computing the volume of ter- 
minal pyramids and wedges (see § 89 and Fig. 43), by the method 
of averaging end areas, it must be remembered that, although 
the gross volume is comparatively small, the prismoidal correc- 
tion is relatively very large. 

112. Correction for level sections. Absolutely level sections 
are practically imknown, and the error involved in assuming any 
given sections as truly level will ordinarily be greater than the 
computed correction. If greater accuracy is required, more 
points should be obtained in the cross-sectioning, which will 
generally show that the sections are not truly level. But it 
may be easily computed that the correction equals 

12 a 

The squares of the differences of center depth of consecutive 
sections are always positive, regardless of whether the differences 
are positive or negative. Therefore the correction is always 
negative, showing that the method of averaging end areas, when 
the sections are level, always gives too large results. 

113. Prismoidal correction for " equivalent sections." It is 
a simple although tedious problem in mathematics to compute 
algebraically the true and approximate volumes of a prismoid 
when the areas are determined on the basis of " equivalent 
sections," § 104, and from thence to derive a formula for the 
prismoidal correction, but it is generally true that the errors 
due to such an approximate method of getting the area are so 
great that it is a needless refinement to compute the correction. 

114. Prismoidal correction for three-level sections. The 
prismoidal correction may be obtained by applying Eq. 53 to 
each side in turn. For the left side we have 



132 RAILROAD CONSTRUCTION. § 115. 

—[(a+d')-(a-{-dn]{wr-w/), which equals 

^id'-d")(.wi"-wi'). 
For the right side we have, similarly, 

. -^(,d'-d"){w/'-Wr'). 

The total correction therefore equals 

■^(d'-d") [{Wi" +Wr") - {wi' +Wr')\ 

^-^{d'-d"){w"-w'). 

Reduced to cubic yards, and with I — 100, 

Pris. Corr.=ff(d'-d'OK'-ti^'). • • . (54) 
Applying this formula to the numerical problem worked out in 
§ 105, the several values of (d' — d") and w" — w') are computed 
as given in the firsl two columns under Prismoidal Correction. 
Then, for example, 

-20 = ff(d'-d'0(^''-^0=ff(2.6-8.1)(42.8— 31.1) 
=ff(-5.5)(+11.7). 

For the next line, -3=3^[ff (-2.6)(+8.7)], and similarly 
for the rest. For this typical case j the correction is over 2 % of the 
volume and is, as usual, negative, or in other words, the approx- 
imate method, if used without correction, allows a contractor 
in this case 2% too much. 

115. Prismoidal correction; irregular sections. For reasons 
given in the next article, the correction is computed as if the 
sections were ^^ three-level " sections. This method was used 
in the numerical problem worked out in § 109. Instead of con- 
sidering the heights and widths of the separate triangles, the 
center height and total width for each section is recorded in two 
columns and the differences {d' —d") and {w" —w') are computed. 
(-3.4) X ( + 13.6) -r 3.24= -14, which would be the correction 
for a section 100 feet long. For 42 feet the correction is 42% 
of —14 or —6. Note that the total prismoidal correction for 
this stretch of 300 feet is negative, as is usual, and that it is a 
little less than 2%, about the same as the numerical problem of 
§ 105. 



§ 116. EARTHWORK. 133 

ii6. Magnitude of the probable error of this method. In 

previous editions of this work, methods were given for com- 
puting the mathematically exact volume of a prismoid whose 
ends coincide with the " irregular sections " as measured, and 
whose upper surfaces are assumed to coincide with the actual 
surface of the ground. As in the previous methods, the *' ap- 
proximate volunie" is computed by averaging end areas and 
then a correction is applied. If the end sections have the same 
number of intermediate points on each side, and if it can be as- 
sumed that the corresponding lines in each section are connected 
by plane or warped surfaces, which coincide with the surface of 
the ground, then the mathematically exact or *Hrue" correc- 
tion may be obtained by dividing the volume into elementary 
triangular prismoids, finding the correction for each and adding 
the results. Although such a method appears very complicated, 
it is readily possible to develop a law by means of which the 
true prismoidal correction may be written out (similarly to 
writing out the formula for the area, Eq. (50)) without any 
preliminary calculation. Such a law has a mathematical 
fascination, but it should be remembered that when the ground 
surface is so broken up that the cross-sections are '' irregular '' 
it is in general correspondingly rough and irregular between 
the cross-sections, especially when those sections are 100 feet 
apart. It is also true that the cross-sections do not usually 
have the same number of intermediate points on corresponding 
sides of the center. In such a case, unless the actual form of 
the ground between the cross-sections is observed and measured, 
the exact method cannot be used. An extra point in one cross- 
section implies an extra ridge (or hollow) which ^^ runs out " 
or disappears by the time the adjoining section is reached. 
Theoretically a cross-section should be taken at the point where 
such a ridge or hollow runs out. In general this point will not 
be at an even 100-foot station. The attempt to compute the 
exact prismoidal correction usually gives merely a false appear- 
ance of extreme accuracy to the work which is not justified 
by the results. It should not be forgotten that it is readily 
possible to spend an amount of time on the surveying and 
computing which is worth more than the few cubic yards of 
earth which represents the additional accuracy of the more 
precise method. The accuracy of the office computation should 
be kept proportionate to the accuracy of the cross-sectioning 



134 



EAILROAD CONSTRtJCTION. 



§117. 



in the field. The discussion of the magnitude of the prismoidal 
correction in §§ 110-115 shows that it is small except when the 
two ends of the prismoid are very dissimilar. The dissimilarity 
between the two ends of a prismoid tvould be substantially the 
same whether the ends were actually '^ irregular ''or had ''three- 
leveP' sections, which for each end had the same slope stakes 
and center heights as the irregular sections. Experience proves 
that the approximate prismoidal correction, computed by 
considering the ground as three-level, is so nearly equal to the 
true prismoidal correction that the difference is perhaps no 
greater than the probable difference between the true volume 
of earth and the volume of the geometrical prismoid which is 
assumed to represent that volume. The experienced surveyor 
will take his cross-sections at such places and so close together 
that the warped surfaces joining the sections will lie very nearly 
in the surface or at least will so average the errors that they 
will substantially neutralize each other. 

117. Numerical illustration of the accuracy of the approxi- 
mate rule. The " true '' prismoidal correction for the numerical 
case given in § 109 was computed by the method outlined above, 
and on the basis of certain figures as to the vanishing of the 
ridges and valleys found in one section and not found in the 
adjacent sections. The various quantities for the volumes 
between the cross-sections have been tabulated as shown. 





1 


2 


3 


4 


5 


6 


7 




-• M 







05.213 


c<i 


1— J Ui -4 


CO 


Sections. 


OX. vo 
eragin 
areas. 




13 
> 


ox. pri 
on has 
-ee-lev 
und. 


rl 


Approx. vo 

computed 

from cente 

and side 

heights onl] 


§"? 




Appr 

by ay 

end 


H^8 


2 


< a 


3 


3 


16 16 + 42 


378 


- 5 


373 


- 6 


-1 


396 


-23 


16+42.. 17 


584 


- 3 


581 


- 6 


-3 


577 


+ 4 


17 18 


528 


-16 


512 


-17 


-1 


463 


+ 49 


IS 19 


177 


- 3 


174 


- 1 


+2 


147 


+ 27 




1667 


-27 


1640 


-30 


-3 


1583 


+ 57 



There has also been shown in the last two columns the error 

involved if the '' intermediate points" had been ignored in 

the cross-sectioning. From the tabular form we may learn that 

1. The differences between the *Hrue" and approximate 



§ 118. EARTHWORK. 135 

corrections is so small that it is probably swallowed up by errors 
resulting from inaccurate cross-sectioning. 

2. The error which would have been involved in ignoring 
the intermediate points is so very large in comparison with 
the other corresponding errors that (although it proves nothing 
absolutely definite, being an individual case) the prohabilities 
of the relative error from these sources are clearly indicated. 

ii8. Cross-sectioning irregular sections. The slope stake 
should preferably be determined first, and then the "breaks*' 
between the slope stake and the center. When, as is usual, 
the ground is not even between the cross-section just taken 
and the section at the next 100-foot station, a point should be 
selected for a cross-section such that the lines to the previous 
section should coincide with the actual surface of the ground as 
closely as the accuracy of the work demands. § 125 gives 
a numerical illustration of the magnitude of some of these 
errors. Although it is possible for a skillful surveyor to so 
choose his cross-sections in rough and irregular ground that 
the positive and negative errors will nearly balance, it requires 
exceptional skill. Frequently the work may be simplified by 
computing separately the volume of a mound or pit, the 
existence of which has been ignored in the regular cross- 
sectioning, 

1 19. Side-hill work. When the natural slope cuts the roadbed 
there is a necessity for both cut and fill at the same cross-section. 



Fig. 55. 

When this occurs the cross-sections of both cut and fill are often 
so nearly triangular that they may be considered as such without 
great error, and the volumes may be computed separately as 
triangular prismoids without adopting the more elaborate form 



136 



RAILROAD CONSTRUCTION. 



§119. 



of computation so necessary for complicated irregular sections. 
When the ground is too irregular for this the best plan is to 
follow the uniform system. In computing the cut, as in Fig. 55, 
the left side would be as usual; there would be a small center 
cut and an ordinate of zero at a short distance to the right of the 
center. Then, ignoring the fill, and applying Eq. 56 strictly, 
we have two terms for the left side, one for the right, and the 
term involving J6, which will be ^hhi in this case, since hr = 0, 
and the equation becomes 

AxQ2^ = \[xiki-\-yi{d—hi)+Xrd+^hhi[. 

The area for fill may also be computed by a strict application 
of Eq. 50, but for Fig. 56 all distances for the left side are zero 
and the elevation for the first point out is zero, d also must be 




Fig. 56. 



considered as zero. Following the rule, § 107, literally, the equa- 
tion becomes 

Area(Fill) =h\.^rkr-\-yr{o—hr) +Zr{o—kr) + J?>(o + Ar)], 



which reduces to 



^[Xrkr—yrhr — Zrkr + ihhr]. 



(Note that Xr, hr, etc., have different significations and values 
in this and in the preceding paragraphs.) The '' terminal 
pyramids" illustrated in Fig. 43 are instances of side-hill work 
for very short distances. Since side-hill work always implies 
both cut and fill at the same cross-section, whenever either the 
cut or fill disappears and the earthwork becomes wholly cut or 
wholly fill, that point marks the end of the ''side-hill work," 
and a cross-section should be taken at this point. 



§ 120. EARTHWORK. 137 

120. Borrow-pits. The cross-sections of borrow-pits will vary 
not only on account of the undulations of the surface of the 




mmrnm 

Fig. 57. 

ground, but also on the sides, according to whether they are 
made by widening a convenient cut (as illustrated in Fig. 57) 
or simply by digging a pit. The sides should always be prop- 
Trly sloped and the cutting made cleanly, so as to avoid un- 
sightly roughness. If the slope ratio on the right-hand side 
(Fig. 57) is 5, the area of the triangle is ^sm^. The area of the 
section is i[ug-i-{g-\-h)v-{-{h-i-j)x + {j-\-k)y + {k+m)z — sm^]. If 
all the horizontal measurements were referred to one side as 
an origin, a formula similar to Eq. 50 could readily be devel- 
oped, but little or no advantage would be gained on account of 
any simphcity of computation. Since the exact volume of the 
earth borrowed is frequently necessary, the prismoidal correc- 
tion should be computed; and since such a section as Fig. 57 
does not even approximate to a three-level section, the method 
suggested in § 108 cannot be employed. It will then be neces- 
sary to employ the more exact method of dividing the volume 
into triangular prismoids and taking the summation of their 
correction, found according to the general method of § 110. 

121. Correction for curvature. The volume of a sohd, gen- 
erated by revolving a plane area about an axis lying in the 
plane but outside of the area, equals the product of the given 
area times the length of the path of the center of gravity of the 
area. If the centers of gravity of all cross-sections lie in the 
center of the road, where the length of the road is measured, 
there is absolutely no necessary correction for curvature. If all 
the cross-sections in any given length were exactly the same and 
therefore had the same eccentricity, the correctien for curvature 
would be very readily computed according to the above prin- 
ciple. But when both the areas and the eccentricities vary 
from point to point, as is generally the case, a theoretically exact 



138 KAILROAD CONSTRUCTION. § 121. 

solution is quite complex, both in its derivation and application. 
Suppose, for simplicity, a curved section of the road, of uniform 
cross-sections and with the center of gravity of every cross- 
section at the same distance e from the center line of the road 
The length of the path of the center of gravity will be to the 
length of the center line as R±e : R. Therefore we have 

R±e 
True vol.: nominal vol. :: R±e : R. .*. True vol.=lA for 

K 

a volume of uniform area and eccentricity. For any other area 

R±e^ 
and eccentricity we have, similarly, True vol/ =IA' — ^— . This 

It 

shows that the effect of curvature is the same as increasing (or 

diminishing) the area by a quantity depending on the area and 

eccentricity, the increased (or diminished) area being found by 

R±e 
multiplying the actual area by the ratio —^—. This being 

independent of the value of I, it is true for infinitesimal lengths. 
If the eccentricity is assumed to vary uniformly between tw^o 
sections, the equivalent area of a cross-section located midway 

between the two end cross-sections would be Am- ^ ~ o 

Therefore the volume of a solid which, when straight, would be 
—(A^ + 4:Am + A"), would then become 

Truevol.=~rA'(R±e')+4Am(R±^-^^^ 

Subtracting the nominal volume (the true volume when the 
prismoid is straight), the 



Correction =±~\ (A' + 2Ant)e' + (2A^+A'0e'' L . (55) 



6R 



Another demonstration of the same result is given by Prof. 
C. L. Crandall in his ^'Tables for the Computation of Railway 
and other Earthwork," in which is obtained by calculus methods 
the summation of elementary volumes having variable areas 
with variable eccentricities. The exact application of Eq. 55 
requires that Am be known, which requires laborious computa- 



§ 122. EARTHWORK. 139 

tions, but no error worth considering is involved if the equation 
is written approximately 

Ci^n;.corr.=;r^(AV + A'VO, .... (56) 

which is the equation generally used. The approximation con- 
sists in assuming that the difference between A' and A^ equals 
the difference between Am and A" but with opposite sign. The 
error due to the approximation is always utterly insignificant. 
122. Eccentricity of the center of gravity. The determination 
of the true positions of the centers of gravity of a long series of 
irregular cross-sections would be a very laborious operation, 
but fortunately it is generally sufficiently accurate to consider 
the cross-sections as three-level ground, or, for side-hill work, to 



Xr 







Fig. 58. 



be triangular, ]or the purpose of this correction. The eccentricity 
of the cross-section of Fig. 58 (including the grade triangle) may 
be written 



(a-\-d)xiXi (a-\-d)XrXr 



O i^ Xj — Xr 1 , . /r^\ 

= -^(Xi-Xr). . (57) 



(a-\-d)xi (a-^d)xr 3 Xi + Xr S 



The side toward xi being considered positive in the above 
demonstration, if Xr>xi, e would be negative, i.e., the center 
of gravity would be on the right side. Therefore, for three-level 



140 RMLROAD CONSTRUCTION. § 122. 

ground, the correction for curvature (see Eq. 56) may be written 

Correction = J^[A\xi' -Xr') +A"{xr -a:/')], 
oil 

Since the approximate volume of the prismoid is 
1{A+A')=^A'+^A" = V' + V", 

in which F' and F" represent the number of cubic yards corre- 
sponding to the area at each station, we may write 

Corr, in cub. yds. = :^[V'ixi' -x/) + Y'\x{' '-Xr")\ (58) 

It should be noted that the value of e, derived in Eq. 57, is 
the eccentricity of the whole area including the triangle under 
the roadbed. The eccentricity of the true area is greater than 
this and equals 

true area + \db 

eX — 7 ^- — ^— =^1. 

true area 

The required quantity {A'e' of Eq. 56) equals true areaXei 
which equals {true area-^iab)Xe. Since the value of e is very 
simple, while the value of Ci would, in general, be a complex 
quantity, it is easier to use the simple value of Eq. 57 and add 
iab to the area. Therefore, in the case of three-level ground 
the subtractive term ^ah (§ 105) should not be subtracted in 
computing this correction. For irregular ground, when com- 
puted by the method given in §§ 107 and 108, which does not 
involve the grade triangle, a term ^ah must be added at every 
station when computing the quantities F' and F" for Eq. 58. 

It should be noted that the factor l-=-3i^, which is constant 
for the length of the curve, may be computed with all necessary 
accuracy and without resorting to tables by remembering that 

5730 
K 



degree of curve 



Since it is useless to attempt the computation of railroad 
earthwork closer than the nearest cubic yard, it will frequently 



§ 122. ^ EARTHWORK. 141 

be possible to write out all curvature corrections by a simple 

mental process upon a mere inspection of the computation sheet. 

Eq. 58 shows that the correction for each station is of the form 

V(xi—x ) 

_p ■ . 3R is generally a large quantity — for a 6° curve 

it is 2865. (xi—Xr) is generally small. It may frequently be 
seen by inspection that the product V(xi—Xr) is roughly twice 
or three times 3R, or perhaps less than half of SR, so that the 
corrective term for that station may be written 2, 3, or cubic 
yards, the fraction being disregarded. For much larger absolute 
amounts the correction must be computed with a correspondingly 
closer percentage of accuracy. 

The algebraic sign of the curvature correction is best deter- 
mined by noting that the center of gravity of the cross-section is 
on the right or left side of the center according as a: r is greater 
or less than xi, and that the correction is positive if the center of 
gravity is on the outside of the curve, and negative if on the 
inside. 

It is frequently found that xi is uniformly greater (or uni- 
Cormly less) than Xr throughout the length of the curve. Then 
the curvature correction for each station is uniformly positive or 
negative. But in irregular ground the center of gravity is apt 




Fig. 59. 

to be irregularly on the outside or on the inside of the curve^ 
and the curvature correction will be correspondingly positive or 
negative. If the curve is to the rights the correction will be 
positive or negative according as (xi—Xr) is positive or negative; 
if the curve is to the left, the correction will be positive or nega- 



142 RAILROAD CONSTRUCTION. § 123. 

tive according as (xr—xi) is positive or negative. Therefore 
when computing curves to the right use the form (xi — Xr) in 
Eqs. 58 and 60; when computing curves to the left use the form 
(xr—xi) in these equations; the algebraic sign of the correction 
will then be strictly in accordance with the results thus obtained. 
123. Center of gravity of side-hill sections. In computing the 
correction for side-hill work the cross-section would be treated 
as triangular unless the error involved would evidently be too. 
great to be disregarded. The center of gravity of the triangle 
lies on the line joining the vertex with the middle of the base 
and at J of the length of this line from the base. It is therefore 
equal to the distance from the center to the foot of this line plus 
J of its horizontal projection. Therefore 

*=[2-2(2+^'-)j+i[^'-(|-|(2+^^))J 

h Xr XI b Xr 
h XI Xr 

4[| + (..-.)]. . .......... (59) 

By the same process as that used in § 122 the correction equation 
may be written 

Corr. in cub. yds. = 3^[^'(|4 (.-/ - x/') + W^(^+(xi" -x/') )]. (60) 

It should be noted that since the grade triangle is not used in 
this computation the volume of the grade prism is not involved 
in computing the quantities F' and V. 

The eccentricities of cross-sections in side-hill work are never 
zero, and are frequently quite large. The total volume is gen- 
erally quite small. It follows that the correction for curvature 
is generally a vastly larger proportion of the total volimae than 
in ordinary three-level or irregular sections. 

If the triangle is wholly to one side of the center, Eq. 59 can 
still be used. For example, to compute the eccentricity of the 
triangle of fill, Fig. 59, denote the two distances to the slope- 



§ 124. EARTHWORK. 143 



stakes by yr and —yi (note the minus sign). Applying Eq. 5^ 

literally (noting that — must here be considered as negative in 

J/ 

order to make the notation consistent) we obtain 



which reduces to 



= 4[^ + 2/.+2/.] (61) 



As the algebraic signs tend to create confusion in these 
formulae, it is more simple to remember that for a triangle 
lying on hoth sides of the center e is always numerically equal 

to--i ~^{xi^x^ L and for a triangle entirely on one side, e is 

1 r^ 

numerically equal to— ^+ the numerical sum of the two dis- 
tances out]. The algebraic sign of e is readily determinable as 

in § 122. 

124. Example of curvature correction. Assume that the fill in 

§ 105 occurred on a 6° curve to the right. -— = . The 

3R 2865 

quantities 210, 507, etc., represent the quantities V\ F", etc,^ 

since they include in each case the 61 cubic yards due to the 

grade prism. Then 



V(xir^xr) _ 210(22.9-8.2) ^ 3101.7 ^ 
3R " 2865 2865 "^ "^ 



The sign is plus, since the center of gravity of the cross-sec- 
tion is on the left side of the center and the road curves to the 
right, thus making the true volume larger. For Sta. 18 the 
correction, computed similarly, is +3, and the correction for 
the whole section is 1+3=4. For Sta. 18 + 40 the correction 
! is computed as 6 yards. Therefore, for the 40 feet, the correc- 
tion is tVo(3 + ^) =3.6, which is called 4. Computing the others 
similarly we obtain a total correction of + 16 cubic yards. 



144 RAILROAD CONSTRUCTION. § 125. 

125. Accuracy of earthwork computations. The preceding 

methods give the precise volume (except where approximations 
are distinctly admitted) of the prismoids which are supposed to 
represent the volume of the earthwork. To appreciate the 
accuracy necessary in cross-sectioning to obtain a given accuracy 
in volume, consider that a fifteen-foot length of the cross-section, 
which is assumed to be straight, really sags 0.1 foot, so that the 
cross-section is in error by a triangle 15 feet wide and 0.1 foot 
high. This sag 0.1 foot high would hardly be detected by the 
eye, but in a length of 100 feet in each direction it would make 
an error of volume of 1.4 cubic yards in each of the two pris- 
moids, assuming that the sections at the other ends were perfect. 
If the cross-sections at both ends of a prismoid were in error by 
this same amount, the volume of that prismoid would be in error 
by 2.8 cubic yards if the errors of area were both plus or both 
minus. If one were plus and one minus, the errors would 
neutralize each other, and it is the compensating character of 
these errors which permits any confidence in the results as 
obtained by the usual methods of cross-sectioning. It demon- 
strates the utter futility of attempting any closer accuracy than 
the nearest cubic yard. It will thus be seen that if an error 
really exists at any cross-section it involves the prismoids on 
both sides of the section, even though all the other cross-sections 
are perfect. As a further illustration, suppose that cross-sec- 
tions were taken by the three-level method (§ 105), and that a 
cross-section, assumed as uniform from center to side, sags 0.4 
foot in a width of 20 feet. Assume an equal error (of same 
sign) at the other end of a 100-foot section. The error of 
volume for that one prismoid is 38 cubic yards, 

The computations further assume that the warped surface, ^ 
passing through the end sections, coincides with the surface of 
the ground. Suppose that the cross-sectioning had been done 
with mathematical perfection; and, to assume a simple case, 
suppose a sag of 0.5 foot between the sections, which causes an 
error equal to the volume of a pyramid having a base of 20 feet 
(in each cross-section) times 100 feet (between the cross-sec- 
tions) and a height of 0.5 foot. The volume of this pyramid is 
K20X100)X0.5=333 cub. ft. = 12 cub. yds. And yet this sag 
or hump of 6 inches would generally be utterly unnoticed, or 
at least disregarded. 

When the ground is very rough and broken it is sometimes 



I 



§126. 



EARTHWOBK. 



145 



I 



practically impossible, even with frequent cross-sections, to 
locate warped surfaces which will closely coincide with all the 
sudden irregularities of the ground. In such cases the compu- 
tations are necessarily more or less approximate and dependence 
must be placed on the compensating character of the errors. 

126. Approximate computations from profiles. When a 
"paper location" has been laid out on a topographical map 
having contours, it is possible to compute approximately the 
amount of earthwork required by some very simple and rapid 
calculations. A profile may be readily drawn by noting the 
intersections of the proposed center line with the various con- 
tours and plotting the surface line on profile paper. Drawing 
the grade-line on the profile, the depth of cut or fill may be 
scaled off at any point. When it is only desired to obtain 




Fig. 60. 



very quickly an approximate estimate of the amount of earth- 
work required on a suggested line, it may be done by the method 
described in § 103, or by the use of Table XVII. But the 
assumption that the surface of the ground at each cross-section 
is level invariably has the effect that the estimated volumes 
are not as large as those actually required. The difference 
between the ^4evel section" hkms and the actual slope section 
hknq equals the difference between the triangles mon and oqs, 
and this difference equals the shaded area mpn. The excess 
volume is proportional to the area of the triangle mpn. This 
area may be expressed by the formula, 

A 0/17 , 7 . r)N2sin^« sin/?cos>5 

Area mpn = 2(ib + d cot S) 2 ^ — -^. 

^ '^ cos 2a — cos 2^ 



146 RAILROAD CONSTRUCTION. § 126. 

The percentage of this excess area to the nominal area Jikms 
therefore depends on the dimensions h and d and the angles a 
and /3. A solution of this equation for ninety different com- 
binations of various numerical values for these four variables 
is included in Table XVII for the purpose of making cor- 
rections. A study of this correction table points conclusively 
to the following laws, a thorough understanding of which will 
enable an engineer to appreciate the degree of accuracy which 
is attainable by this approximate method: 

(a) Increasing the width of the roadbed (6), the other three 
factors remaining constant, increases the percentage of error, 
but the increase is comparatively small. 

(h) Increasing the depth of cut or fill ((i), decreases the per- 
centage of error, but the decrease is almost insignificant. 

(c) Increasing the angle of the side slopes (^) decreases the 
percentage of error, the decrease being very considerable. 

(d) Increasing the angle of the slope of the ground (a), 
increases the percentage of error, the percentage rapidly in- 
creasing to infinity as the value of a approaches that of /?. 
This is another method of stating the fact that a must always 
be less than ^ and, practically, must be considerably less, so 
that the slope stake shall be within a reasonable distance from 
the center. 

Since the above value for the corrective area is a function of 
the angle a, which is usually variable and whose value is fre- 
quently known only approximately, it is useless to attempt 
to apply the correction with great precision, and the following 
rules will usually be found amply accurate, considering the 
probable lack of precision in the data used. 

1. For embankments or cuts, having a slope of 1.5:1, and 
with a surface slope of 5° (nearly 9%) the excess of true area 
over nominal area is about 2%. There is only a slight varia- 
tion from this value for all ordinary depths (d) and widths (b) 
of roadbed. Therefore the nominal volume would be about 2% 
too small. On the other hand, the effect of the prismoidal 
correction is such that, even with truly level sections, the 
nominal volume is too large. See §§ 103 and 104. The amount 
of the prismoidal correction depends on the differences between 
successive center depths. In the very ordinary numerical 
case given in § 104, the correction was nearly 3%, which more 
than neutralizes the error due to surface slope. Therefore in 



§ 126. EARTHWORK. 147 

many cases on slightly sloping ground the error due to the 
surface slope will so nearly neutralize the prismoidal correc- 
tion that the quantities taken directly from the tables (without 
correction for either cause) will equal the true volume with as 
close an approach to accuracy as the precision of the surveying 
will permit. 

2. For a cut with a slope of 1:1, and with a surface slope of 
5° the error is about 1%. This will be neutralized by still 
smaller prismoidal corrections. Therefore, for surface slopes 
of 5° or less, no allowance should be made for this error unless 
the prismoidal correction is also considered. 

3. When the surface slope is 10^ (nearly 18%) the error for 
a 1.5:1 slope is from 7% to 10% and for a 1:1 slope from 3% 
to 5%. 

4. For a 30° surface slope and 1.5:1 side slopes the excess 
volume is three or four times the nominal volume. Such a 
steep surface slope implies the probability of ''side-hill work'* 
to which the above corrective rules are not applicable. When 
the surface slopes are very steep careful work must be [done 
to avoid excessive error. For a 1:1 side slope, the errors are 
from 50% to 80%. 

A still closer approximation, especially for the steeper surface 
slopes, may be obtained by using, directly or by interpolation, 
figures from the corrective tabular form which forms part of 
Table XVII. Unless the surface slope angle is known accurately 
(especially when large) no great accuracy in the final result is 
possible. Close accuracy would also require the determination 
of the prismoidal correction. But if such close accuracy is 
deemed essential, it can be most easily obtained by ac- 
curate cross-sectioning at each station and the adoption of 
other methods of computation — such as are given in §§108 
and 109, 

When the contours have been drawn in for a sufiicient 
distance on either side to include the position of both slope 
stakes at every station, as will usually be the case, cross-sections 
may be obtained by drawing lines on the map at each station 
perpendicular to the center line — see Fig. 4. The intersection 
of these lines with the contours will furnish the distances for 
drawing on cross-section paper the transverse profile at each 
station. Drawing on the same cross-section the lines repre- 
senting the roadbed and the side slopes, the cross-section of 



148 RAILROAD CONSTRUCTION. § 126. 

cut (or fill) is complete and its area may be obtained by scaling 
from the cross-section paper. If the contours have been 
located on the map with sufficient accuracy, such a method 
Will determine the cross-sectional area very closely. When 
cross-sections have been taken with a wye- or hand-level, as 
described in § 12, the cross-sections as plotted will probably 
be more accurate than when the contours are run in from 
points determined by the stadia method. In fact this semi- 
graphical method is frequently used, in place of the purely 
numerical methods described in previous sections, to make 
final estimates of the volume of earthwork. 

As a numerical example, an assumed location line was laid 
out on the contours given in Fig. 4. The volume of cut, as 
determined by Table XVII for a roadbed 20 feet wide, with 
side slopes of 1:1, was 5746 cubic yards. The surface slope 
varied from 3° to 11°. Computing the corrections by a careful 
interpolation from the corrective table, the total correction was 
found to be 128 cubic yards, or an average of a little over 2%. 
On the other hand the negative prismoidal correction amounts 
to 72 cubic yards, which leaves a net correction of 56 cubic 
yards — about 1%. It so happens that in this case a correction 
for curvature would tend further to wipe out this correction. 
These figures merely verify numerically the general conclusions 
stated above, although it should not be forgotten that in indi- 
vidual cases the figures taken from Table XVII require ample 
correction. 

The following approximate rule, for which the author is 
indebted to Mr. W. H. Edinger, is exceedingly useful when it 
is desired to rapidly determine the approximate volume of 
earthwork between two points along the road. Its great merit 
lies in the fact that it only means the memorizing of a com- 
paratively simple rule which will make it possible to make 
such computations in the field, without the use of tables. The 
rule is based on the fact that the area of any level section equals 
bd-hsd^; and therefore, 

S(vol.) = (6Sd+sSd2)^, 

in which L is usually 100 feet. For strict accuracy this would 
dnly be the volume provided the total length was an even num- 
ber of hundred feet, and the various values of d represented 



§ 127. EARTHWORK. 149 

the depths which were iiniform for hundred foot sections. It 
makes no allowance for the comparatively large prismoidal 
error of the pyramidal and wedge-shaped sections usiaally found 
at each end of a cut or fill, but where an approximate estimate 
is desired, in which this inaccuracy may be neglected, the 
method is very useful. The method of applying this rule with- 
out tables may best be illustrated by a simple numerical exr 
ample. Assume that the levels on a stretch of fairly level 
ground, which is about 500 feet long, have been taken, the depths 
being taken at points 100 feet apart, the first and last points 
being about 40 or 50 feet from the ends of the cut, or fill. The 
depths are as given in the first column in the tabular form 
below; the slope is 1.5:1, and the breadth (Jb) is 14 feet. 



d 




d^ 


1.6 




2.56 


2.8 




7.84 


4.5 




20.25 


3.1 




9.61 


0.9 




.81 


Zd = 12.9 


ScZ2 


= 41.07 


14 




20.53 


62^ = 180.6 


sSd2 


= 61.60 


61.60 






242.2 






24220^27 = 


= 897 cubic yarc 


is. 



The 180.6 is the hXd and the 61.6 is s2d2; adding these and 
moving the decimal point two places to multiply by 100, we 
only have to divide by 27 to obtain the value in cubic yards. 
Although the above rule requires more work than the employ- 
ment of earthwork tables, yet it is a very convenient method 
of estimating the approximate volume of a short section of 
earthwork when no tables are at hand. 

FORMATION OF EMBANKMENTS. 

127. Shrinkage of earthwork. The statistical data indicating 
the amount of shrinkage is very conflicting, a fact which is 
probably due to the following causes: 

1. The various kinds of earthy material act very differently 
as respects shrinkage. There is a great lack of uniformity in 



150 RAILROAD CONSTRUCTION. § 127. 

the classification of earths in the tests and experiments which 
have been made. 

2. Very much depends on the method of forming an embank- 
ment (as will be shown later). Different reports have been 
based on different methods — often without mention of the 
method. 

3. An embankment requires considerable time to shrink to 
its final volume, and therefore much depends on the time 
elapsed between construction and the measurement of what is 
supposed to be the settled volume. 

4. A soft subsoil will frequently settle under the weight of a 
high embankment and apparently indicate a far greater shrink- 
age than the actual reduction in volume. 

5. An embankment of very soft material will sometimes 
"mush^' or widen at the sides, with a consequent settling of 
the top, due to this cause alone. 

This subject has called forth much discussion in the technical 
press and literature. Quotations can be made of figures cover- 
ing a large range of values, but space will only permit the 
statement of the conclusions which may be drawn from the 
large mass of testimony which has been presented. 

1. Volume of loose material. When material of any character 
is excavated and deposited loosely in a pile, its volume is 
always largely in excess of the volume of the excavation. 
Solid rock will occupy from 60% to 80% more space when 
broken up than when solid. A soft earth will have an excess 
volume of about 20% to 25%. 

2. Effect of method of depositing. When material is de- 
posited loosely, as from a trestle, the excess of volume when 
the embankment is just completed is very large. The time 
required for final settlement is also very great. .When an 
embankment is formed by the wheelbarrow method, the initial 
expansion is about as great as when the material is merely 
dumped from cars. When the material is deposited in small 
increments from wagons and each layer is subjected to com- 
pression from horses' hoofs and from wheels, the contraction 
during construction is far greater and the additional shrinkage 
is comparatively small. Wheeled scrapers and drag scrapers 
will produce even more initial compression. 

3. Time required for final settlement. This depends partly 
on the method of formation and also on the character of the 



§ 128. EARTHWORK. 151 

material. When a soft loamy soil is deposited loosely, the dry- 
ing out of the soil during the first long dry season will develop 
large cracks. Subsequent rains will close these cracks by a 
general contraction of the whole mass. When the embank- 
ment is loosely formed it may take two years before additional 
settlement becomes inappreciable, but when the method of 
deposition ensures compression during construction the subse" 
quent shrinkage is less in time as well as amount. 

4. Classification of soils with respect to shrinkage. Loose 
vegetable surface soil will expand very greatly when excavated 
and first deposited, but will subsequently shrink to considerably 
less than its original volume. Clay soils are next in order 
and the sandy and gravelly soils come at the other end of 
the list of earthy materials. Rock expands very greatly when 
first broken up and deposited and there is no appreciable sub- 
sequent shrinkage. 

128. Proper allowance for shrinkage. Specifications for the 
Mississippi River levees require that there shall be a 10% 
shrinkage allowance for embankments formed by team work 
and 25% allowance for wheelbarrow work. It is contended 



A- N ^ 






Fig. 61. 

that such figures are only justified because the subsoil settles 
or because the embankments mush out at the sides, and that 
if these effects do not occur the levees are permanently higher 
than designed. 

It is usual to require that embankments shall be constructed 
higher than their desired ultimate, as shown in Fig. 61. Since 
the base does not contract, the contraction may be said to 
be all vertical. Since a high embankment will unquestionably 
shrink a greater total amount than a low embankment (what- 
ever the percentage), it follows that an embankment having 



L-- 



152 HAlLrOAD CONSTRUCTION. § 128. 

variable heights (as usual) should have an initial grade-line 
-somewhat hke the dotted hne adc in Fig. 62. Although some 
such method is essential if there is to be no ultimate sag below 
the desired grade-line, the policy is sharply criticized. The 
grade ad^ even though temporary, may prove objectionable 
from an operating standpoint. Frequently the allowance is 
made too great or the shrinkage is not as much as anticipated, 
and it becomes necessary to cut off the top of the bank. On 
the other hand, the expense of raising the track after the road 
is in operation and the inevitable loss of ballast is so great 
that the danger of being required to fill up a sag should be 
avoided if possible* 



^ A sharp and clear distinction should be made between the 
coefficient of extra height of an embankment and the coefficient 
of shrinkage which determines how many cubic yards of settled 
embankment may be made from a definite volume of earth or 
rock measured in the excavation. The values quoted above 
for the Mississippi levees (from 10% to 25%) refers usuallji 
to a very soft soil and includes the effects other than actual 
contraction of volume. From 8% to 15% is usually quoted as 
the required extra height of embankments, although it is 
strenuously claimed by many that 3% or 2% is sufficient, 
©r even that no allowance should be made. 

The coefficients to determine the amount of settled embank- 
ment which may be made from a given volume of earth or 
rock measured in the excavation, are necessarily subject to 
variation on account of the method employed and the amount 
of compression and settlement which will take place during 
the progress of the work. The following figures have the 
weight of considerable authority but, if in error, the coefficients 
are probably high rather than low: 



§129. 



EARTHWORK. 



153 



Gravel or sand about ayo 

Clay '' 10% 

Loam " 12% 

Loose vegetable surface soil ** 15% 



It may be noticed from the above table that the harder and 
cleaner the material the less is the contraction. Perfectly clean 
gravel or sand would not probably change volume appreciably. 
The above coefficients of shrinkage and expansion may be used 
to form the following convenient table: 



Material. 



To make 1000 cubic 

yards of embankment 

will require 



1000 cubic yards 
measured in exca- 
vation will make 



Gravel or sand 

Clay 

Loam 

Loose vegetable soil 
Rock, large pieces . . 
' * small * ' 



1087 cubic yards 

1111 

1136 

1176 

714 

625 
measured in excavation 



920 cubic yards 

900 

880 

850 
1400 
1600 
of embankment. 



Since writing the above the following values have been 
adopted by the American Railway Engineering Association as 
representing standard practice : 

Coefficients of Shrinkage Allowance for Depositing 

Earthwork. 





Trestle filling. 


Raising imder traffic. 


Black dirt 


15% 

10% 

6% 


5% 


Clay 


5% 


Sand 


5% 







129. Methods of forming embankments. Embankments of 
moderate height are sometimes formed by scraping material 
with drag scrapers from ditches at the sides, especially if there 
is little or no cutting to be done in the immediate vicinity. 
Over a low level swampy stretch this method has the double 
advantage of building an embankment which is well above 
the general level and also provides generous drainage ditches 
which keep the embankment dry. Wheeled scrapers may be 
used economically up to a distance of 4Q0 feet to excavate 



154 RAILROAD CONSTRUCTION. § 129. 

cuts and deposit the material on low embankments. Such 
methods have the advantage of compacting the embankments 
during construction and reducing future shrinkage. 

When carts are used, an embankment of any height may be 
formed by ^^ dumping over the end'' and building to the full 
height (or even higher to allow for shrinkage) as the embank- 
ment proceeds. The method is especially applicable when the 
material comes from a place as high as or higher than the 
grade-line, so that no up-hill hauling is necessary. Only a 
small contractor's plant is required for all of these methods. 

Trestles capable of carrying carts, or even cars and loco- 
motives, from which excavated material may be dropped, are 
found to be economical in spite of the fact that their cost is a 
construction expense. There is the disadvantage that such 
embankments require a long time to settle, but there are the 
advantages that the earth may be hauled by the train load 
from a distance of perhaps several miles, dumped from the 




Fig. 63. 

cars by train ploughs, or automatically dumped when the 
material is carried in patent dumping-cars, and all at a com- 
paratively small cost per cubic yard. The disadvantages of 
slow settlement may be obviated, although at some additional 
cost, by making the trestle sufficiently strong to support regular 
traffic until the settlement is complete. 

During recent years cableways have been utilized to fill 
comparatively narrow but deep ravines from material obtain- 
able on either side of the ravine. This method obviates the 
construction of an excessively high trestle which might other- 
wise be considered necessary. 

When an embankment is to be placed on a steep side hill 
which has a slippery clay surface, the embankment will some- 



I 



§ 130. EARTHWORK. 155 

times slide down the hill, unless means are taken to prevent it. 
Some sort of bond between the old surface and the new material 
becomes necessary. This has sometimes been provided by 
cutting out steps somewhat as is illustrated in Fig. 63. It is 
possible that a deep ploughing of the surface would accom- 
plish the result just as effectively and much cheaper. The 
tendency to slip is generally due not only to the nature of the 
soil but also to the usual accompanying characteristic that the 
soil is wet and springy. The sub-surface drainage of such a 
place with tile drains will still further prevent such slipping, 
which often proves very troublesome and costly. 

COMPUTATION OF HAUL. 

130. Nature of subject. As will be shown later when analyz- 
ing the cost of earthwork, the most variable item of cost is that 
depending on the distance hauled. As it is manifestly imprac- 
ticable to calculate the exact distance to which every individual 
cartload of earth has been moved, it becomes necessary to devise 
a means which will give at least an equivalent of the haulage of 
all the earth moved. Evidently the average haul for any mass 
of earth moved is equal to the distance from the center of grav- 
ity of the excavation to the center of gravity of the embank- 
ment formed by the excavated material. As a rough approxi- 
mation the center of gravity of a cut (or fill) may sometimes be 
considered to coincide with the center of gravity of that part of 
the profile representing it, but the error is frequently very large. 
The center of gravity may be determined by various methods, 
but the method of the ^'mass diagram" accomplishes the same 
ultimate purpose (the determination of the haul) with all-sufh- 
cient accuracy and also furnishes other valuable information. 

131. Mass diagram. In Fig. 64 let A'B\ . .G^ represent 
a profile and grade line drawn to the usual scales. Assume A' 
to be a point past which no earthwork will be hauled. Such 
a point is determined by natural conditions, as, for example, a 
river crossing, or one end of a long level stretch along which 
no grading is to be done except the formation of a low embank- 
ment from the material excavated from ample drainage ditches 
on each side. Above the profile draw an indefinite horizontal line 
{ACn in Fig. 64) which may be called the "zero line." Above 
every station point in the profile draw an ordinate (above or be- 



156 



RAILROAD CONSTRUCTION. 



§131. 



low the zero line) which will represent the algebraic sum of 
the cubic yards of cut and fill 
(calling cut + and fill — ) from 
the point A^ to the point con- 
sidered. The computations of 
these ordinates should first be 
made in tabular form as shown 
below. In doing this shrinkage 
must be allowed for by consider- 
ing how much embankments 
would actually be made by so 
many cubic yards of excavation 
of such material. For example, 
it will be found that 1000 cubic 
yards of sand or gravel, measured 
in place (see § 128) will make 
about 920 cubic yards of embank- 
ment; therefore all cuttings in 
sand or gravel should be dis- 
counted in about this propor- 
tion. Excavations in rock should 
be increased in the proper 
ratio. In short, all excavations 
should be valued according to the 
amount of settled embankment 
that could be made from them. 
Place in the first column a list 
of the stations; in the second 
column,the number of cubicyards 
of cut or fill between each station 
and the preceding station; in 
the third and fourth columns, the kind of material and the proper 
shrinkage factor; in the fifth column, a repetition of the quan- 
tities in cubic yards, except that the excavations are diminished 
(or increased, in the case of rock) to the number of cubic yards 
of settled embankment which may be made from them. In 
the sixth column place the algebraic sum of the quantities in the 
fifth column (calling cuts + and fills — ) from the starting- 
point to the station considered. These algebraic sums at each 
station will be the ordinates, drawn to some scale, of the mass 
curve. The scale to be used will depend somewhat on whether 




^si^ 



§132. 



EARTHWORK. 



157 



the work is heavy or light, but for ordinary cases a scale of 
5000 cubic yards per inch may be used. Drawing these ordi- 
nates to scale, a curve A, B, . . . G may be obtained by joining 
the extremities of the ordinates. 



Sta. 


Yards] -fl 


Material. 


Shrinkage 
factor. 


Yards, 

reduced 

for 

shrinkage. 


Ordinate 

in mass 

curve. 


46 + 70 













47 

48 

+ 60 
49 


+ 195 
-h 1792 
4- 614 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 177 
+ 180 

- 52 

- 71 
-f 276 
+ 1242 
+ 1302 


Clayey soil 


- 10 per cent 

- 10 

- 10 


4- 175 
4- 1613 
4- 553 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 283 
4- 289 

- 52 

- 71 
4- 249 
+ 1118 
4- 1172 


4- 175 
4- 1788 
4 2341 
4 2198 


50 






-f- 1292 


51 






- 693 


52 






— 2414 


4- 30 






- 2526 


53 

+ 70 
54 




Hard rock 


4- 60 per cent 
4- 60 


- 2243 

- 1954 

- 2006 


+ 42 






- 2077 


55 
56 
57 


Clayey soil 


- 10 per cent 

- 10 
~ 10 


- 1828 

- 710 
4- 462 



132. Properties of the mass curve. 

1. The curve will be rising while over cuts and falling while 
over fills. 

2. A tangent to the curve will be horizontal (as at 5, Z), E, 
F, and G) when passing from cut to fill or from fill to cut. 

3. When the curve is below the ^'zero line" it shows that 
material must be drawn backward (to the left) ; and vice versa^ 
when the curve is above the zero line it shows that material 
must be drawn forward (to the right) . 

4. When the qurve crosses the zero line (as at A and C) it 
shows (in this instance) that the cut between A' and B^ will just 
provide the material required for the fill between B^ and C, and 
that no material should be hauled past C\ or, in general, past 
any intersection of the mass curve and the zero line. 

5. If any horizontal line be drawn (as ab), it indicates that 
the cut and fill between a' and 6' will just balance. 

6. When the center of gravity of a given volume of material 
is to be moved a given distance, it makes no difference (at least 
theoretically) how far each individual load may be hauled or 
how any individual load may be disposed of. The summation 



158 RAILROAD CONSTRUCTION. § 132. 

of the products of each load times the distance hauled will be a 
constant, whatever the method, and will equal the total volume 
times the movement of the center of gravity. The average 
haul J which is the movement of the center of gravity, will there- 
fore equal the summation of these products divided by the total 
volume. If we draw two horizontal parallel lines at an infini- 
tesimal distance dx apart, as at abj the small increment of cut 
dx at a' will fill the corresponding increment of fill at h' , and 
this material must be hauled the distance ah. Therefore the 
product of ah and dx, which is the product of distance times 
volume, is represented by the area of the infinitesimal rectangle 
at ah, and the total area ABC represents the summation of 
volume times distance for all the earth movement between A' 
and C. This summation of products divided by the total 
volume gives the avei-age haul. 

7. The horizontal line, tangent at E and cutting the curve 
at e, f, and g, shows that the cut and fill between e' and E^ will 
just balance, and that a possible method of hauling (whether 
desirable or not) Avould be to " borrow" earth for the fill between 
C and e', use the material between D^ and £" for the fill betw^een 
e' and D', and similarly balance cut and fill betw^een £" and /' 
and also between /' and g\ 

8. Similarly the horizontal line hklm may be drawn cutting 
the curve, w^hich will show another possible method of hauling. 
According to this plan, the fill between C and h^ would be 
made by borrowing; the cut and fill between h' and ^' would 
balance; also that between /c' and V and between V and m'. 
Since the area ehDkE represents the measure of haul for the 
earth between e' and E% and the other areas measure the corre- 
sponding hauls similarly, it is evident that the sum of the areas 
eHDkE and ElFmf, which is the measure of haul of all the 
material between e^ and f , is largely in excess of the sum of 
the areas hDk, kEl, and IFm, plus the somewhat uncertain 
measures of haul due to borrowing material for e'h' and wasting 
the material between m' and f . Therefore to make the meas- 
ure of haul a minimum a line should be drawn which will make 
the sum of the areas between it and the mass curve a minimum. 
Of course this is not necessarily the cheapest plan, as it implies 
more or less borrowing and wasting of material, which may 
cost more than the amount saved in haul. The comparison of 
the two methods is quite simple, however. Since the amount 



§ 133. EARTHWORK. 159 

of fill between e' and h^ is represented by the difference of the 
ordinates at e and h, and similarly for m' and /', it follows that 
the amount to be borrowed between e' and h' will exactly equal 
the amount wasted between m' and /'. By the first of the above 
methods the haul is excessive^ but is definitely known from the 
mass diagram, and all of the material is utilized ; by the second 
method the haul is reduced to about one-half, but there is a 
known quantity in cubic 3^ards wasted at one place and the same 
quantity borrowed at another. The length of haul necessary 
for the borrowed material would need to be ascertained; also 
the haul necessary to waste the other material at a place where 
it would be unobjectionable. Frequently this is best done by 
widening an embankment beyond its necessary width. The 
computation of the relative cost of the above methods will be 
discussed later (§ 148). 

9. Suppose that it were deemed best, after drawing the mass 
curve, to introduce a trestle between s' and v' , thus saving an 
amount in fill equal to tv. If such had been the original design, 
the mass carve would have been a straight horizontal line between 
s and t and would continue as a curve which would be at all 
points a distance tv above the curve vFmzfGg. If the line Ef is 
to be used as a zero line, its intersection with the new curve at x 
will show that the material between E^ and z^ will just balance 
if the trestle is used, and that the amount of haul will be meas- 
ured by the area between the line Ex and the broken line Estx, 
The same computed result may be obtained without drawing 
the auxiliary curve txn ... by drawing the horizontal line zy 
Sit a distance xz{ =tv) below Ex. The amount of the haul can 
then be obtained by adding the triangular area between Es and 
the horizontal line Ex, the rectangle between st and Ex, and the 
'irregular area between vFz and y . . . z (which last is evidently 
equal to the area between tx and E . . . x). The disposal of the 
material at the right of z' would then be governed by the indica- 
tions of the profile and mass diagram which would be found at 
the right of g\ In fact it is difficult to decide with the best of 
judgment as to the proper disposal of material without having 
a mass diagram extending to a considerable distance each side 
of that part of the road under immediate consideration. 

133. Area of the mass curve. The area may be computed 
most readily by means of a planimeter, which is capable with 
reasonable care of measuring such areas with as great accuracy 



L 



160 RAILROAD CONSTRUCTION. § 134. 

as is necessary for this work. If no such instrument is obtain- 
able, the area may be obtained by an application of '' Simpson's 
rule/' The ordinates will usually be spaced 100 feet apart. 
Select an even number of such spaces, leaving, if necessary, one 
or more triangles or trapezoids at the ends for separate and 
independent computation. Let y^ , . .ynhe the ordinates, i.e., 
the number of cubic yards at each station of the mass curve, or 
the figures of "column six'' referred to in § 13 L Let the uni- 
form distance between ordinates (^100 feet) be called 1, i.e., 
one station. Then the imits of the resulting area will be cubic 
yards hauled one station. Then the 

Area = i[2/o+ 4(2/1 +2/3-'- • • .2/(«-l)+ 2(2/2 +2/4+ • • .2/(»_2)+2/m]- (62) 

When an ordinate occurs at a substation, the best plan is to 
ignore it at first and calculate the area as above. Then, if the 
difference involved is too great to be neglected, calculate the 
area of the triangle having the extremity of the ordinate at the 
substation as an apex, and the extremities of the ordinates at the 
adjacent stations as the ends of the base. This may be done by 
finding the ordinate at the substation that would be a propor- 
tional between the ordinates at the adjacent full stations. Sub- 
tract this from the real ordinate (or vice versa) and multiply the 
difference by §Xl. An inspection will often show that the 
correction thus obtained would be too small to be worthy of con- 
sideration. If there is more than one substation between two 
full stations, the corrective area will consist of two triangles and 
one or more trapezoids which may be similarly computed, if 
necessary. 

When the zero line (Fig. 64) is shifted to eEj the drop from 
AC (produced) to E is known in the same units, cubic yards. 
This constant may be subtracted from the numbers ("column 
6," § 131) representing the ordinates, and will thus give, with- 
out any scaling from the diagram, the exact value of the modi- 
fied ordinates. 

134. yalue of the mass diagram. The great value of the mass 
diagram lies in the readiness with which different plans for the 
disposal of material may be examined and compared. When 
the mass curve is once drawn, it will generally require only a 
shifting of the horizontal line to show the disposal of the material 
by any proposed method. The mass diagram also shows the 



§ 135. EARTHWORK. 161 

extreme length of haul that will be required by any proposed 
method of disposal of material. This brings into consideration 
the ''limit of profitable haul/' which will be fully discussed in 
§ 148. For the present it may be said that with each method 
of carrying material there is some limit beyond which the expense 
of hauling will exceed the loss resulting from borrowing and 
wasting. With wheelbarrows and scrapers the limit of profit- 
able haul is comparatively short, with carts and tram-cars it is 
much longer, while with locomotives and cars it may be several 
miles, if, in Fig. 64, eE or Ef exceeds the limit of profitable 
haul, it shows at once that some such line as hklm should be 
drawn and the material disposed of accordingly. 

135. Changing the grade line. The formation of the mass 
curve and the resulting plans as to the disposal of material are 
based on the mutual relations of the grade line and the surface 
profile and the amounts of cut and fill which are thereby im- 
plied. If the grade line is altered, every cross-section is altered, 
the amount of cut and fill is altered, and the mass curve is also 
changed. At the farther limit of the actual change of the grade 
line the revised mass curve will have (in general) a different 
ordinate from the previous ordinate at that point. From that 
point on, the revised mass curve will be parallel to its former 
position, and the revised curve may be treated similarly to the 
case previously mentioned in w^hich a trestle was introduced. 
Since it involves tedious calculations to determine accurately 
how much the volume of earthw^ork is altered by a change in 
grade line, especially through irregular country, the effect on 
the mass curve of a change in the grade line cannot therefore 
be readily determined except in an approximate way. Raising 
the grade line will evidently increase the fills and diminish the 
cuts, and vice versa^ Therefore if the mass curve indicated, for 
example, either an excessively long haul or the necessity for 
borrowing material (implying a fill) and wasting material 
farther on (implying a cut), it w^ould be possible to diminish the 
fill (and hence the amount of material to be borrowed) by lower- 
ing the grade line near that place, and diminish the cut (and 
hence the amount of material to be wasted) by raising the 
grade line at or near the place farther on. Whether the advan- 
tage thus gained would compensate for the possibly injurious 
effect of these changes on the grade line would require patient 
investigation. But the method outlined shows how the mass 



162 



RAILROAD CONSTRUCTION, 



§136. 



curve might be used to indicate a possible change in grade Hne 
which might be demonstrated to be profitable. 

136. Limit of free haul. It is sometimes specified in con- 
tracts for earthwork that all material shall be entitled to free 
haul up to some specified limit, say 500 or 1000 feet, and that 
all material drawn farther than that shall be entitled to an 
allowance on the excess of distance. It is manifestly imprac- 
ticable to measure the excess for each load, as much so as to 
measure the actual haul of each load. The mass diagram also 
solves this problem very readily. Let Fig. 65 represent a pro- 




FiG. 65. 



file and mass diagram of about 2000 feet of road, and suppose 
that 800 feet is taken as the limit of free haul. Find two points, 
a and h, in the mass curve which are on the same horizontal line 
and which are 800 feet apart. Project these points down to a' 
and h\ Then the cut and fill between a' and ¥ will just balance, 
and the cut between A ' and a' will be needed for the fill between 
V and C. In the mass curve, the area between the horizontal 
line ah and the curve aBh represents the haulage of the material 
between a' and 6', which is all free. The rectangle dbmn repre- 
sents the haulage of the material in the cut A^a' across the 800 
feet from a' to h\ This is also free. The sum of the two areas 
Aam and hnC represents the haulage entitled to an allowance, 
since it is the summation of the products of cubic yards times 
the excess of distance hauled. 

If the amount of cut and fill was symmetrical about the point 



I 



§ 137. EARTHWORK. 163 

B^j the mass curve would be a symmetrical curve about the 
vertical line through B, and the two limiting lines of free haul 
would be placed symmetrically about B and B\ In general 
there is no such symmetry, and frequently the difference is con- 
siderable The area aBbnm will be materially changed accord- 
ing as the two vertical lines am and bn, always 800 feet apart, 
are shifted to the right or left. It is easy to show that the area 
aBbnm is a maximum when ab is horizontal. The minimum 
value would be obtained either when m reached A or n reached 
C, depending on the exact form of the curve. Since the posi- 
tion for the minimum value is manifestly unfair, the best definite 
value obtainable is the maximium, which must be obtained as 
above described. Since aBbnm is made maximum, the remainder 
of the area, which is the allowance for overhaul, becomes a mini- 
mum. The areas Aam and bCn may be obtained as in § 102. 
If the whole area AaBbCA has been previously computed, it 
may be more convenient to compute the area aBbnm and sub- 
tract it from the total area. 

Since the intersections of the mass curve and the ^^zero line" 
mark limits past which no material is drawn, it follows that 
there will be no allowance for overhaul except where the dis- 
tance between consecutive intersections of the zero line and mass 
curve exceeds the limit of free haul. 

Frequently all allowances for overhaul are disregarded; the 
profiles, estimates of quantities, and the required disposal of 
material are shown to bidding contractors, and they must then 
make their owm allowances and bid accordingly. This method 
has the advantage of avoiding possible disputes as to the amount 
of the overhaul allowance, and is popular with railroad com- 
panies on this account. On the other hand the facility with 
which different plans for the disposal of material may be studied 
and compared by the mass-curve method facilitates the adoption 
of the most economical plan, and the elimination of uncertainty 
will frequently lead to a safe reduction of the bid, and so the 
method is valuable to both the railroad company and the con- 
tractor. 

ELEMENTS OF THE COST OF EARTHW^ORK. 

137. Analysis of the total cost into items. The variation in 
the total cost of excavating earthwork, hauling it a greater or 
less distance, and forming with it an embankment of definite 



164 RAILROAD CONSTRUCTION. § 138. 

form or wasting it on a spoil bank, is so great that the only 
possible method of estimating the cost under certain assumed 
conditions is to separate the total cost into elementary items. 
Ellwood Morris was perhaps the first to develop such a method 
— see Journal of the Franklin Institute, September and October, 
1841. Trautwine used the same general method with some 
modifications.- The following analysis will follow the same 
general plan, will quote some of the figures given by Morris 
and by Trautwine, but will also include facts and figures better 
adapted to modern conditions. Since every item of cost (except 
interest on cost of plant and its depreciation) is a direct function 
of the current price of common labor, all calculations will be 
based on the simple unit of $1 per day. Then the actual cost 
may be obtained by multiplying the calculated cost imder the 
given conditions by the current price of day labor. When 
possible, figiu-es will be quoted giving the cost of all items of 
work on a loose sandy soil which is the easiest to work and also 
for the cost olthe heaviest soils, such as stiff clay and hard pan. 
These represent the extremes, excluding rock, which will be 
treated separately. The cost of intermediate grades may be 
interpolated between the extreme values according to the 
judgment of the engineer as to the character of the soil. 

The possible division into items varies greatly according to 
the method adopted, but the differentiation into items given 
below (which is strictly applicable to the old fashioned simpler 
methods of work) can usually be applied to any other method 
by merely combining or eliminating some of the items. The 
items are 

1. Loosening the natural soil. 

2. Loading the soil into whatever carrier may be used. 

3. Hauling excavated material from excavation to embank- 

ment or spoil bank. 

4. Spreading or distributing the soil on the embankment. 

5. Keeping roadways or tracks in good running order. 

6. Trimming cuts to their proper cross-section (sometimes 

called ^ ' sandpapering ' . 

7. Repairs, wear, depreciation, and interest on cost of plant. 

8. Superintendence and incidentals. 

138. Item I. Loosening, (a) Ploughs. Very light sandy 
soils can frequently be shovelled without any previous loosen- 
ing, but it is generally economical, even with very light material. 



§ 138. EARTHWORK. 165 

to use a plough. Morris quotes, as the results of experiments, 
that a three-horse plough would loosen from 250 to 800 cubic 
yards of earth per day, which at a valuation of $5 per day 
would make the cost per yard vary from 2 cents to 0.6 cent. 
Trautwine estimates the cost on the basis of two men handling 
a two-horse plough at a total cost of $3.87 per day, being $1 
each for the men, 75 c. for each horse, and an allowance of 
37 c.'for the plough, harness, etc. From 200 to 600 cubic yards 
is estimated as a fair day's work, which makes a cost of 1.9 c. 
to 0.65 c. per yard, which is substantially the same estimate 
as above. Extremely heavy soils have sometimes been loosened 
by means of special ploughs operated by traction-engines. 

Gillette estimates that "a two-horse team with a driver and 
a man holding the plough will loosen 25 cubic yards of fairly 
tough clay, or 35 cubic yards of gravel and loam per hour." 
For ten hours per day this would be 250 to 350 cubic yards 
per day. These values are neither as high nor as low as the 
extremes above noted. It is probably very seldom that a soil 
will be so light that a two-horse (or three-horse) plough can 
loosen as much as 600 (or 800) cubic yards per day. 

It is sometimes necessary to plough up a macadamized street. 
This may be done by using as a plough a pointed steel bar 
which is fastened to a very strong plough frame. A prelimi- 
nary hole must be made which* will start the bar under the 
macadam shell. Then, as the plough is drawn ahead, the shell 
is ripped up. Four or six horses, or even a traction-engine, 
are used for such work. Gillette quotes two such cases w^here 
the cost of such loosening was 2 c. and 6 c. per cubic yard, 
with common labor at 15 c. per hour. Two-thirds of such 
figures will reduce them to the $1 per day basis. The cost for 
ploughing on the $1 per day basis may therefore be summarized 
as follows: 

For very loose sandy soils. 0.6 c. per cubic yard 

'' '' heavy clay '' 2.0 c. '' " '* 

** hard pan and macadam, up to . . . 4.0 c. ** " *' 

(b) Picks. When picks are used for loosening the earth, as 
is frequently necessary and as is often done when ploughing 
would perhaps be really cheaper, an estimate * for a fair day's 

* Trautwine. 



166 RAILROAD CONSTRUCTION. § 139. 

work is from 14 to 60 cubic yards, the 14 yards being the esti- 
mate for stiff clay or cemented gravel, and the 60 yards the esti- 
mate for the lightest soil that would require loosening. At $1 
per day this means about 7 c. to 1.7 c. per cubic yard, which is 
about three times the cost of ploughing. Five feet of the face 
is estimated * as the least width along the face of a bank that 
should be allowed to enable each laborer to work with freedom 
and hence economically. 

(c) Blasting. Although some of the softer shaly rocks may 
be loosened with a pick for about 15 to 20 c. per yard, yet rock 
in general, frozen earth, and sometimes even compact clay are 
most economically loosened by blasting. The subject of blast- 
ing will be taken up later, §§ 149-155. 

(d) Steam-shovels. The items of loosening and loading 
merge together with this method, which will therefore be treated 
in the next section. 

139. Item 2. Loading, (a) Hand-shovelling. Much depends 
on proper management, so that the shovellers need not wait un- 
duly either for material or carts. With the best of management 
considerable time is thus lost, and yet the intervals of rest 
need not be considered as entirely lost, as it enables the men to 
work, while actually loading, at a rate which it would be physi- 
cally impossible for them to maintain for ten hours. Seven 
shovellers are sometimes allowed for each cart; otherwise there 
should be five, two on each side and one in the rear. Economy 
requires that the number of loads per cart per day should be 
made as large as possible, and it is therefore Vi^ise to employ as 
many shovellers as can work without mutual interference and 
without wasting time in waiting for material or carts. The 
figures obtainable for the cost of this item are unsatisfactory on 
account of their large disagreements. The following are quoted 
as the number of cubic yards that can be loaded into a cart by 
an average laborer in a working day of ten hours, the lower 
estimate referring to heavy soils, and the ^higher to light sandy 
soils: 10 to 14 cubic yards (Morris), 12 to 17 cubic yards (Has- 
koll), 18 to 22 cubic yards (Hurst), 17 to 24 cubic yards (Traut- 
wine), 16 to 48 cubic yards (Ancelin). As these estimates are 
generally claimed to be based on actual experience, the discrep- 
ancies are probably due to differences of management. If the 



* Hurst. 



§ 139. EAKTHWORK. 167 

average of 15 to 25 cubic yards be accepted, it means, on the 
basis of $1 per day, 6.7 c. to 4 c. per cubic yard. These esti- 
mates apply only to earth. Rockwork costs more, not only 
because it is harder to handle, but because a cubic 3^ard of solid 
rock, measured in place, occupies about 1.8 cubic yards when 
broken up, while a cubic yard of earth will occupy about 1.2 
cubic yards. Rockwork will therefore require about 50% more 
loads to haul a given volume, measured in place, than will the 
same nominal volume of earthwork. The above authorities give 
estimates for loading rock varying from 6.9 c. to 10 c. per cubic 
yard. The above estimates apply only to the loading of carts 
or cars Avith shovels or by hand (loading masses of rock). The 
cost of loading wheelbarrows and the cost of scraper work will 
be treated under the item of hauling. 

(b) Steam-shovels.* Whenever the magnitude of the work 
will warrant it there is great economy in the use of steam-shovels. 
These have a '^ bucket" or ^^ dipper" on the end of a long beam, 
the bucket having a capacity varying from J to 2J cubic yards. 
Steam-shovels handle all kinds of material from the softest 
earth to shale rock, earthy material containing large boulders, 
tree-stumps, etc. The record of work done varies from 200 to 
1000 cubic yards in 10 hours. They perform all the work of 
loosening and loading. Their economical working requires that 
the material shall be hauled away as fast as it can be loaded, 
which usually means that cars on a track, hauled by horses or 
mules, or still better by a locomotive, shall be used. The ex- 
penses for a steam-shovel, costing about $5000, will average 
about $1000 per month. Of this the engineer may get $100; the 
fireman $50 ; the cranesman $90 ; repairs perhaps $250 to $300 ; 
coal, from 15 to 25 tons, cost very variable on account of expen- 
sive hauling; water, a very uncertain amount, sometimes costing 
$100 per month ; about five laborers and a foreman, the laborers 
getting $1.25 per day and the foreman $2.50 per day, which will 
amount to $227.50 per month. This gang of laborers is employed 
in shifting the shovel when necessary, taking up and relaying 

* For a thorough treatment of the capabilities, cost, and management 
of steam-shovels the reader is referred to " Steam-shovels and Steam-shovel 
Work," by E. A. Hermann. D. Van Nostrand Co., New York. 

This book is now out of print. " Earthwork and its Cost," by H. P. Gil- 
lette, to which the student is referred for a more elaborate exposition of the 
subject, has used many of Hermann's cuts. 



168 RAILROAD CONSTRUCTION. § 139. 

tracks for the cars, shifting loaded and unloaded cars, etc. In 
shovelling through a deep cutj the shovel is operated so as to 
undermine the upper parts of the cut* which then fall down 
within reach of the shovel, thus increasing the amoiuit of material 
handled for each new position of the shovel. If the material is 
too tough to fail down by its own weight, it is sometimes found 
economical to employ a gang of men to loosen it or even blast it 
rather than shift the shovel so frequently. Non-condensing 
engines of 50 horse-power use so much water that the cost of 
water-supply becomes a serious matter if water is not readily 
obtainable. The lack of water facilities will often justify the 
construction of a pipe line from some distant source and the 
installation of a steam-pump. Hence the seemingly large 
estimate of $100 per month for water-supply, although imder 
favorable circumstances the cost may almost vanish. The larger 
steam-shovels will consume nearly a ton of coal per day of 10 
hours. The expense of hauling this coal from the nearest rail- 
road or canal to the location of the cut is often a very serious 
item of expense and may easily double the cost per ton. Some 
steam-shovels have been constructed to be operated by electrioity 
obtained from a plant perhaps several miles away. Such a 
method is especially advantageous when fuel and water are diffi- 
cult to obtain. 

The following general requirements and specifications were 
recommended in 1907 by the American Railway Engineering 
Association : 

Three important cardinal points should be given careful 
attention in the selection of a steam-shovel. These are in their 
order 

(1) Care in the selection, inspection and acceptance of all 
material that enters into every part of the machine. 

(2) Design for strength. 

(3) Design for production. 

GENERAL SPECIFICATIONS. 

Weight of shovel: Seventy (70) tons. 
Capacity of dipper: Two and one-half (2^) yards. 
Steam pressure: One hundred and twenty (120) pounds. 
Clear height above rail of shovel track at which dipper should 
unload: Sixteen (16) feet. 



§ 140. EARTHWORK. 169 

Depth below rail of shovel track at which dipper should dig 
Four (4) feet. 

Number of movements of dipper per minute from time of 
entering bank to entering bank: Three (3). 

Character of hoist: Cable. 

Character of swing: Cable. 

Character of housing: Permanent for all employes. 

Capacity of tank: Two thousand (2000) gallons. 

Capacity of coal-bunker: Four (4) tons. 

Spread of jack arm: Eighteen (18) feet. A special short arm 
should be provided. 

Form of steam-shovel track: "T'* rails on ties. 

Length of rails for ordinary work: Six (6) feet. 

Form of rail joint: Strap. 

Manufacturers of steam-shovels will fjometimes "guarantee" 
that certain of their shovels will excavate, say 3000 cubic yards 
of earth per day of ten hours. Even if it were possible for a 
shovel to fill a car at the rate of 5 cubic yards per minute, it is 
always impracticable to maintain such a speed, since a shovel 
must always wait for the shifting of cars and for the frequent 
shifting of the shovel itself. There are also delays due to 
adjustments and minor breakdowns. The best shovel records 
are made when the cars are large — other things being equal. 
The item of interest and depreciation of the plant is very large 
in steam-shovel work. This will be discussed further later. 
The cost of loading alone will usually come to between 3 and 
4 c. per cubic yard. The cost of shifting the cars so as to 
place them successively under the shovel, haul them to the 
dumping place, diunp them and haul them back, will generally 
be as much more. Gillette quotes five jobs on one railroad 
where the total cost for loading and hauling varied from 5.9 c. 
to 11.4 c. per cubic yard. But as these figures are based on 
car measurement, the cost per cubic yard in place measure- 
ment must be increased about one-fourth, or from 7.4 c. to 
14.2 c. 

i4o. Item 3. Hauling. The cost of hauling depends on 
the number of roimd trips per day that can be made by each 
vehicle employed. As the cost of each vehicle is practically the 
same whether it makes many trips or few, it becomes important 
that the number of trips should be made a maximum, and to that 
end there should be as little delay as possible in loading and 



170 RAILROAD CONSTRUCTION. § 140. 

unloading. Therefore devices for facilitating the passage of the 
vehicles have a real money value. 

(a) Carts. The average speed of a horse hauling a two- 
wheeled cart has been found to be 200 feet per minute, a little 
slower when hauling a load and a little faster when returning 
empty. This figure has been repeatedly verified. It means an 
allowance of one minute for each 100 feet (or '^ station'') of 
"lead — the lead being the distance the earth is hauled." The 
time lost in loading, dumping, waiting to load, etc., has been 
found to average 4 minutes per load. Representing the mmi- 
ber of Nations (100 feet) of lead by s, the number of loads 
handled in 10 hours (600 minutes) would be 600 --(s + 4). The 
number of loads per cubic yard, measured in the bank, is differ- 
entiated by Morris into three classes, viz. : 

3 loads per cubic yard in descending hauling; 
3K' " " " " level hauling; and 

4 ^ ^ ' ' ' ' ' * ' ' ascending hauling. 

Attempts have been made to estimate the effect of the grade 
of the roadway by a theoretical consideration of its rate, and of 
the comparative strength of a horse on a level and on various 
grades. While such computations are always practicable on a 
railway (even on a temporary construction track), the traction 
on a temporary earth roadway is always very large and so very 
variable that any refinements are useless. On railroad earth- 
work the hauling is generally nearly level or it is descending — 
forming embankments on low ground with material from cuts in 
high ground. The only common exception occurs when an 
embankment is formed from borrow-pits on low ground. One 
method of allowing for ascending grade is to add to the hori- 
zontal distance 14 times the difference of elevation for work 
with carts and 24 times the difference of elevation for work 
with wheelbarrows, and use that as the lead. For example, 
using carts, if the lead is 300 feet and there is a difference of 
elevation of 20 feet, the lead would be considered equivalent to 
300 + (14X20) =580 feet on a level. 

Trautwine assumes the average load for all classes of work 
to be J cubic yard, which figure is justified by large experience. 
Using one figure for all classes of work simplifies the calculations 
and gives the number of cubic yards carried per day of 10 hours 

equal to — —. Dividing the cost of a cart per day by the 

(J\S -f-4:) 



§ 140. EARTHWORK. 171 

number of cubic yards carried gives the cost of hauling pel 
yard. In computing the cost of a cart per day, Trautwine 
refers to the practice of having one driver manage four carts, 
thus making a charge of 25 c. per day for each cart for the driver. 
Although this might be an economical method when the haul is 
very long, it is not economical for short hauls. A safer estimate 
is to allow not more than two carts per driver and in many 
cases a driver for each cart. Some contractors employ a driver 
for each cart and then require that the drivers shall assist in 
loading. The policy to be adopted is sometimes dependent on 
labor union conditions, which may demand that drivers must 
not assist in loading. The supply of labor and the amount of 
work on hand have a great influence on the methods of work 
which a contractor may adopt, for a strike will often disarrange 
all plans. 

The cost of a horse and cart must practically include a 
charge for the time of the horse on Sundays, rainy days and 
holidays. The cost of repairs of cart and harness is generally 
included in this item for simplicity, but, under a strict applica- 
tion of the analysis suggested in § 138, it should properly be 
included under Item 7, Repairs, etc. 

Since the time required for loading loose rock is greater 
than for earthwork, less loads will be hauled per day. The time 
allowance for loading, etc.,* is estimated by Trautwine as 6 
minutes instead of 4 as for earth. Considering the great ex- 
pansion of rock when broken up (see § 128), one cubic yard of 
solid rock, measured in place, would furnish the equivalent of 
five loads of earthwork of J cubic yard. Therefore, on the 
basis of five loads per cubic yard, the number of cubic yards 

handled per day per cart would be ■=-, — -ttt. 

5(s + 6) 

Let C represent the daily cost of a horse and cart and of 

the proportional cost of the driver (according to the number of 

carts handled by one driver), then the cost per cubic yard, 

measured in the cuU for hauling may be given by the formula: 



Cost per cu. yd. of hauling earth in carts = ^^^r — - 

\ . (63) 
u it ic u it ic „^pT. ct (c CX5(s + 6) [ 



U I 



172 RAILROAD CONSTRUCTION. § 140 

(b) Wagons. For longer leads (i.e., from | to § of a mile) 
wagons drawn by two (or three) horses are more economical. 
The old-style wagons (about 0.8 cu. yd.) have bottoms of loose 
thick narrow boards. Raising them individually deposits the 
load underneath. Modern dump wagons contain from 1.0 to 
2.0 cu. yds. The daily cost may be estimated on the same prin- 
ciple as the cost of carts. 

The number of wagon trips per 10 hours will depend some- 
what on the management of the shovellers. Too many shovel- 
lers per wagon is not economical, measured in yards shovelled 
per man, although it may reduce the time consumed in loading 
any one wagon. At an average figure of 20 cubic yards, 
measured in place, per shoveller per 10 hours, seven shovellers 
would load 14 cubic yards per hour or one cubic yard in 4.3 
minutes. This would be the allowance for a wagon with a 
capacity of about IJ yards of loose earth. Adding time for 
unloading, waiting to load and other possible " lost time/' there 
is probably a total of six minutes. This figure will vary very 
considerably according to the number of shovellers per wagon, 
the capacity of the wagon, the type of wagon (whether self- 
dumping) and' other details in the method of management. 
Adopting six minutes as the time used for loading, unloading, 
and other "lost time," the formula becomes. 

Cost per cubic yard of hauling in wagons =-^^r^^ — ^, .... (64) 

in which C is the cost of the wagon, team and driver per day 
of 10 hours; s is the distance hauled in stations of 100 feet, 
and c is the capacity of the wagon in cubic yards, place meas- 
urement, which should be about three fourths of the nominal 
capacity of the wagon for earth and about sixty per cent when 
handling rock. 

(c) Wheelbarrows. Gillette has computed from observa- 
tions that a man will trundle a wheelbarrow at the rate of 250 
feet per minute or 1.25 stations of lead per minute for the round 
trip. The time required for loading is estimated at 2^ minutes 
and for unloading, adjusting wheeling planks, short rests, etc., 
I minute, or a total of three minutes per trip for all purposes 
except hauling. Gillette allows for a load only 1/15 cubic yard, 



§ 140. EARTHWORK. 173 

measured in place, or about 1/11 yard, 2,5 cubic feet, on the 
wheelbarrow. With notation as before 

Cost per cubic yard of loading and \ CX 15(1.255 + 3) ^q^\ 
hauling earth in wheelbarrows / ~" 600 * 

In this equation C is the cost of both loading and hauling, and 
usually includes the allowance (Item 7) for the cost, repairs 
and depreciation of the wheelbarrows, whose service is very 
short lived. Trautwine estimates this at five cents per day or 
a total of SI. 05 for labor and wheelbarrow. 

The number of wheelbarrow loads required for a cubic yard 
of rock, measured in place, is about twenty-four. The time 
required for loading should also be increased about one fourth; 
the time required for all purposes except hauling is therefore 
about 3.75 minutes, and the corresponding equation becomes 

Cost per cubic yard of loading and ^ _ Cx24(1.25s + 3.75) (qq) 
hauling rock in wheelbarrows J "" 600 

(d) Scrapers, These are made in three general ways, *'buck'* 
Bcrapers, *'drag" scrapers and ^'wheeled" scrapers. The buck 
scraper in its original form consisted merely of a wide plank, 
shod with an iron strap on the lower edge and provided with 
a pole and a small platform on which the driver may stand to 
weight it down. The earth is not loaded on to any receptacle 
and carried, but is merely pushed over the ground. Notwith- 
standing the apparent inefficiency of the method, its extreme 
simplicity has caused its occasional adoption for the construc- 
tion of canal embankments out of material from the bed of the 
canal. The occasions are rare when their use for railroad work 
would be practicable, and even then drag scrapers would prob- 
ably be preferable. 

A drag scraper is an immense "scoop shovel" about three feet 
long and three feet wide. There are usually two handles and a 
bail in front by which it is dragged by a team of horses. The 
nominal capacity varies from 7.5 cubic feet for the largest sizes, 
down to 3 cubic feet for the ''one-horse^' size, but these figures 
must be discounted by perhaps 40 or 50% for the actual average 
volume (as measured in the cut) loaded on during one scoop. 
The expansion of the earth during loosening is alone respons- 



174 RAILROAD CONSTRUCTION. § 140. 

ible for a discount of 25%. These scrapers cost from $10 to 

$18. 

A wheeled scraper is essentially an extra-large drag scraper 
which may be raised by a lever and carried on a pair of large 
wheels. Their nominal capacity ranges from 10 to 17 cubic feet, 
which should usually be liberally discounted when estimating 
output. They are loaded by dropping the scoop so that it 
scrapes up its load. The lever raises the scoop so that the load 
is carried on wheels instead of being dragged. At the dump the 
scoop is tipped so as to unload it. The movement of the 
scraper is practically continuous. They cost from $40 to 
$75. Their advantages over drag scrapers consist (1) in their 
greater capacity, (2) in the economy of transporting the load 
on wheels instead of by dragging, and (3) in the far greater 
length of haul over which the earth may be economically 
handled. 

Morris estimated the speed of drag scrapers to be 140 feet per 
minute, or 70 feet of lead per minute. The "lead" should be 
here interpreted as the average distance from the center of the 
pit to the center of the dump. Gillette declares the speed to be 
220 feet per minute. Some of this variation may be due to dif- 
ferences in the method of measuring the distance actually trav- 
elled, especially when the lead is very short, since the scraper 
teams must always travel a considerable extra distance at each 
end in order to turn around most easily. This extra distance is 
practically constant whether the lead is long or short. Gillette 
quotes an instance where the length of lead was actually about 
20 feet, but the scraper teams travelled about 150 feet for each 
load carried. On this account Gillette adopts a minimum of 
75 feet of lead no matter how short the lead actually may be. 
Of course the speed depends considerably on how strictly the 
men are kept to their work and also on the care which may be 
taken to obtain a full load for each scraper. As a compromise 
between Morris's and Gillette's estimates we may adopt the con- 
venient rate of speed of 200 feet per minute, or 100 feet of lead 
per minute. There should also be allowed for the time lost 
in loading and unloading and for travelling the extra distance 
travelled by the teams in making the circuit, If minutes. Allow- 
ing the average value of seven loads per cubic yard and letting 
C represent the cost of scraper team and driver per day, we 
liave for the cost as follows: 



§ 140. EARTHWORK. 175 

Cost per cubic yard of loading and I CX7(g+l|) .^„. 

hauling earth in drag scrapers ( ^^^ 



600 

In this formula C should include the cost of not only the 
driver, team, and scraper, but also the proper proportion of 
the wages of an extra man) who assists each driver in loading 
his scraper, and whose wages should be divided among the two 
(or three) scrapers to which he is assigned. Scraper work 
nearly always implies ploughing, the cost of which should be 
computed as under Item 1. 

When a low embankment is formed from borrow-pits on each 
side of the road, it may be done with scrapers, which move from 
one borrow-pit to the other, taking a load alternately from each 
side to the center and making but one half turn for each load 
carried. This reduces the time lost in turning by one third of a 
minute and reduces the constant in the numerator in Eq. (67) 
from IJ to 1. In this case the lead will usually be not greater 
than 75 feet, and therefore, if we consider this as a minimum 
value, s will ordinarily equal .75 and the quantity in the paren- 
thesis will equal 1.75. 

When using wheeled scrapers the catalogue capacity, which 
varies from 9 or 10 feet for a No. 1 scraper to 16 or 17 feet for 
a No. 3 scraper, must be reduced to 5 loads per cubic yard 
(place measurement) for a No. 1 scraper and to 2 J loads per 
cubic yard for a No. 3, not only on account of the expansion of 
the earth during loosening, but also on account of the imprac- 
ticability of loading these scrapers to their maximum nominal 
capacity. When the haul or lead for wheeled scrapers is 300 
feet or over, it will be justifiable to employ shovellers to fill up 
the bowl of the shovel, especially when the soil is tough and 
when it is impracticable to fill the shovel even approximately 
full by the ordinary method. A snatch team to assist in load- 
ing the scrapers it also economical, especially with the larger 
scrapers. The proportionate number of snatch teams to the 
total number of scrapers of course depends on the length of 
haul. The cost of these extra shovellers and extra snatch teams 
must be divided proportionally among the number of scrapers 
assisted, in determining the value C in the formula given below. 
The extra time to be allowed on account of turning, loading, 
and dumping is about IJ minutes. The speed is considered 
one station of lead per minute as before. If we call C the average 



176 RAILROAD CONSTRUCTION. § 140. 

daily cost of one scraper and n the capacity of the scraper, or 
the number of loads per cubic yard, we may write the following 
formula: 



Cost per cubic yard of loading and \ CXn(s-\-H) 



hauling earth in wheeled scrapers J 600 



(68) 



(e) Cars and horses. The items of cost by this method are 
(a) charge for horses employed, (b) charge for men employed 
strictly in hauling, (c) charge for shifting rails when necessary, 
(d) repairs, depreciation, and interest on cost of cars and track. 
Part of this cost should strictly be classified under items 5 and 
7, mentioned in § 137, but it is perhaps more convenient to 
estimate them as follows: 

The traction of a car on rails is so very small that grade 
resistance constitutes a very large part of the total resistance 
if the grade is 1% or more. For all ordinary grades it is 
sufficiently accurate to say that the grade resistance is to 
the gross weight as the rise is to the distance. If the distance 
is supposed to be measured along the slope, the proportion is 
strictly true; i.e., on a 1% grade the grade resistance is 1 lb. 
per 100 of weight or 20 lbs. per ton. If the resistance on a 
level at the usual velocity is y-|-g^, a grade of 1:120 (0.83%) will 
exactly double it. If the material is hauled down a grade of 
1:120, the cars will run by gravity after being started. The 
work of hauling will then consist practically of hauling the 
empty cars up the grade. The grade resistance depends only 
on the rate of grade and the weight, but the tractive resistance 
will be greater per ton of weight for the unloaded than for the 
loaded cars. The tractive power of a horse is less on a grade 
than on a level, not only because the horse raises his own weight 
in addition to the load, but is anatomically less capable of 
pulling on a grade than on a level. In general it will be pos- 
sible to plan the work so that loaded cars need not be hauled up 
a grade, linless an embankment is to be formed from a low 
borrow-pit, in which case another method would probably be 
advisable. These computations are chiefly utilized in design- 
ing the method of work — the proportion of horses to cars. An 
example may be quoted from English practice (Hurst), in which 
the cars had a capacity of 3 J cubic yards, weighing 30 cwt. 
empty. Two horses took five "wagons" i of a mile on a level 



11 



§ 140. EARTHWORK. 177 

railroad and made 15 journeys per day of 10 hours, i.e., they 
handled 250 yards per day. In addition to those on the 
"straight road/' another horse was employed to make up 
the train of loaded wagons. With a short lead the straight- 
road horses were employed for this purpose. In the above 
example the number of men required to handle these cars, 
shift the tracks, etc., is not given, and so the exact cost of the 
above work cannot be analyzed. It may be noticed that the 
two horses travelled 22^ miles per day, drawing in one direction 
a load, including the weight of the ears, of about 57,300 lbs., 
or 28.65 net tons. Allowing yl-^ as the necessary tractive 
force, it would require a pull of 477.5 lbs., or 239 lbs. for each 
horse. With a velocity of 220 feet per minute this would amount 
to IJ horse-power per horse, exerted for only a short time, 
however, and allowing considerable time for rest and for draw- 
ing only the empty cars. Gillette claims that the rolling re- 
sistance for such cars on a contractor's track should be con- 
sidered as 40 lbs. per ton (the equivalent of a 2% grade) and 
quotes many figures to support the assertion. Unquestionably 
the resistance on tracks with very light rails, light ties with 
wide spacing and no tamping, would be very great and might 
readily amount to 40 lbs. per ton. In the above ease, the 
resistance could not have been much if any over y^Q-. A re- 
sistance of 40 lbs. per ton would have required each horse to 
pull about 573 lbs. for nearly five hours per day, beside pulling 
the empty cars the rest of the time. This is far greater exertion 
than any ordinary horse can maintain. The cars generally used 
in this country have a capacity of 1 J cubic yards and cost about 
$65 apiece. Besides the shovellers and dumping-gang, several 
men and a foreman will be required to keep the track in order 
and to make the constant shifts that are necessary. Two trains 
are generally used, one of which is loaded while the other is run 
to the dump. Some passing-place is necessary, but this is 
generally provided by having a switch at the cut and running 
the trains on each track alternately. This insures a train of 
ears always at the cut to keep the shovellers employed. The 
cost of hauling per cubic yard can only be computed when the 
number of laborers, cars, and horses employed are known, and 
these will depend on the lead, on the character of the excavation, 
on the grade, if any, etc., and must be so proportioned that the 
shovellers need not wait for cars to fill, nor the dumping-gang 



178 RAILROAD CONSTRUCTION. § 140. 

for material to handle, nor the horses and drivers for cars to 
haul. Much skill is necessary to keep a large force in smooth 
running order. 

(f) Cars and locomotives. 30-lb. rails are the lightest that 
should be used for this work, and 35- or 40-lb. rails are better. 
One or two narrow-gauge locomotives (depending on the length 
of haul), costing about $2500 each, will be necessary to handle 
two trains of about 15 cars each, the cars having a capacity of 
about 2 cubic yards and costing about $100 each. Some cars 
can be obtained as low as $70. A force of about five men and 
a foreman will be required to shift the tracks. The track- 
shifters, except the foreman, may be common laborers. The 
dumping-gang will require about seven men. Even when the 
material is all taken down grade the grades may be too steep for 
the safe hauling of loaded cars down the grade, or for hauling 
empty cars up the grade. Under such circumstances temporary 
trestles are necessary to reduce the grade. When these are 
used, the uprights and bracing are left in the embankment — 
only the stringers being removed. This is largely a necessity, 
but is partially compensated by the fact that the trestle forms a 
core to the embankment which prevents lateral shifting during 
settlement. The average speed of the trains may be taken as 
10 miles per hour or 5 miles of lead per hour. The time lost 
in loading and unloading is estimated (Trautwine) as 9 minutes 
or .15 of an hour. The number of trips per day of 10 hours 

^.jj ^ ^^j 10 ^^ 50 ^^ 

^ -i-(miles of lead) + .15 (miles of lead) 4- .75* 

course this quotient must be a whole number. Knowing the 
number of trains and their capacity, the total number of cubic 
yards handled is known, which, divided into the total daily cost 
of the trains, will give the cost of hauling per yard. The daily 
cost of a train will include 

(a) Wages of engineer, who frequently fires his own engine; 

(6) Fuel, about } to 1 ton of bituminous coal, depending on 
work done ; 

(c) Water, a very variable item, frequently costing $3 to $5 
per day; 

{d) Repairs, variable, frequently at rate of 50 to 60% per year; 

(c) Interest on cost and depreciation, 16 to 40%. 

To these must be added, to obtain the total cost of haul, 

(/) Wages of the gang employed in shifting track. 



§ 141. EARTHWORK. 179 

The above calculation for the number of train loads depends 
on the assumption that 9 minutes is total time lost by a 
locomotive for each round trip. If the haul is very short it 
may readily happen that a steam-shovel cannot fill one train 
of cars before the locomotive has returned with a load of empties 
and is ready to haul a loaded train away. The estimation of 
the number of train loads is chiefly useful in planning 
the work so as to have every tool working at its high- 
est efficiency. Usually the capacity of the steam-shovel 
or the ability to promptly "spot" the cars under the 
shovel is the real limiting agent which determines the daily 
output. 

141. Choice of method of haul dependent on distance. In 
light side-hill work in which material need not be moved more 
than 12 or 15 feet, i.e., moved laterally across the roadbed, 
the earth may be moved most cheaply by mere shovelling. 
Beyond 12 feet scrapers are more economical. At about 100 
feet drag-scrapers and wheelbarrows are equally economical. 
Between 100 and 200 feet wheelbarrows are generally cheaper 
than either carts or drag-scrapers, but wheeled scrapers are 
always cheaper than wheelbarrows. Beyond 500 feet two- 
wheeled carts become the most economical up to about 1700 
feet; then four-wheeled wagons become more economical up to 
3500 feet. Beyond this cars on rails, drawn by horses or by 
locomotives, become cheaper. The economy of cars on rails 
becomes evident for distances as small as 300 feet provided the 
volume of the excavation will justify the outlay. Locomotives 
will always be cheaper than horses and mules, providing the 
work to be done is of sufficient magnitude to justify the piu*- 
chase of the necessary plant and risk the loss in selling the plant 
ultimately as second-hand equipment, or keeping the plant on 
hand and idle for an indefinite period waiting for other 
work. Horses will not be economical for distances much 
over a mile. For greater distances locomotives are more 
economical, but the question of "limit of profitable haul*' 
(§ 148) must be closely studied, as the circumstances are cer- 
tainly not common when it is advisable to haul material much 
over a mile. 

142. Item 4. Spreading. The cost of spreading varies 
with the method employed in dumping the load. When the earth 



180 RAILROAD CONSTRUCTION. § 14S. 

is tipped over the edge of an embankment there is little if any 
necessary work. Trautwine allows about J c. per cubic yard 
for keeping the dumping-places clear and in order. This would 
represent the wages of one man at $1 per day attending to the 
unloading of 1200 two-wheeled carts each carrying J cubic yard. 
1200 carts in 10 hours would mean an average of two per minute, 
which implies more rapid and efficient work than may be de- 
pended on. The allowance is probably too small. When the 
material is dumped in layers some levelling is required, for 
which Trautwine allows 50 to 100 cubic yards as a fair day's 
work, costing from 1 to 2 cents per cubic yard. The cost of 
spreading will not ordinarily exceed this and is frequently 
nothing — all depending on the method of unloading. It should 
be noted that Mr. Morris's examples and computations (Jour. 
Franklin Inst., Sept. 1841) disregard altogether any special 
charge for this item. 

143. Item 5. Keeping Roadways in order. This feature 
is irtiportant as a measure of true economy, whatever the system 
of transportation, but it is often neglected. A petty saving in 
such matters will cost many times as much in increased labor 
in hauling and loss of time. With some methods of haul the 
cost is best combined with that of other items. 

(a) Wheelbarrows. Wheelbarrows should generally be run 
on planks laid on the ground. The adjusting and shifting of 
these planks is done by the wheelers, and the time for it is 
allowed for in the "| minute for short rests, adjusting the 
wheeling plaiik, etc." The actual cost of the planks must be 
added, but it would evidently be a very small addition per cubic 
yard in a large contract. When the wheelbarrows are run on 
planks placed on '^horses'' or on trestles the cost is very appre- 
ciable; but the method is frequently used with great economy. 
The variations in the requirements render any general estimate 
of such cost impracticable. 

(b) Carts and wagons. The cost of keeping roadways in 
order for carts and wagons is sometimes estimated merely as so 
much per cubic yard, but it is evidently a function of the lead. 
The work consists in draining off puddles, filling up ruts, pick- 
ing up loose stones that may have fallen off the loads, and in 
general doing everything that will reduce the traction as much 
as possible. Temporary inclines, built to avoid excessive grade 



§ 144. EARTHWORK. 181 

• 

at some one pointy are often measures of true economy. Traut- 
wine suggests -^^ c. per cubic yard per 100 feet of lead for earth- 
work and Y^Q- c. for rockwork, as an estimate for this item when 
carts are used. 

(c) Cars. When cars are used a shifting-gang, consisting 
of a foreman and several men (say five), are constantly em- 
ployed in shifting the track so that the material may be loaded 
and unloaded where it is desired. The average cost of this 
item may be estimated by dividing the total daily cost of this 
gang by the number of cubic yards handled in one day. 

144. Item 6. Trimming cuts to their proper cross- 
section. This process, often called "sand-papering," must 
be treated as an expense, since the payment received for the 
very few cubic yards of earth excavated is wholly inadequate 
to pay for the work involved. Gillette quotes bids of 2 cents 
per square yard of surface trimmed, and from this argues that, 
for average excavations, it adds to the cost four cents per cubic 
yard of the total excavation. The shallower the cut the greater 
is the proportionate cost. Of course the actual cost to the 
contractor will depend largely on the accuracy of outline de- 
manded by the engineer or inspector. 

145. Item 7. Repairs, wear, depreciation, and interest 
ON COST OF PLANT. The amount of this item evidently depends 
upon the character of the soil — ^the harder the soil the worse the 
wear and depreciation. The interest on cost depends on the 
current borrowing value of money. The estimate for this item 
has already been included in the allowances for horses, carts, 
ploughs, harness, wheelbarrows, steam-shovels, etc. Trautwine 
estimates J c. per cubic yard for picks and shovels. Deprecia- 
tion is generally a large percentage of the cost of earth-working 
jtools, the life of all being limited to a few years, and of many 
tools to a few months. 

146. Item 8. Superintendence and incidentals. The 
incidentals include the cost of water-boys, timekeepers, watch- 
men, blacksmiths, fences, and other precautions to protect the 
public from possible injury, cost of casualty insurance for 
workmen, etc. Although the cost of some of these sub-items 
may be definitely estimated, others are so uncertain that it is 
only possible to make a lump estimate and add say 5 to 7% 
of the sum of the previous items for this item. 



182 RAILROAD CONSTRUCTION. § 147. 

147. Contractor's profit and contingencies. The word " con- 
tingencies" here refers to the abnormal expenses caused by 
freshets, continued wet weather, and ''hard luck," as dis- 
tinguished from mere incidentals which are really normal 
expenses. They are the expenses which literally cannot be 
foreseen, and on which the contractor must 'Hake chances." 
They are therefore included with the expected profit. The 
allowance for these two elements combined is variously esti- 
mated up to 25% of the previously estimated cost of the work,' 
according to the sharpness of the competition, the contractor's 
confidence in the accuracy of his estimates, and the possible un- 
certainty as to true cost owing to unfavorable circumstances. 
The contractor's real profit may vary considerably from this.. 
He often pays clerks, boards and lodges the laborers in shan- 
ties built for the purpose, or keeps a supply-store, and has 
various other items both of profit and expense. His profit 
is largely dependent on skill in so handling the men that all 
can work effectively without interference or delays in wait- 
ing for others. An unusual season of bad weather will often 
affect the cost very seriously. It is a common occurrence 
to find that two contractors may be working on the same kind 
of material and under precisely similar conditions and at the 
same price, and yet one niay be making money and the 
other losing it — all on account of difference of manage- 
ment. 

148. Limit of profitable haul. As intimated in §§ 134 and 
141, there is with every method of haul a limit of distance 
beyond which the expense for excessive hauling will exceed the 
loss resulting from borrowing and wasting. This distance is 
somewhat dependent on local conditions, thus requiring an inde- 
pendent solution for each particular case, but the general prin- 
ciples involved will be^about as follows : Assume that it has been 
determined, as in Fig. 64, that the cut and fill will exactly bal- 
ance between two points, as between e and x, assuming that, as 
indicated in § 132 (9), a trestle has been introduced between s 
and t, thus altering the mass curve to Estxn . . . Since there 
is a balance between A' and C, the material for the fill between 
C and e' must be obtained either by " borrowing " in the im- 
mediate neighborhood or by transportation from the excavation 
between ^' md n\ If cut and fill bave been approximately 



§ 148. EARTHWORK. 183 

between 2' and n'. If cut and fill have been approximately 
balanced in the selection of grade line, as is ordinarily done, 
borrowing material for the fill CV implies a wastage of material 
at the cut z'n' . To compare the two methods, we may place 
against the plan of borrowing and wasting, (a) cost, if any, of 
extra right of wa}^ that may be needed from which to obtain 
earth for the fill C'e'; (6) cost of loosening, loading, hauling 
a distance equal to that between the centers of gravity of the 
borrow-pit and of the fill, and the other expenses incidental to 
borrowing M cubic yards for the fill C'e'\ (c) cost of loosening, 
loading, hauling a distance equal to that between the centers 
of gravity of the cut z'n' and of the spoil-bank, and the other 
expenses incidental to wasting M cubic yards at the cut z'm/: 
(d) cost, if any, of land needed for the spoil-bank. The cost of 
the other plan will be the cost of loosening, loading, hauling (the 
hauling being represented by the trapezoidal figure Cexn), and 
the other expenses incidental to making the fill Ce' with the 
material from the cut z'n' , the amount of material being M cubic 
yards, which is represented in the figure by the vertical ordi- 
nate from e to the line Cn. The difference between these costs 
will be the cost, if any, of land for borrow-pit and spoil-bank 
'plus the cost of loosening, loading, etc. (except hauling and 
roadways) of M cubic yards, minus the difference in cost of the 
excessive haul from Ce to xn and the comparatively short hauls 
from borrow-pit and to spoil-bank. 

As an illustration, taking some of the estimates previously 
given for operating with average material, the cost of all items, 
except hauling and roadways, would be about as follows: 
loosening, with plough, 1.2 c, loading 5.0 c, spreading 1.5 c, 
wear, depreciation, etc., .25 c, superintendence, etc., 1.5 c; 
total 8.95 c. Suppose that the haul for both borrowing and 
wasting averages 100 feet or 1 station. Then the cost of haul 
per yard, using carts, would be (§140, a) [125X3(l+4)]-^600 
= 3.125 c. The cost of roadways would be about 0.1 c. per yard, 
making a total of 3.225 c. per cubic yard. Assume 11^ = 10000 
cubic yards and the area (7exn = 180000 yards-stations or the 
equivalent of 10000 yards hauled 1800 feet. This haul would 
cost [125X3(18 + 4)]--600 = 13.75 c. per cubic yard. The cost 
of roadways will be 18 X .1 or 1.8 c, making a total of 15.55 c. for 
hauling and roadways. The difference of cost of hauling and 
roadways will be 15.55-(2X3.225) =9.10 c. per yard or $910 



184 RAILROAD CONSTRUCTION. § 149. 

for the 10000 yards. Offsetting this is the cost of loosening, etc., 
10000 yards, at 8.95 c, costing $895. These figures may be 
better compared as follows : 

f Loosening, etc., 10000 yards, @ 8.95 c. $ 895. 

L Had \ ^^^^^^S, ** 10000 " @, 15.55 c. 1555. 

J $2450. 

f Loosening, etc., 10000 yards (borrowed), @, 8.95 c. $895. 
10000 ** (wasted), ® 8.95 c. 895. 



BoEROWiNG I Hauling, etc., 10000 •* (borrowed), (^ 3.225 c. 322.50 

;. 322.50 

$2435.00 



WasSng. ^ " " ^^^^^ " (wasted), ^ 3.225 c. 322.50 



I 



These costs are practically balanced, but no allowance has 
been made for right of way. If any considerable amount had 
to be paid for that, it would decide this particular case in favor 
of the long haul. This shows that under these conditions 1800 
feet is about the limit of profitable haul, the land costing nothing 
extra. 

BLASTING. 

149. Explosives. The effect of blasting is due to the ex- 
tremely rapid expansion of a gas which is developed by the 
decomposition of a very small amount of solid matter. Blasting 
compounds may be divided into two general classes, (a) slow- 
burning and {h) detonating. Gunpowder is a type of the slow- 
burning compounds. These are generally ignited by heat; the 
ignition proceeds from grain to grain; the heat and pressure 
produced are comparatively low. Nitro-glycerine is a type of 
the detonating compounds. They are exploded by a shock 
which instantaneously explodes the whole mass. The heat and 
pressure developed are far in excess of that produced by the 
explosion of powder. Nitro-glycerine is so easily exploded 
that it is very dangerous to handle. It was discovered that if 
the nitro-glycerine was absorbed by a spongy material like infu- 
sorial earth, it was much less liable to explode, while its power 
when actually exploded was practically equal to that of the 
amount of pure nitro-glycerine contained in the dynamite, which 
is the name given to the mixture of nitro-glycerine and infusorial 
earth. Nitro-glycerine is expensive; many other explosive 
chemical compounds which properly belong to the slow-burning 



§ 150. EARTHWORK. 18 



r. 



class are comparatively cheap. It has been conclusively demon- 
strated that a mixture of nitro-glycerine and some of the cheaper 
chemicals has a greater explosive force than the sum of the 
strengths of the component parts when exploded separately. 
Whatever the reason, the fact seems established. The reason is 
possibly that the explosion of the nitro-glycerine is sufficiently 
powerful to produce a detonation of the other chemicals, which 
is impossible to produce by ordinary means, and that this explo- 
sion caused by detonation is more powerful than an ordinary 
explosion. The majority of the explosive compounds and 
"powders'^on the market are of this character — a mixture of 
20 to 60 per cent, of nitro-glycerine with variable proportions of 
one or more of a great variety of explosive chemicals. 

The choice of the explosive depends on the character of the 
rock. A hard brittle rock is most effectively blasted by a 
detonating compound. The rapidity " with which the full force 
of the explosive is developed has a shattering effect on a brittle 
substance. On the contrary, some of the softer tougher rocks 
and indurated clays are but little affected by dynamite. The 
result is but little more than an enlargement of the blast-hole. 
Quarrying must generally be done with blasting-powder, as the 
quicker explosives are too shattering. Although the results 
obtained by various experimenters are very variable, it may be 
said that pure nitro-glycerine is eight times as powerful as black 
powder, dynamite (75% nitro-glycerine) six times, and gun- 
cotton four to six times as powerful. For open work where 
time is not particularly valuable, black powder is by far the 
cheapest, but in tunnel-headings, whose progress determines the 
progress of the whole work, dynamite is so much more effective 
and so expedites the work that its use becomes economical. 

150. Drilling. Although many very complicated forms of 
drill-bars have been devised, the best form (with slight modifi- 
cations to suit circumstances) is as shown in Fig. 66 (a), and (6). 
The width should flare at the bottom (a) about 15 to 30%. For 
hard rock the curve of the edge should be somewhat flatter and 
for soft rock somewhat more curved than shown, Fig. 66, (a). 
Sometimes the angle of the two faces is varied from that given, 
Fig. 66, (6) and occasionally the edge is purposely blunted so 
as to give a crushing rather than a cutting effect. The drills 
will require sharpening for each 6 to 18 inches depth of hole, 

and will require ^ iiew edge tQ be worked every 2 to 4 days. 



186 



RAILROAD CONSTRUCTION. 



151. 



For drilling vertical holes the churn-drill is the most econom- 
ical. The drill-bar is of iron, about 6 to 8 feet long, IJ" in 
diameter, weighs about 25 to 30 lbs., and is shod with a piece 
of steel welded on. The bar is lifted a few inches between each 
blow, turned partially around, and allowed to fall, the impact 
doing the work. From 5 to 15 feet of holes, depending on the 
character of the rock, is a fair day's work — 10 hours. In very 
soft rocks even more than this may be done. This method is 





Fig. 66. 



inapphcable for inclined holes or even for vertical holes in con- 
fined places, such as tunnel-headings. For such places the only 
practical hand method is to use hammers. This may be done 
by light drills and light hammers (one-man work), or by heavier 
drills held by one man and struck by one or two men with heavy 
hammers. The conclusion of an exhaustive investigation as to 
the relative economy of light or heavy hammers is that the light- 
hammer method is more economical for the softer rocks, the 
heavy-hammer method is more economical for the harder rocks, 
but that the light-hammer method is always more expeditious 
and hence to be preferred when time is important. 

The subject of machine rock-drills is too vast to be treated 
here. The method is only practicable when the amount of 
work to be done is large, and especially when time is valuable. 
The machines are generally operated by compressed air for tun- 
nel-work, thus doing the additional service of supplying fresh 
air to the tunnel-headings where it is most needed. The cost 
per foot of hole drilled is quite variable, but is usually some- 
what less than that of hand-drilling — sometimes but a small 
fraction of it. 

151. Position and direction of drill-holes. As the cost of 
drilling holes is the largest single item in the total cost of blast- 
ing, it is necessary that skill and judgment should be used in so 



§152. 



EARTHWORK. 



187 



locating the holes that the blasts will be most effective. The 
greatest effect of a blast will evidently be in the direction of the 
"line of least resistance." In a strictly homogeneous material 
this will be the shortest hne from the center of the explosive to 
the surface. The variations in homogeneity on account of 
laminations and seams require that each case shall be judged 
according V to experience. In open-pit blasting it is generally 
easy to obtain two and sometimes three exposed faces to the 

rock, making it a simple matter 
to drill holes so that a blast will 
do effective work. When a solid 
face of rock must be broken into, 
as in a tunnel-heading, the work 
is necessarily ineffectual and ex- 
pensive. A conical or wedge- 
shaped mass will first be blown 
out by simultaneous blasts in 
the holes marked 1, Fig. 67; 
blasts in the holes marked 2 and 
3 will then complete the cross- 
section of the heading. A great saving in cost may often be 
secured by skilfully taking advantage of seams, breaks, and irreg- 
ularities. When the work is economically done there is but little 
noise or throwing of rock, a covering of old timbers and branches 
of trees generally sufficing to confine the smaller pieces which 
would otherwise fly up. 

152. Amount of explosive. The amount of explosive required 
varies as the cube of the line of least resistance. The best 
results are obtained when the line of least resistance is f of the 
depth of the hole ; also when the powder fills about ^ of the hole. 
For average rock the amount of powder required is as follows : 




DRILL HOLES IN TUNNEL HEADING 
Fig. 67. 



Line of least resistance 


2 ft. 
i lb. 


4 ft. 
2 lbs. 


6 ft. 
61 lbs. 


8 ft. 


Weight of powder 


16 lbs. 







Strict compliance with all of the above conditions would re- 
quire that the diameter of the hole should vary for every case. 
While this is impracticable, there should evidently be some 
variation in the size of the hole, depending on the work to be 
done. For example, a 1" hole, drilled 2' 8" deep, with its 
line of least resistance 2'. and loaded with J lb, of powder, would 



188 RAILROAD CONSTRUCTION. § 153. 

be filled to a depth of 9J'', which is nearly i of the depth. A 
3'' hole, drilled 8' deep^ with its line of least resistance 6', and 
loaded with 6| lbs. of powder, would be filled to a depth of over 
28'', which is also nearly J of the depth. One pound of blasting- 
powder will occupy about 28 cubic inches. Quarrying necessi- 
tates the use of numerous and sometimes repeated light charges of 
powder, as a heavy blast or a powerful explosive like dynamite 
is apt to shatter the rock. This requires more powder to the 
cubic yard than blasting for mere excavation, which may usually 
be done by the use of J to |^ lb. of powder per cubic yard of easy 
open blasting. On account of the great resistance offered by 
rock when blasted in headings in tunnels, the powder used per 
cubic yard will run up to 2, 4, and even 6 lbs. per cubic yard. 
As before stated, nitro-glycerine is about eight times (and 
dynamite about six times) as powerful as the same weight of 
powder. 

153. Tamping. Blasting-powder and the slow-burning ex- 
plosives require thorough tamping. Clay is probably the best, 
but sand and fine powdered rock are also used. Wooden plugs, 
inverted expansive cones, etc., are periodically reinvented by 
enthusiastic inventors, only to be discarded for the simpler 
methods. Owing to the extreme rapidity of the development 
of the force of a nitro-glycerine or dynamite explosion, tamping 
is not so essential with these explosives, although it unquestion- 
ably adds to their effectiveness. Blasting under water has been 
effectively accomplished by merely pouring nitro-glycerine into 
the drilled holes through a tube and then exploding the charge 
without any tamping except that furnished by the superincum- 
bent water. It has been found that air-spaces about a charge 
make a material reduction in the effectiveness of the explosion. 
It is therefore necessary to carefully ram the explosive into a 
solid mass. Of course the liquid nitro-glycerine needs, no ram- 
ming, but dynamite should be rammed with a wooden rammer. 
Iron should be carefully avoided in ramming gunpowder. A 
copper bar is generally used. 

154. Exploding the charge. Black powder is generally ex- 
ploded by means of a fuse which is essentially a cord in which 
there is a thin vein of gunpowder, the cord being protected by 
tar, extra linings of hemp, cotton, or even gutta-percha. The 
fuse is inserted into the middle of the charge, and the tamping 
carefully packed around it so that it will not be injured. To 



^i. 



§ 155. EARTHWORK. 189 

produce the detonation required to explode nitro-glycerine and 
dynamite, there must be an initial explosion of some easily 
ignited explosive. This is generally accomplished by means of 
caps containing fulminating-powder which are exploded by 
electricity. The electricity (in one class of caps) heats a very 
fine platinum wire to redness, thereby igniting the sensitive 
powder, or (in another class) a spark is caused to jump through 
the powder between the ends of two wires suitably separated. 
Dynamite can also be exploded by using a small cartridge of 
gunpowder which is itself exploded by an ordinary fuse. 

155. Cost. Trautwine estimated the cost of blasting (for 
mere excavation) as averaging 45 cents per cubic yard, falling 
as low as 30 cents for easy but brittle rock, and running up to 
60 cents and even $1 when the cutting is shallow, the rock 
especially tough, and the strata unfavorably placed. Increased 
costs of labor and material may add 50 to 100% to these esti- 
•mates. 

156, Classification of excavated material. The classification 
of excavated material is a fruitful source of dispute between 
contractors and railroad companies, owing mainly to the fact 
that the variation between the softest earth and the hardest rock 
is so gradual that it is very difficult to describe distinctions 
between different classifications which are unmistakable and 
indisputable. The classification frequently used is (a) earth, 
(6) loose rock, and (c) solid rock. As blasting is frequently 
used to loosen ^' loose rock" and even ^' earth" (if it is frozen), 
the fact that blasting is employed cannot be used as a criterion, 
especially as this would (if allowed) lead to unnecessary blasting 
for the sake of classifying material as rock. 

Earth. This includes clay, sand, gravel, loam, decomposed 
rock and slate, boulders or loose stones not greater than 1 cubic 
foot (3 cubic feet, P. R. R.), and sometimes even ^'hard-pan.'' 
In general it will signify material which can be loosened by a 
plough with two horses, or with which one picker can keep one 
shoveller busy. 

Loose rock. This includes boulders and loose stones of more 
than one cubic foot and less than one cubic 3^ard; stratified rock, 
not more than six inches thick, separated by a stratum of clay; 
also all material (not classified as earth) which may be loosened 
by pick or bar and which ^' can be quarried without blasting, 
although blasting may occasionally be resorted to," 



190 RAILROAD CONSTRUCTION. § 157. 

Solid rock includes all rock found in masses of over one cubic 
yard which cannot be removed except by blasting. 

It is generally specified that the engineer of the railroad 
company shall be the judge of the classification of the material, 
but frequently an appeal is taken from his decisions to the 
courts. 

157. Specifications for earthwork. The following specifica- 
tions, issued by the Norfolk and Western R, R., represent the 
average requirements. It should be remembered that very 
strict specifications invariably increase the cost of the work, 
and frequently add to the cost more than is gained by improved 
quality of work. 

1. The grading will be estimated and paid for by the cubic 
yard, and will include clearing and grubbing, and all open ex- 
cavations, channels, and embankments required for the forma- 
tion of the roadbed, and for turnouts and sidings; cutting all 
ditches or drains about or contiguous to the road; digging the 
foundation-pits of all culverts, bridges, or walls; reconstructing 
turnpikes or common roads in cases where they are destroyed or 
interfered with; changing the course or channel of streams; and 
all other excavations or embankments connected with or incident 
to the construction of said Railroad. 

2. All grading, except where otherwise specified, whether 
for cuts or fills, will be measured in the excavations and will be 
classified under the following heads, viz.: Solid Rock, Loose 
Rock, Hard-pan, and Earth. 

Solid Rock shall include all rock occurring in masses which, 
in the judgment of the said Engineer Maintenance of Way, may 
be best removed by blasting. 

Loose Rock shall include all kinds of shale, soapstone, and 
other rock which, in the judgment of the said Engineer Main- 
tenance of Way, can be removed by pick and bar, and is soft and 
loose enough to be removed without blasting, although blasting 
may be occasionally resorted to ; also, detached stone of less than 
one (1) cubic yard and more than one (1) cubic foot. 

Hard-pan shall consist of tough indurated clay or cemented 
gravel, which requires blasting or other equally expensive 
means for its removal, or which cannot be ploughed with less 
than four horses and a railroad plough, or which requires two 
pickers to a shoveller, the said Engineer Maintenance of Way 
to be the judge of these conditions. 



I 



§ 157. EARTHWORK. 191 

Earth shall include all material of an earthy nature, of what- 
ever name or character, not unquestionably loose rock or hard- 
pan as above defined. 

Powder. The use of powder in cuts will not be considered 
as a reason for any other classification than earth, unless the 
material in the cut is clearly other than earth under the above 
specifications. 

3. Earth, gravel, and other materials taken from the exca- 
vations, except when otherwise directed by the said Engineer 
Maintenance of Way or his assistant, shall be deposited in the 
adjacent embankment; the cost of removing and depositing 
which, when the distance necessary to be hauled is not more 
than sixteen hundred (1600) feet, shall be included in the price 
paid for the excavation. 

4. Extra Haul will be estimated and paid for as follows: 
whenever material from excavations is necessarily hauled a 
greater distance than sixteen hundred (1600) feet, there shall be 
paid in addition to the price of excavation the price of extra 
haul per 100 feet, or part thereof, after the first 1600 feet; the 
necessary haul to be determined in each case by the said Engi- 
neer Maintenance of Way or his assistant, from the profile and 
cross-sections, and the estimates to be in accordance therewith. 

5. All embankments shall be made in layers of such thick- 
ness and carried on in such manner as the said Engineer Mainte- 
nance of Way or his assistant may prescribe, the stone and heavy 
materials being placed in slopes and top. And in completing 
the fills to the proper grade such additional heights and fulness 
of slope shall be given them, to provide for their settlement, as 
the said Engineer Maintenance of Way, or his assistant, may 
direct. Embankments about masonry shall be built at such 
times and in such manner and of such materials as the said Engi- 
jaeer Maintenance of Way or his assistant may direct. 

6. In procuring materials for embankments from without 
the line of the road, and in wasting materials from cuttings, the 
place and manner of doing it shall in each case be indicated by 
the Engineer Maintenance of Way or his assistant; and care 
must be taken to injure or disfigure the land as little as possible. 
Borrow-pits and spoil-banks must be left by the Contractor in 
Tegular and sightly shape. 

7. The lands of the said Railroad Company shall be cleared 
to the extent required by the said Engineer Maintenance of 



192 RAILROAD CONSTRUCTION. § 157. 

Way, or his assistant, of all trees, brushes, logs, and other perish- 
able materials, which shall be destroyed by burning or deposited 
in heaps as the said Engineer Maintenance of Way, or his assist- 
ant, may direct. Large trees must be cut not more than two 
and one-half (2 J) feet from the ground, and under embank- 
ments less than four (4) feet high they shall be cut close to the 
ground. All small trees and bushes shall be cut close to the 
ground. 

8. Clearing shall be estimated and paid for by the acre or 
fraction of an acre. 

9. All stumps, roots, logs, and other obstructions shall be 
grubbed out, and removed from all places where embankments 
occur less than two (2) feet in height; also, from all places where 
excavations occur and from such other places as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

10. Grubbing shall be estimated and -paid for by the acre or 
fraction of an acre, 

11. Contractors, when directed by the said Engineer Main- 
tenance of Way or his assistant in charge of the work, will deposit 
on the side of the road, or at such convenient points as may be 
designated, any stone, rock, or other materials that they may 
excavate; and all materials excavated and deposited as above, 
together with all timber removed from the line of the road, will 
be considered the property of the Railroad Company, and the 
Contractors upon the respective sections will be responsible for 
its safe-keeping until removed by said Railroad Company, or 
until their work is finished. 

12. Contractors will be accountable for the maintenance of 
safe and convenient places wherever public or private roads are 
in any way interfered with by them during the progress of the 
work. They will also be responsible for fences thrown down, 
and for gates and bars left open, and for all damages occasioned 
thereby. 

13. Temporary bridges and trestles^ erected to facilitate the 
progress of the work, in case of delays at masonry structures 
from any ^ause, or for other reasons, will be at the expense of 
the Contractor. 

14. The line of road or the gradients may be changed in any 
manner, and at any time, if the said Engineer Maintenance of 
Way or his assistant shall consider such a change necessary or 
expedient; but no claim for an increase in prices of excavation 



ii 



§ 157. EARTHWORK. 193 

or embankment on the part of the Contractor will be allowed 
or considered unless made in writing before the work on that 
part of the section where the alteration has been made shall have 
been commenced. The said Engineer Maintenance of Way or 
his assistant may also, on the conditions last recited, increase or 
diminish the length of any section for the purpose of more nearly 
equahzing or balancing the excavations and embankments, or 
for any other reason. 

15. The roadbed will be graded as directed by the said En- 
gineer Maintenance of Way or his assistant, and in conformity 
with such breadths, depths, and slopes of cutting and filling as 
he may prescribe from time to time, and no part of the work 
will be finally accepted until it is properly completed and dressed 
off at the required grade. 



CHAPTER IV. 

TRESTLES. 

158. Extent of use. Trestles constitute from 1 to 3% of the 
length of the average railroad. It was estimated in 1889 that 
there was then about 2400 miles of single-track railway trestle 
in the United States, divided among 150,000 structures and esti- 
mated to cost about $75,000,000. The annual charge for main- 
tenance, estimated at J of the cost, therefore amounted to about 
$9,500,000 and necessitated the annual use of perhaps 300,000,000 
ft. B. M. of timber. The corresponding figures at the present 
time must be somewhat in excess of this. The magnitude of 
this use, which is causing the rapid disappearance of forests, has 
resulted in endeavors to limit the use of timber for this purpose. 
Trestles may be considered as justifiable under the following 
conditions : 

a. Permanent trestles. 

1. Those of extreme height — then called viaducts and 
frequently constructed of steel, as the Kinzua viaduct, 302 
feet high. 

2. Those across waterways — e.g., that across Lake Pontchar- 
train, near New Orleans, 22 miles long. 

3. Those across swamps of soft deep mud, or across a river- 
bottom, liable to occasional overflow. 

b. Temporary trestles. 

1. To open the road for trafiic as quickly as possible — often 
a reason of great financial importance. 

2. To quickly replace a more elaborate structure, destroyed 
by accident, on a road already in operation, so that the inter- 
ruption to traffic shall be a minimum. 

3. To form an earth embankment with earth brought from 
a distant point by the train-load, when such a measure would 
cost less than to borrow earth in the immediate neighborhood. 

4. To bridge an opening temporarily and thus allow time to 
learn the regimen of a stream in order to better proportion the 

194 



§159. TRESTLES. 195 

size of the waterway and also to facilitate bringing suitable stone 
for masonry from a distance. In a new country there is always 
the double danger of either building a culvert too small, requir- 
ing expensive reconstruction, perhaps after a disastrous washout, 
or else wasting money by constructing the culvert unnecessarily 
large. Much masonry has been built of a very poor quality of 
stone because it could be conveniently obtained and because 
good stone was unobtainable except at a prohibitive cost for 
transportation. Opening the road for traffic by the use of 
temporary trestles obviates both of these difficulties. 

159. Trestles vs. embankments. Low embankments are very 
much cheaper than low trestles both in first cost and mainte- 
nance. Very high embankments are very expensive to con- 
struct, but cost comparatively little to maintain. A trestle of 
equal height may cost much less to construct, but will be expen- 
sive to maintain — perhaps J of its cost per year. To determine 
the height beyond which it will be cheaper to maintain a trestle 
rather than build an embankment, it will be necessary to allow 
for the cost of maintenance. The height will also depend on 
the relative cost of timber, labor, and earthwork. At the pres- 
ent average values, it will be found that for less heights than 
25 feet the first cost of an embankment will generally be less 
than that of a trestle; this implies that a permanent trestle 
should never be constructed with a height less than 25 feet except 
for the reasons given in § 158. The height at which a permanent 
trestle is certainly cheaper than earthwork is more uncertain. 
A high grade line joining two hills will invariably imply at least 
a culvert if an embankment is used. If the culvert is built of 
masonry, the cost of the embankment will be so increased that 
the height at which a trestle becomes economical will be mate- 
rially reduced. The cost of an embankment increases much 
more rapidly than the height — with very high embankments 
more nearly as the square of the height — while the cost of 
trestles does not increase as rapidly as the height. Although 
local circumstances may modify the application of any set rules, 
it is probably seldom that it will be cheaper to build an embank- 
ment 40 or 50 feet high than to permanently maintain a wooden 
trestle of that height. A steel viaduct would probably be the 
best solution of such a case. These are frequently used for 
permanent structures, especially when very high. The cost of 
maintenance is much less than that of wood, which makes the 



HHWtf 



196 RAILROAD CONSTRUCTION. § 160. 

use of steel preferable for permanent trestles unless wood is 
abnormally cheap. Neither the cost nor the construction of 
steel trestles will be considered in this chapter. 

1 60. Two prmcipal types. There are two principal types of 
wooden trestles — pile trestles and framed trestles. The great 
objection to pile trestles is the rapid rotting of the portion of the 
pile which is underground, and the difficulty of renewal. The 
maximum height of pile trestles is about 30 feet, and even this 
height is seldom reached. Framed trestles have been con- 
structed to a height of considerably over 100 feet They are 
frequently built in such a manner that any injured piece may be 
readily taken out and renewed without interfering with traffic. 
Trestles consist of two parts — the supports called ^^ bents," and 
the stringers and floor system. As the stringers and floor system 
are the same for both pile and framed trestles, the " bents '' are 
all that need be considered separately. 



PILE TRESTLES. 

161. Pile bents. A pile bent consists generally of four piles 
driven into the ground deep enough to afford not only sufficient 
vertical resistance but also lateral resistance. On top of these 
piles is placed a horizontal ''cap." The caps are fastened to 
the tops of the piles by methods illustrated in Fig. 68. The 
method of fastening shown in each case should not be considered 
as applicable only to the particular type of pile bent used to illus- 
trate it. Fig. 68 (a and d) illustrates a mortise- joint with a hard« 
wood pin about IJ'' in diameter. The hole for the pin should 
be bored separately through the cap and the mortise, and the 
hole through the cap should be at a slightly higher level than 
that through the mortise, so that the cap will be drawn down 
tight when the pin is driven. Occasionally an iron dowel (an 
iron pin about 1^" in diameter and about 6" long) is inserted 
partly in the cap and partly in the pile. The use of driff-bolts, 
shown in Fig. 68 (b), is cheaper in first cost, but renders repairs 
and renewals very troublesome and expensive. '' Split caps," 
shown in Fig. 68 (c), are formed by bolting two half -size strips 
on each side of a tenon on top of the pile. Repairs are very 
easily and cheaply made without interference with the traffic 
and without injuring other pieces of the bent. The smaller 
pieces are more easily obtainable in a sound condition; the 



§161, 



TRESTLES= 



197 



decay of one does not affect the other, and the fu'st cost is but 
Uttle if any greater than the method of using a single piece. For 
further discussion, see § 170. 

For very Hght traffic and for a height of about 5 feet three 
vertical piles will suffice, as shown in Fig, 68 (a). Up to a height 




of 8 or 10 feet four piles may be used without sway-bracing, as 
in Fig. 68 (b), if the piles have a good bearing. For heights 
greater than 10 feet sway-bracing is generally necessary. The 
outside piles are frequently driven with a batter varying from 
1 : 12 to 1 : 4. 

Piles are made, if possible, from timber obtained in the 
vicinity of the work. Durability is the great requisite rather 
than strength, for almost any timber is strong enough (except 
as noted below) and will be suitable if it will resist rapid decay. 
The following list is quoted as being in the order of preference 
on account of durability ! 



1. Red cedar 

2. Red cypress 

3. Pitch-pine 

4. Yellow pine 



5. White pine 

6. Redwood 

7. Elm 

8. Spruce ^ 



9. White oak 

10. Post-oak 

11. Red oak 



12. Black oak 

13. Hemlock 

14. Tamarac 



Red-cedar piles are said to have an average life of 27 years 
with a possible maximum of 50 years, but the timber is rather 



198 RAILROAD CONSTRUCTION. § 162. 

weak, and if exposed in a river to flowing ice or driftwood is 
apt to be injured. Under these circumstances oak is prefer- 
able, although its life may be only 13 to 18 years. 

162. Methods of driving piles. The following are the prin- 
cipal methods of driving piles : 

a, A hammer weighing 2000 to 3000 lbs. or more, sliding 
in guides, is drawn up by horse-power or a portable engine, and 
allowed to fall freely. ^ 

h. The same as above except that the hammer does not fall 
freely, but drags the rope and revolving drum as it falls and is 
thus quite materially retarded. The mechanism is a little more 
simple, but is less effective, and is sometimes made deliberately 
deceptive by a contractor by retarding the blow, in order to 
apparently indicate the requisite resistance on the part of 
the pile. 

The above methods have the advantage that the mechanism 
is cheap and can be transported into a new country with com- 
parative ease, but the work done is somewhat ineffective and 
costly compared with some of the more elaborate methods 
given below. 

c. Gunpowder pile-drivers, which automatically explode a 
cartridge every time the hammer falls. The explosion not only 
forces the pile down, but throws up the hammer for the next 
blow. For a given height of fall the effect is therefore doubled. 
It has been shown by experience, however, that when it is at- 
tempted to use such a pile-driver rapidly the mechanism be- 
comes so heated that the cartridges explode prematurely, and the 
method has therefore been abandoned. 

d. Steam pile-drivers, in which the hammer is operated 
directly by steam. The hammer falls freely a height of about 
40 inches and is raised again by steam. The effectiveness is 
largely due to the rapidity of the blows, which does not allow 
time between the blows for the ground to settle around the pile 
and increase the resistance, which does happen when the blows 
are infrequent. ^^The hammer-cylinder weighs 5500 lbs., and 
with 60 to 75 lbs. of steam gives 75 to 80 blows per minute. 
With 41 blows a large unpointed pile was driven 35 feet into a 
hard clay bottom in half a minute." Such a driver would cost 
about $800. 

The above four methods are those usual for dry earth. In 
very soft wet or sandy soils, where an unlimited supply of water 



^. 



§163. 



TRESTLES. 



199 




is available, the water-jet is sometimes employed. A pipe is 
driven along the side of the pile and extends to the pile-point. 
If water is forced through the pipe, it loosens the sand around 
the point and, rising along the sides, decreases the side resist- 
ance so that the pile sinks by its own weight, aided perhaps by 
extra weights loaded on. This loading may be accompHshed by 
connecting the top of the pile and the pile-driver by a block 
and tackle so that a portion of the weight of the pile-driver is 
continually thrown on the pile. 

Excessive driving frequently fractures the pile below the 
surface and thereby greatly weakens its bearing power. To 
prevent excessive *' brooming '' of the top of the 
pile, owing to the action of the hammer, the top 
should be protected by an iron ring fitted to the 
top of the pile. The ^'brooming" not only ren- 
ders the driving ineffective and hence uneconomi- 
cal, but vitiates the value of any test of the bearing 
power of the pile by noting the sinking due to a 
given weight falling a given distance. If the pile 
is so soft that brooming is unavoidable, the top 
should be adzed off frequently, and especially 
should it be done just before the final blows which are to test its 
bearing-power. 

In a new country judgment and experience will be required 
to decide intelligently whether to employ a simple drop-hammer 
machine, operated by horse-power and easily transported but 
uneconomical in operation, or a more complicated machine 
working cheaply and effectively after being transported at 
greater expense. 

163. Pile-driving formulae. If E = the resistance of a pile, 
and s the set of the pile during the last blow, w the weight of 
the pile-hammer, and h the fall during the last blow, then we 

mav state the approximate relation that Rs=wh, or R = — . 

This is the basic principle of aU rational formulae, but the maxi- 
rrnim weight which a pile will sustain after it has been driven 
some time is by no means the same as the resistance of the pile 
during the last blow. There are also many other modifying 
elements which have been variously allowed for in the many 
proposed formulae. The formulae range from the extreme of 
empirical simplicity to very comphcated attempts to allow 



Fig. 69. 



200 RAILROAD CONSTRUCTION. § 163. 

properly for all modifying causes. As the simplest rule, the 
A. R. E. A. specifications require that the piles shall be driven 
until the pile will not sink more than 2i inches under five consecu- 
tive blows of a 3000-lb. hammer falling 15 feet. The ^'Engineering 

News formula" * gives the safe load as ~—-rj in which w = 

s + 1 

weight of hammer, /i=fall in feet, s = set of pile in inches under 

the last blow. This formula is derived from the above basic 

formula by calling the safe load ^ of the final resistance, and 

by adding (arbitrarily) 1 to the final set (s) as a compensation 

for the extra resistance caused by the settling of earth around 

the pile between each blow. This formula is used only for 

ordinary hammer-driving. When the piles are driven by a 

steam pile-driver the formula becomes safe load = — -. For 

s + 0.1 

the "gunpowder pile-driver/' since the explosion of the cartridge 

drives the pile in with the same force with which it throws the 

hammer upward, the effect is twice that of the fall of the hammer, 

^wh 
and the formula becomes safe load = —-7:—. In these last two 

s + 0.1 

formulae the constant in the denominator is changed from s + 1 

to s + 0.1. The constant (1.0 or 0.1) is supposed to allow, as 

before stated, for the effect of the extra resistance caused by the 

earth settling around the pile between each blow. The more 

rapid the blows the less the opportunity to settle and the less 

the proper value of the constant. 

The above formulae have been given on account of their 
simplicity and their practical agreement with experience. Many 
other formulae have been proposed, the majority of which are 
more complicated and attempt to take into account the weight of 
the pile, resistance of the guides, etc. While these elements, 
as well as many others, have their influence, their effect is so 
overshadowed by the indeterminable effect of other elements — 
as, for example, the effect of the settlement of earth around the 
pile between blows — that it is useless to attempt to employ any- 
thing but a purely empirical formula. 

Examples. 1. A pile was driven with an ordinary hammer 
weighing 2500 pounds until the sinking under five consecutive 
*blows was 15 J inches. The fall of the hammer during the last 

* Engineering News, Nov. 17, 1892. 



^. 



§164. 



TKESTLES. 



201 



blows was 24 feet. What was the safe bearing power of the 
pile? 



2wh 2X2500X24 120000 
s + l~(iXl5.5)+l~ 4.1 



29300 pounds. 



2. Piles are being driven into a firm soil with a steam pile- 
driver until they show a safe bearing power of 20 tons. The 
hammer weighs 5500 pounds and its fall is 40 inches. What 
should be the sinking under the final blow? 



40000 = 



2wh 2X5500X3.33 



s= 



s + 0.1 
36667 



40000 



s + 0.1 ' 
0.1 =.81 inch. 



164. Pile-pomts and pile-shoes. Piles are generally sharpened 
to a blunt point. If the pile is liable to strike boulders, sunken 

logs, or other obstructions which are 
liable to turn the point, it is necessary 
to protect the point by some form of 
shoe. Several forms in cast iron have 
been used, also a wrought-iron shoe, 
ha\dng four '^ straps" radiating from 
the apex, the straps being nailed on to 
the pile, as shown in Fig. 70 (h). The 
cast-iron form shown in Fig. 70 (a) 
has a base cast around a drift-bolt. 
The recess on the top of the base re- 
FiG. 70. ceives the bottom of the pile and pre- 

vents a tendency to spht the bottom of the pile or to force the 
shoe off laterally. 

165. Details of design. No theoretical calculations of the 
strength of pile bents need be attempted on account of the ex- 
treme complication of the theoretical strains, the uncertainty as 
to the real strength of the timber used, the variabihty of that 
strength with time, and the insignificance of the economy that 
wouLd be possible even if exact sizes could be computed. The 
caps are generally 14 feet long (for single track) with a cross- 
section 12"X12" or 12"X14". ''Split caps" would consist 




202 RAILROAD CONSTRUCTION. § 166. 

of two pieces 6"X12". The sway-braces, never used for less 
heights than 6', are made of 3'' X 12'' timber, and are spiked on 
with f" spikes 8" long. The floor system will be the same as 
that described later for framed trestles. 

i66. Specifications for timber piles (Adopted 1909 by 
Amer. Rwy. Eng. Assoc). 1. This grade [railroad heart grade] 
includes white, burr, and post oak; longleaf pine, Douglas 
fir, tamarack, Eastern white and red cedar, chestnut. Western 
cedar, redwood and cypress. 2. Piles shall be cut from sound 
trees; shall be close-grained and solid, free from defects, such as 
injurious ring shakes, large and unsound or loose knots, decay 
or other defects, which may materially impair their strength or 
durability. In Eastern red or white cedar a small amount of 
heart rot at the butt, which does not materially injure the 
strength of the pile, will be allowed. 3. Piles must be butt cut 
above the ground swell and have a uniform taper from butt to 
tip. Short bends will not be allowed. A line drawn from the 
center of the butt to the center of the tip shall lie within the body 
of the pile. 4. Unless otherwise allowed, piles must be cut when 
sap is down. Piles must be peeled soon after cutting. All 
knots shall be trimmed close to the body of the pile. 5. The 
minimum diameter at the tips of round piles shall be 9 inches 
for lengths not exceeding 30 feet; 8 inches for lengths over 30 
feet but not exceeding 50 feet, and 7 inches for lengths over 50 
feot The minimum diameter at one-quarter of the length from 
the butt shall be 12 inches and the maximum diameter at the 
butt 20 inches. 6. The minimum width of any side of the tip 
of a square pile shall be 9 inches for lengths not exceeding 30 feet; 
8 inches for lengths over 30 feet but not exceeding 50 feet and 7 
inches for lengths over 50 feet. The minimum width of any side 
at one-quarter of the length from the butt shall be 12 inches. 7. 
Square piles shall show at least 80% heart ou each side at any 
cross-section of the stick, and all round piles shall show at least 
lOi inches diameter of heart at the butt. 

The second grade {'' Railroad falsework grade ") includes 
other woods which '^ will stand driving " and which cannot pass 
the specification for proportion of heart; also, they are usually 
not peeled. 

167. Pile driving — principles of practice. As adopted by the 
Amer. Rwy. Eng. Assoc. 1911 and revised 1915. 

1. A thorough exploration of the soil by borings, or prehminary 



§167. TRESTLES. 203 

test piles, is the most important prerequisite to the design and 
construction of pile foundations. 

2. Soil consisting wholly or chiefly of sand is most favorable 
to the use of the water-jet. 

3. In harder soils containing gravel the use of the jet 
may be advantageous, if sufficient volume and pressure be 
provided. 

4. In clay it may be economical to bore several holes in the 
soil with the aid of the jet before driving the pile, thus securing 
the accurate location of the pile, and its lubrication while being 
driven. 

5. In general, the water-jet should not be attached to the pile, 
but handled separately. 

6. Two jets will often succeed where one fails. In special 
cases a third jet extending a part of the depth aids materially in 
keeping loose the material around the pile. 

7. Where the material is of such a porous character that the 
water from the jets may be dissipated and fail to come up in the 
immediate vicinity of the pile, the utility of the jet is uncertain, 
except for a part of the penetration. 

8. A steam or drop hammer should be used in connection 
with the water- jet, and used to test the final rate of penetra- 
tion. 

9. The use of the water jet is one of the most effective means 
of avoiding injury to piles by overdriving. 

10. There is danger from overdriving when the hammer 
begins to bounce. Overdriving is also indicated by the bending, 
kicking or staggering of the pile. 

11. The brooming of the head of the pile dissipates a part, 
and in some cases all, of the energy due to the fall of the 
hammer. 

12. The steam hammer is usually more effective than 
the drop hammer in securing the penetration of a wooden pile 
without injury, because of the shorter interval between 
blows. 

13. Where shock to surrounding material is apt to prove 
detrimental to the structure, the steam hammer should always 
be used- instead of the drop hammer. This is especially true 
in the case of sheet piling which is intended to prevent the 
passage of water. In some cases also the jet should not be 
used. 



204 EAILROAD CONSTRUCTION. § 168. 

14. In general, the resistance of piles, penetrating soft mate- 
rial, depending solely upon skin friction, is materially increased 
after a period of rest. This period may be as short as fifteen 
minutes, and rarely exceeds twelve hours. 

15. Where a pile penetrates muck or a soft yielding material 
and bears upon a hard stratum at its foot, its strength should be 
determined as a column or beam; omitting the resistance, if any, 
due to skin friction. 

16. Unless the record of previous experience at the same site 
is available, the approximate bearing power may be obtained 
by loading test piles. The results of loading test piles should be 
used with caution, unless their condition is fairly comparable 
with that of the piles in the proposed foundation. 

17. In case the piles in a foundation are expected to act as 
columns, the results of loading test piles should not be depended 
upon unless they are sufficient in number to insure their action in 
a similar manner; and unless they are stayed against lateral 
motion. 

18. Before testing the penetration of a pile in a soft material 
where its bearing power depends principally, or wholly, upon 
sliin friction, the pile should be allowed to rest for 24 hours after 
driving. 

19. Where the resistance of piles depends mainly upon skin 
friction it is possible to diminish the combined strength, or bear- 
ing capacity, of a group of piles, by driving additional piles 
within the same area. 

20. Where piles will foot in a hard stratum, investigation 
should be made to determine that this stratum is of sufficient 
depth and strength to carry the load. 

21. Timber piles may be advantageously pointed, in some 
cases, to a 4-inch or 6-inch square at the end. 

22. Piles should not be pointed when driven into soft 
material. 

23. Shoes should be provided for piles when the driving is 
very hard, especially in riprap or shale. These shoes should be 
so constructed as to form an integral part of the pile. 

24. The use of a cap is advantageous in distributing the impact 
of the hammer more uniformly over the head of the pile, as well 
as in holding it in position during driving. 

1 68. Cost of pile trestles. The cost, per linear foot, of piling 
depends on the method of driving, the scarcity of suitable timber, 



^. 



§ 169. TRESTLES. ' 205 

the price of labor, the length of the piles, and the amount of 
shifting of the pile-driver required. The cost of soft-wood 
piles varies from 8 to 15 cents per lineal foot, and the cost of oak 
piles varies from 10 to 30 cents per foot, according to the length, 
the longer piles costing more per foot. The total cost of putting 
the piles in place is so dependent on other items than the cost of 
driving, such as the cost of shifting the driver, getting the piles 
into the leaders, straightening and bracing them, leveling and 
nailing guide strips for sawing them off, and then the actual 
sawing, that there is a wide variation in the figures that 
are obtainable for the cost of such work. Of course the 
cost per pile of driving is also dependent on the total num- 
ber of piles in the job. The cost per pile of placing a dozen 
piles for a single foundation would be far greater than the 
cost per pile for several hundred piles in one job. Among a 
large number of obtainable figures the average figure of $1.54 
per pile for driving 1267 piles in 46 days is typical. Another 
quoted figure is $2.88 each, for driving 391 piles in 32 working 
days. On another job it cost $150 to drive thirty 30-foot piles, 
or an average of $5 each. In this case the piles cost $1.50 each 
or only 5 cents per hneal foot. The above cost figures are taken 
from Gillette's " Handbook of Cost Data '^ to which the student 
is referred for numerous examples of the cost of piles and pile- 
driving, as well as innumerable other cost analyses. 

Specifications generally say that the piling will be paid for 
per lineal foot of piling left in the work. The wastage of the tops 
of piles sawed off is always something, and is frequently very 
large. Sometimes a small amount per foot of piling sawed off is 
allowed the contractor afe compensation for his loss. This 
reduces the contractor's risk and possibly reduces his bid by 
an equal or greater amount than the extra amount actually 
paid him. 

FRAMED TRESTLES 

169. Typical design. A typical design for a framed trestle 
bent is given in Fig. 71. This represents, with sHght variations 
of detail, the plan according to which a large part of the framed 
trestle bents of the country have been built — i.e., of those less 
than 20 or 30 feet in height, not requiring multiple story con- 
struction. 

170. Joints, (a) The mortise-and-tenon joint is illustrated in 



206 RAILROAD CONSTRUCTION. § 170. 

Fig. 71 and also in Fig. 68 (a). The tenon should be about 



i^ 





Fig. 71. 

thick, 8" wide, and 5J" long. The mortise should be cut 
a little deeper than the tenon. ''Drip-holes^' 
from the mortise to the outside will assist in 
draining off water that may accumulate in the 
joint and thus prevent the rapid decay that 
would otherwise ensue. These joints are very 
troublesome if a single post decays and requires 
renewal. It is generally required that the mor- 
FiG. 72. tise and tenon should be thoroughly daubed 
with paint before putting them together. This will tend to 
make the joint water-tight and prevent decay from the accu- 
mulation and retention of water in the joint. 

(b) The plaster joint. This joint is made by bolting and 
spiking a 3''Xl2" plank on 
both sides of the joint. The 
cap and sill should be 
notched to receive the posts. 
Repairs are greatly facili- 
tated by the use of these 
joints. This method has been 
used by the DelaAvare and 
Hudson Canal Co. [R. R.]. 

(c) Iron plates. An iron plate of the form shown in Fig. 74 




Fig. 73. 



§171. 



TRESTLES. 



207 




,\-^ 



L::'-.>'(a) 



,'"'" 


o o 


6 

c 


o 
o 




o 
o 


a 


o o 


(b) " 



Fig. 74. 



apply with even greater force to 



(6) is bent and used as shown in Fig. 74 (a). Bolts passing 
through the bolt-holes ^.'T""""'- 

shown secure the plates 
to the timbers and make 
a strong joint which may- 
be readily loosened for re- 
pairs. By slight modifi- 
cations in the design the 
method may be used for 
inclined posts and compli- 
cated joints. 

(d) Split caps and sills. 
These are described in 
§ 161. Their advantages 
framed trestles. 

(e) Dowels and drift-bolts. These joints facilitate cheap and 
rapid construction^ but renewals and repairs are very difficult, it 
being almost impossible to extract a drift-bolt, which has been 
driven its full length, without splitting open the pieces contain- 
ing it. Notwithstanding this objection they are extensively 
used, especially for temporary work which is not expected to 
be used long enough to need repairs. 

171. Multiple-story construc- 
tion. Single-story framed trestle 
bents are used for heights up 
to 18 or 20 feet and exception- 
ally up to 30 feet. For greater 
heights some such construction as 
is illustrated in a skeleton design 
in Fig. 75 is used. By using split 
sills between each story and sepa- 
rate vertical and batter posts in 
each story, any piece may readily 
be removed and renewed if neces- 
sary. The height of these stories 
varies, in different designs, from 
15 to 25 and even 30 feet. In 
some designs the structure of each 
story is independent of the stories 
above and below. This greatly 
facilitates both the original construction and subsequent repairs. 




Fig. 75. 



208 



RAILROAD CONSTRUCTION. 



§172, 



In other designs the verticals and batter-posts are made con- 
tinuous through two consecutive stories. The structure is 
somewhat stiffer, but is much more difficult to repair. 

Since the bents of any trestle are usually of variable height 
and those heights are not always an even multiple of the uniform 
height desired for the. stories, it becomes necessary to make the 



I 




Fig. 76. 

upper stories of uniform height and let the odd amount go to the 
lowest story, as shown in Figs. 75 and 76. 

172. Span. The shorter the span the greater the number of 
trestle bents; the longer the span the greater the required strength 
of the stringers supporting the floor. Economy demands the 
adoption of a span that shall make the sum of these require- 




FiG. 77. 



ments a minimum. The higher the trestle the greater the cost 
of each bent, and the greater the span that would be justifiable. 
Nearly all trestles have bents of variable height, but the advan- 
tage of employing uniform standard sizes is so grea^ that many 



.1. 



§173. 



TRESTLES, 



209 




roads use the same span and sizes of timber not only for the 
panels of any given trestle, but also for all trestles regardless of 
height. The spans generally used vary from 10 to 16 feet. The 
Norfolk and Western R. R. uses a span of 12' 6" for all single- 
story trestles, and a span of 25' for all multiple-story trestles. 
The stringers are the same in both cases, but when the span is 
25 feet, knee-braces are run from the sill of the first story below 
to near the middle of each set of stringers. These knee-braces 
are connected at the top by a " straining-beam '^ on which the 
stringers rest, thus supporting the stringer in the center and vir- 
tually reducing the span about one-half. 

173. Foundations, (a) Piles. Piles are frequently used as a 
foundation, as in Fig. 78, particularly in soft ground, and also 
for temporary structures. These 
foundations are cheap, quickly 
constructed, and are particularly 
valuable when it is financially 
necessary to open the road for 
traffic as soon as possible and 
with the least expenditure of 
money; but there is the disad- 
vantage of inevitable decay 
mthin a few years unless the piles are chemically treated, as will 
be discussed later. Chemical treatment, however, increases the 
cost so that such a foundation would often cost more than a 
foundation of stone. A pile should be driven under each post 
as shown in Fig. 78. 

(b) Mud-sills. Fig. 79 illustrates the use of mud-sills as 

built by the Louisville and 
Nashville R. R. Eight blocks 
12"X12"X6' are used under 
each bent. When the ground 
is very soft, two additional 
timbers (12" X 12" X length of 
bent-sill), as shown by the 
dotted lines, are placed under- 
neath. The number required 
evidently depends on the na- 

^^^- '^^- ture of the ground. 

(c) Stone foundations. Stone foundations are the best and 
the most expensive. For very high trestles the Norfolk and 



Fia. 78. 



n \\n w 



SILL 



s 



EJ^^^SB 



212 



RAILROAD CONSTRUCTION. 



§178. 



caps having a width of 12") and also the stress due to trans- 
verse strain are kept approximately constant for the variable 
gross load on these varying spans. 



Clear span. 


No. of pieces 
under each rail. 


Width. 


Depth. 


10 feet 
12 " 
14 ** 


2 
2 
3 


8 inches 
10 " 
10 ** 


16 inches 
16 " 
16 •• 



178. Corbels. A corbel (in trestle-work) is a stick of timber 
(perhaps two placed side by side)^ about 3' to 6' long, placed 
underneath and along the stringers and resting on the cap. 
There are strong prejudices for and against their use, and a 
corresponding diversity in practice. They are bolted to the 
stringers and thus stiffen the joint. They certainly reduce the 
objectionable crushing of the fibers at each end of the stringer, 
but if the corbel is no wider than the stringers, as is generally 
the case, the area of pressure between the corbels and the cap is 




Fig. 82. 

no greater and the pressure per square inch on the cap is no less 
than the pressure on the cap if no corbels were used. If the 
corbels and cap are made of hard wood, as is recommended by 
some, the danger of crushing is lessened, but the extra cost and 
the frequent scarcity of hard wood, and also the extra cost and 
labor of using corbels, may often neutralize the advantages 
obtained by their use. 

179. Guard-rails. These are frequently made of 5"X8'' stuff, 
notched 1" for each tie. The sizes vary up to 8"X8", and the 
depth of notch from f" to IJ". They are generally bolted to 
every third or fourth tie. It is frequently specified that they 
shall be made of oak, white pine, or yellow pine. The joints 
are made over a tie, by halving each piece, as illustrated in Fig. 
83. The joints on opposite sides of the trestle should be ^^stag- 



§ 180. TRESTLES. 213 

gered." Some roads fasten every tie to the guard-rail, using a 
bolt, a spike, or a lag-screw. 

Guard-rails were originally used with the idea of preventing 
the wheels of a derailed truck from running off the ends of the 
ties. But it has been found that an outer guard-rail alone (with- 
out an inner guard-rail) becomes an actual element of danger, 
since it has frequently happened that a derailed wheel has caught 
on the outer guard-rail, thus causing the truck to slew around 




Fig. 83. 

and so produce a dangerous accident. The true function of the 
outside guard-rail is thus changed to that of a tie-spacer, which 
keeps the ties from spreading when a derailment occurs. The 
inside guard-rail generally consists of an ordinary steel rail 
spiked about 10 inches inside of the running rail. These inner 
guard-rails should be bent inward to a point in the center of the 
track about 50 feet beyond the end of the bridge or trestle. If 
the inner guard-rails are placed with a clear space of 10 inches 
inside the nmning rail, the outer guard-rails should be at least 
6' 10" apart. They are generally much farther apart than this. 

i8o. Ties on trestles. If a car is derailed on a bridge or 
trestle, the heavily loaded wheels are apt to force their way be- 
tween the ties by displacing them unless the ties are closely 
spaced and fastened. The clear space between ties is generally 
equal to or less than their width. Occasionally it is a little more 
than their width. 6"X8" ties, spaced 14" to 16" from center 
to center, are most frequently used. The length varies from 
9' to 12' for single track. They are generally notched J" deep 
on the under side where they rest on the stringers. Oak ties 
are generally required even when cheaper ties are used on the 
other sections of the road. Usually every third or fourth tie is 
bolted to the stringers. When stringers are placed underneath 
the guard-rails, bolts are run from the top of the guard-rail to 
the under side of the stringer. The guard-rails thus hold down 
the whole system of ties, and no direct fastening of the ties to 
'the stringers is needed. 

i8i. Superelevation of the outer rail on curves. The location 

of curves on trestles should be avoided if possible, especially 

j when the trestle is high. Serious additional strains are intro-' 



214 



RAILROAD CONSTRUCTION. 



§181. 



duced especially when the curvature is sharp or the speed high. 
Since such curves are sometimes practically unavoidable, it is 
necessary to design the trestle accordingly. If a train is stopped 
on a curved trestle, the action of the train on the trestle is 
evidently vertical. If the train is moving with a considerable 
velocity, the resultant of the weight and the centrifugal action 
is a force somewhat inclined from the vertical. Both of these 
conditions may be expected to exist at times. If the axis of 
the system of posts is vertical (as illustrated in methods a, h, c, d, 
and e), any lateral force, such as would be produced by a mov- 
ing train, will tend to rack the trestle bent. If the stringers are 
set vertically, a centrifugal force likewise tends to tip them 
side wise. If the axis of the system of posts (or of the stringers) 
is inclined so as to coincide with the pressure of the train on the 
trestle when the train is moving at its normal velocity, there is 
no tendency to rack the trestle when the train is moving at that 
velocity, but there will be a tendency to rack the trestle or 
twist the stringers when the train is stationary. Since a moving 
train is usually the normal condition of affairs, as well as the 
condition which produces the maximum stress, an inclined axis 
is evidently preferable from a theoretical standpoint ; but what- 
ever design is adopted, the trestle should evidently be suffi- 
ciently cross-braced for either a moving or a stationary load, 
and any proposed design must be studied as to the effect of both 
of these conditions. Some of the various methods of securing 
the requisite superelevation may be described as follows : 

(a) Framing the outer posts longer than the inner posts, so 

that the cap is inclined at the 
proper angle; axis of posts verti- 
cal. (Fig. 84.) The method re» 
quires more work in framing the 
trestle, but simplifies subsequent 
track-laying and maintenance, un- 
less it should be found that the 
superelevation adopted is unsuit- 
able, in which case it could be cor- 
rected by one of the other methods 
given below. The stringers tend 
^^^' ^^- to twist when the train is sta- 

tionaiy. 

(b) Notching the cap so that the stringers are at a different 




§181 



TRESTLES. 



215 




Fig. 85. 



elevation. (Fig. 85.) This weakens the cap and requires that 
all ties shall be notched to a 
bevelled surface to fit the string- 
ers, which also weakens the ties. 
A centrifugal force will tend to 
twist the stringers and rack the 
trestle. 

(c) Placing wedges underneath 
the ties at each stringer. These 
wedges are fastened with two ' 
bolts. Two or more wedges will 
be required for each tie. The ad- 
ditional number of pieces required 
for a long curve will be immense, and the work of inspection and 
keeping the nuts tight will greatly increase the cost of main- 
tenance. 

(d) Placing a wedge under the outer rail at each tie. This 
requires but one extra piece per tie. There is no need of a 
wedge under the inner tie in order to make he rail normal to 
the tread. The resulting inward inclination is substantially that 
produced by some forms of rail-chairs or tie-plates. The spikes 
(a Httle longer than usual) are driven through the wedge into 
the tie. Sometimes '^lag-screws" are used instead of spikes. 
If experience proves that the superelevation is too much or too 
little, it may be changed by this method with less work than 
by any other. 

(e) Corbels of different heights. When corbels are used (see 

§ 178) the required in- 
chnation of the floor sys. 
tem may be obtained by 
varying the depth of the 
corbels. 

(f) Tipping the whole 
trestle. This is done by 
placing the trestle on an 
inclined foundation. If 
very much inclined, the 
trestle bent must be se- 
cured against the possi- 
FiG. 86. bility of slipping sidewise, 

for the slope would be considerable with a sharp curve, and the 




216 RAILROAD CONSTRUCTION. § 182. 

vibration of a moving train would reduce the coefficient ol 
friction to a comparatively small quantity. 

(g) Framing the outer posts longer. This case is identical 
with case (a) except that the axis of the system of posts is 
inclined, as in case (/), but the sill is horizontal. 

The above-described plans will suggest a great variety of 
methods which are possible and which differ from the above 
only in minor details. 

' 182. Protection from fire. Trestles are peculiarly subject to 
fire, from passing locomotives, which may not only destroy the 
trestle, but perhaps cause a terrible disaster. This danger is 
sometimes reduced by placing a strip of galvanized iron along 
the top of each set of stringers and also along the tops of the 
caps. Still greater protection was given on a long trestle on the 
Louisville and Nashville R. R. by making a solid flooring of 
timber, covered with a layer of ballast on which the ties and 
rails were laid as usual. 

Barrels of water should be provided and kept near all trestles, 
and on very long trestles barrels of water should be placed every 
two or three hundred feet along its length. A place for the bar- 
rels may be provided by using a few ties which have an extra | 
length of about four feet, thus forming a small platform, which 
should be surrounded by a railing. The track- walker should be | 
held accountable for the maintenance of a supply of water in i 
these barrels, renewals being frequently necessary on account of ■ 
evaporation. Such platforms should also be provided as refuge- i 
BAYS for track-walkers and trackmen working on the trestle. On > 
very long trestles such a platform is sometimes provided with 
sufficient capacity for a hand-car. 

183. Timber. Any strong durable timber may be used when 
the choice is limited, but oak, pine, or cypress are preferred 
when obtainable. When all of these are readily obtainable, 
the various parts of the trestle will be constructed of different 
kinds of wood — the stringers of long-leaf pine, the posts and 
braces of pine or red cypress, and the caps, sills, and corbels (if 
used) of white oak. The use of oak (or a similar hard wood) 
for caps, sills, and corbels is desirable because of its greater ! 
strength in resisting crushing across the grain, which is the 
critical test for these parts. There is no physiological basis to 
the objection, sometimes made, that different species of timber, i 
in contact with each other, will rot quicker than if only one , 




(To face page 216.) 



§ 185. TRESTLES. 217 

kind of timber is used. When a very extensive trestle is io be 
built at a place where suitable growing timber is at hand but 
there is no convenient sawmiU, it will pay to transport a port- 
able sawmill and engine and cut up the timber as desired. 

184. Cost of framed timber trestles. The cost varies widely 
on account of the great variation in the cost of timber. When 
a railroad is first penetrating a new and undeveloped region, the 
cost of timber is frequently small, and when it is obtainable from 
the company's right-of-way the only expense is felling and 
sawing. The work per M, B. M., is small, considering that a 
single stick 12'' X 12'' X 25' contains 300 feet, B. M., and that 
sometimes two hours' work, worth perhaps $1, will finish all 
the work required on it. Smaller pieces will of course require 
more work per foot, B. M. Long-leaf pine can be purchased 
from the mills at from $27 to $45 per M feet, B. M., according 
to the dimensions. To this must be added the freight and labor 
of erection. The cartage from the nearest railroad to the trestle 
may often be a considerable item. Wrought iron will cost 
about 3 cents per pound and cast iron 2 cents, although the prices 
are often lower than these. The amount of iron used depends on 
the detailed design, but, as an average, will amount to $1.50 
to $2 per 1000 feet, B. M., of timber. A large part of the tres- 
tling of the country has been built at a contract price of about 
$30 per 1000 feet, B. M., erected. While the cost will frequently 
rise to $50 and even $60 when timber is scarce, it will drop to 
$13 (cost quoted) when timber is cheap. 

DESIGN OF WOODEN TRESTLES. 

185. Common practice. A great d&al of trestling has been 
constructed without any rational *iesign except that custom and 
experience have shown that certain sizes and designs are probably 

1 safe. This method has resulted occasionally in failures but more 

j frequently in a very large waste of timber. Many railroads 

1 employ a uniform size for all posts, caps, and sills, and a uniform 

I size for stringers, all regardless of the height or span of the 

I trestle. For repair work there are practical reasons favoring 

this. '^To attempt to run a large lot of sizes would be more 

' wasteful in the end than to maintain a few stock sizes only. 

Lumber can be bought more cheaply by giving a general order 

tor * the run of the mill for t^e season,' or ' a cargo lot,^ specif j^ - 



218 RAILROAD CONSTRUCTION. § 186. 

ing approximate percentages of standard stringer size^ of 
12 X 12-inch stuff, 10 X 10-inch stuff, etc., and a liberal propor- 
tion of 3- or 4-inch plank, all lengths thrown in. The 12 X 12- 
inch stuff, etc., is ordered all lengths, from a certain specified 
length up. In case of a wreck, washout, burn-out, or sudden 
call for a trestle to be completed in a stated time, it is much 
more economical and practical to order a certain number of 
carloads of trestle stuff' to the ground and there to select piece 
after piece as fast as needed, dependent only upon the length of 
stick required. When there is time to make the necessary sur- 
veys of the ground and calculations of strength, and to wait for a 
special bill of timber to be cut and delivered, the use of differ- 
ent sizes for posts in a structure would be warranted to a certain 
extent." * For new construction, when there is generally 
sufficient time to design and order the proper sizes, such waste- 
fulness is less excusable, and under any conditions it is both 
safer and more economical to prepare standard designs which 
can be made applicable to varying conditions and which will at 
the same time utilize as much of the strength of the timber as 
can be depended on. In the following sections will be given 
the elements of the preparation of such standard designs, which 
will utilize uniform sizes with as little waste of timber as possible. 
It is not to be understood that special designs should be made 
for each individual trestle. 

1 86. Required elements of strength. The stringers of trestles 
are subject to transverse strains, to crushing across the grain 
at the ends, and to shearing along the neutral axis. The strength 
of the timber must therefore be computed for all these kinds 
of stress. Cajps and sills will fail, if at all, by crushing across 
the grain; although subject to other forms of stress, these could 
hardly cause failure in the sizes usually employed. There is an 
apparent exception to this: if piles are improperly driven and 
an uneven settlement subsequently occurs, it may have the 
effect of transferring practically all of the weight to two or three 
piles, while the cap is subjected to a severe transverse strain - 
which may cause its failure. Since such action is caused gener- 
ally by avoidable errors of construction it may be considered as 
abnormal, and since such a failure will generally occur by a 
gradual settlement, all danger may be avoided by reasonable 
■ < 

* From "Economical Designing of Timber Trestle Bridges." 



§ 187. TRESTLES. 219 

care in inspection. Posts must be tested for their columnar 
strength. These parts form the bulk of the trestle and are the 
parts which can be definitely designed from knoTVTi stresses. 
The stresses in the bracing are more indefinite^ depending on 
indeterminate forces, since the inclined posts take up an un- 
known proportion of the lateral stresses, and the design of the 
bracing may be left to what experience has shown to be safe, 
without involving any large waste of timber. 

187. Strength of timber. Until recently tests of the strength 
of timber have generally been made by testing small, selected, 
w^ell-seasoned sticks of "clear stuff,'' free from knots or imper- 
fections. Such tests would give results so much higher than 
the vaguely known strength of large unseasoned "coromercial" 
timber that very large factors of safety were recommended— 
factors so large as to detract from any confidence in the whole 
theoretical design. Recently the U. S. Government has been 
making a thoroughly scientific test of the strength of full-size 
timber under various conditions as to seasoning, etc. The work 
has been so extensive and thorough as to render possible the 
economical designing of timber structures. 

One important result of the investigation is the determina- 
tion of the great influence of the moisture in the timber and 
the law of its effect on the strength. It has been also shown 
that timber soaked with water has substantially the same 
strength as green timber, even though the timber had once been 
thoroughly seasoned. Since trestles are exposed to the weather 
they should be designed on the basis of using green timber. 
It has been shown that the strength of green timber is very 
regularly about 55 to 60% of the strength of timber in which 
the moisture is 12% of the dry weight, 12% being the proportion 
of moisture usually found in timber that is protected from the 
weather but not heated, as, e.g., the timber in a barn. Since 
the moduli of rupture have all been reduced to this standard of 
moisture (12%), if we take one-eighth of the rupture values, it 
still allows a factor of safety of about five, even on green timber. 
In Table XX there are quoted the values taken from the U= S. 
Government reports on the strength of timber, the tests prob- 
ably being the most thorough and reliable that were ever made. 

In Table XXI are given the '^ working unit stresses for struc- 
tural timber, expressed in pounds per square inch," as recom- 
mended by the committee on " Wooden Bridges and Trestles," 



220 



RAILROAD CONSTRUCTION. 



§188. 



of the American Railway Engineering Association. The report 
was presented at their tenth annual convention, held in Chicago, 
in March, 1909. 



Table XX. moduli of rupture for various timbers. 

[12% moisture.] 
(Condensed from U. S. Forestry Circular, No. 15.) 





Species. 


M 


Cross-bending, 


Crush- 
ing 
end- 
wise. 




Shearing along 
the grain. 


No. 


eg 


Modulus 

of 
Elasticity. 


1 
2 

1 

5 
6 

7 


Long-leaf pine. . . . 
Cuban " .... 

Short-leaf " 

Loblolly " .... 
White " .... 
Red " .... 
Spruce " 


38 
39 
32 
33 
24 
31 
39 


12 600 

13 600 
10 100 
11300 

7 900 

9 100 

10 000 


2 070 000 
2 370 000 

1 680 000 

2 050 000 
1 390 000 
1 620 000 
1 640 000 


8000 
8700 
6500 
7400 
5400 
6700 
7300 


1180 
1220 

960 
1150 

700 
1000 
1200 


700 
700 
700 
700 
400 
500 
800 


8 

9 

10 


Bald cypress 

White cedar 

Douglas sDruce. . . . 


29 
23 
32 


7 900 

6 300 

7 900 


1 290 000 

910 000 

1 680 000 


6000 
5200 
5700 


800 
700 
800 


500 
400 
500 


11 
12 
13 
14 
15 
16 
19 
20 


White oak 

Overcup " 

Post " 

Cow " 

Red " 

Texan " 

Willow " ... 

Spanish ** 


50 
46 
50 
46 
45 
46 
45 
46 


13 100 

11 300 

12 300 

11 500 
11400 

13 100 
10 400 

12 000 


2 090 000 

1 620 000 

2 030 000 
1610 000 
1 970 000 
1 860 000 
1 750 000 
1 930 000 


8500 
7300 
7100 
7400 
7200 
8100 
7200 
7700 


2200 
1900 
3000 
1900 
2300 
2000 
1600 
1800 


1000 

1000 

1100 

900 

1100 

900 

900 

900 


21 
27 
28 
29 
30 


Shagbark hickory. . 
Pignut " 

White elm 

Cedar " 

White ash 


51 
56 
34 
46 
39 


16 000 
18 700 
10 300 
13 500 
10 800 


2 390 000 
2 730 000 
1 540 000 
1 700 000 
1 640 000 


9500 
10900 
6500 
8000 
7200 


2700 
3200 
1200 
2100 
1900 


1100 
1200 
800 
1300 
1100 



1 88. Loading. As shown in § 172, the span of trestles is always 
small, is generally 14 feet, and is never greater than 18 feet 
except when supported by knee-braces. The greatest load that 
will ever come on any one span will be the concentrated loading 
of the drivers of a consolidation locomotive. With spans of 14 
feet or less it is impossible for even the four pairs of drivers to 
be on the same span at once. The weight of the rails, ties, and 
guard-rails should be added to obtain the total load on the string- 
ers, and the weight of these, plus the weight of the stringers, 
should be added to obtain the pressure on the caps or corbels. 



§188. 



TRESTLES. 



221 





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222 



RAILROAD CONSTRUCTION. 



§ 189. 



This dead load is almost insignificant compared with the live 
load and may be included with it. The weight of rails, ties, 
etc., may be estimated at 240 pounds per foot. To obtain the 
weight on the caps the weight of the stringers must be added, 
which depends on the design and on the weight per cubic foot 
of the wood employed. But as the weight of the stringers is 
comparatively small, a considerable percentage of variation in 
weight will have but an insignificant effect on the result. Dis- 
regarding all refinements as to actual dimensions, the ordinary 
maximum loading for standard-gauge railroads may be taken 
as that due to four driving-axles, spaced 5' 0" apart and giving 
a pressure of 40000 pounds per axle. This should be increased 
to 54000 pounds per axle (same spacing) for the heaviest traffic. 
On the basis of 40000 pounds per axle or 20000 pounds per wheel 
the following results have been computed: This loading is 
assumed to allow for impact. 

STRESSES ON VARIOUS SPANS DUE TO MOVING LOADS OP 20000 
POUNDS, SPACED 5' 0'' APART, WITH 120 POUNDS PER FOOT 
OF DEAD LOAD, 



Span in feet. 


Max. moment, 
ft. lbs. 


Max shear. 


Max. load on 

one cap under 

one rail. 


10 
12 
14 
16 
18 


51 500 

82 160 

112 940 

123 840 

164 860 


30 600 
35 720 
39 410 
43 460 

47 747 


41 200 
49 440 
57 680 
65 920 
75 160 



* Although the dead load does not vary in proportion to the 
live load, yet, considering the very small influence of the dead 
load, there will be no appreciable error in assuming the corre- 
sponding values, for a load of 54000 lbs. per axle, to be f o^ of 
those given in the above tabulation. 

189. Factors of safety. The most valuable result of the gov- 
ernment tests is the knowledge that under given moisture condi- 
tions the strength of various species of sound timber is not the 
variable uncertain quantity it was once supposed to be, but that 
its strength can be relied on to a comparatively close percentage. 
This confidence in values permits the employment of lower fac- 
tors of safety than have heretofore been permissible. Stresses, 
which when excessive would result in immediate destruction, 
such as cross-breaking and columnar stresses, should be allowed 
a higher factor of safety— say 6 or 8 for green timber. Other 
stresses, such as crushing across the grain and shearing along the 



I 



190. TRESTLES. 223 



neutral axis, which will be apparent to inspection before it is 
dangerous, may be allowed lower factors — say 3 to 5. 

190. Design of stringers. The strength of rectangular beams 
of equal width varies as the square of the depth; therefore deep 
beams are the strongest. On the other hand, when any cross- 
sectional dimension of timber much exceeds 12'' the cost is 
much higher per M, B. M., and it is correspondingly difficult to 
obtain thoroughly sound sticks, free from wind-shakes, etc. 
Wind-shakes especially affect the shearing strength. Also, if 
the required transverse strength is obtained by using high nar- 
row stringers, the area of pressure between the stringers and the 
cap may become so small as to induce crushing across the grain. 
This is a very common defect in trestle design. As already in- 
dicated in § 172, the span should vary roughly with the average 
height of the trestle, the longer spans being employed when the 
trestle bents are very high, although it is usual to employ the 
same span throughout any one trestle. 

To illustrate, if we select a span of 14 feet, the load on one 
cap will be 57680 lbs. If the stringers and cap are made of 
long-leaf yellow pine, the allowable value, according to Table 
XXI, for " compression across the grain " is 260 pounds per 
square inch; this will require 222 square inches of surface. 
If the cap is 12" wide, this will require a width of 18.5 inches, 
or say 2 stringers under each rail, each 9 inches wide. For 
rectangular beams. 

Moment = |i^'6A2. 

Using for R' the safe value 1300 lbs. per square inch, we have 
112940X12 = iXl300Xl8X/i2, 

from which h = 18''. 7. If desired, the width may be increased 
to 10" and the depth correspondingly reduced, which will give 
similarly /i = 17".7 or say 18". This shows that two beams, 
10"X18", under each rail will stand the transverse bending and 
have more than enough area for crushing. 
The shear per square inch will equal 

3 total shear 3 39410 ,^^ „ 

2 cross-section ==2 2X10X18 = ^^^ ^^^' P^^ '^- ^^^^• 

This is higher than the recommended working value. The com- 
bination suggested in § 177, viz., 3 beams 10"X16" for 14 feet 
span, gives a far safer value. Considering that wooden beams. 



224 RAILROAD CONSTRUCTION. § 191. 

tested to destruction, usually fail by shearing, the three-beam 
combination is safer. 

The deflection should be computed to see if it exceeds the 
somewhat arbitrary standard of ^Jq- of the span. The deflec- 
tion for uniform loading is 



S2hh'E' 



in which Z = length in inches; 

pr= total load, assumed as uniform = 57680; 
£J= modulus of elasticity, given as 1610000 lbs. 

per sq. in. for long-leaf pine, according to Table XXI. Then 

__5X57680X1683_^ 
32X30X163X1610000 

^X168-=0-.84, 

30 that the calculated deflection is well within the limit. Of 
course the loading is not strictly uniform, but even with a lib- 
3ral allowance the deflection is still safe. 

For the heaviest practice (54000 lbs. per axle) these stringer 
dimensions must be correspondingly increased. 

191. Design of posts. Four posts are generally used for 
single-track work. The inner posts are usually braced by the 
cross-braces, so that their columnar strength is largely increased ; 
but as they are apt to get more than their share of work, the ad- 
vantage is compensated and they should be treated as unsup- 
ported columns for the total distance between cap and sill in 
simple bents, or for the height of stories in multiple-story con- 
struction. The caps and sills are assumed to have a width of 12''. 
It facilitates the application of bracing to have the columns of 
the same width and vary the other dimension as required. 

Unfortunately the experimental work of the U. S. Govern- 
ment on timber testing has not yet progressed far enough to 
establish unquestionably a general relation between the strength 
of long columns and the crushing strength of short blocks. The 



§ 192. TRESTLES. 225 

following formula has been suggested, but it cannot be consid- 
ered as established: 

/= allowable working stress per sq. in for long columns; 
F= " '' '' '' '' '' '' short blocks; 

c 

Z= length of column in inches; 

c?= least cross-sectional dimensions in inches. 

The formula recommended by the A. R. E. A. is found in 
Table XXI. For all columns of which the length is less than 15 
times the least diameter, a uniform unit stress is recommended. 
For longer columns, a unit stress is multiplied by the factor 
(1— Z-r-60c?), which is always less than unity. For the above 
case, Z = 240 and d = 12, and the factor = .667, which, multiplied 
by 1300, gives a unit stress of 867 lbs. per square inch for a long- 
leaf yellow pine column of these dimensions. 

867 X 144 = 124848 lbs., the working load for each post. This is 
more than the total load on one trestle btot and illustrates the 
usual great waste of timber. Making the post 8"X12" and 
calculating similarly, we have/ = 650, and the working load per 
column is 650X96=62400 lbs. As considerable must be 
allowed for ^ feathering," which destroys the strength of the 
outer layers of the wood, and also for the dynamic effect of 
the hve load, 8'' X 12'' may not be too great, but it is certainly 
a safe dimension, considered as a column. One method of 
allowing for weathering is to disregard the outer half-inch on 
all sides of the post, i.e., to calculate the strength of a post one 
inch smaller in each dimension than the post actually employed. 
On this basis an 8" X 12" X20' post, computed as a 7'' X 11'' post, 
would have a safe columnar strength of 556 lbs. per square inch. 
With an area of 77 square inches, this gives a working load of 
42812 lbs. for each post, or 171248 lbs. for the four posts. Con- 
sidering that 115360 lbs. is the maximum load on one cap (14 feet 
span), the great excess of strength is apparent. 

.192. Design of caps and sills. The stresses in caps and sills 
are very indefinite, except as to crushing across the grain. As 



226 RAILROAD CONSTRUCTION. § 193. 

the stringers are placed almost directly over the inner posts, and 
as the sills are supported just under the posts, the transverse 
stresses are almost insignificant. In the above case four posts 
have an area of 4 X 12'' X 8'' = 384 sq. in. The total load 115360 
lbs. will then give a pressure of 300 pounds per square inch, 
which is more than the allowable limit. This one feature will 
require the use of 12''X12" (or at least 10''X12'0 posts rather 
than 8"X12" posts, for the smaller posts, although probably 
strong enough as posts, would produce an objectionably high 
pressure. 

193. Bracing. Although some idea of the stresses in the 
bracing could be, found from certain assumptions as to wind- 
pressure, etc., yet it would probably not be found wise to de- 
crease, for the sake of economy, the dimensions which practice 
has shown to be sufficient for the work. The economy that 
would be possible would be too insignificant to justify any risk. 
Therefore the usual dimensions, given in §§ 173 and 174, should 
be employed. 



CHAPTER V. 
TUNNELS. 
SURVEYING. 

I 194. Surface surveys. As tunnels are always dug from each 
end and frequently from one or more intermediate shafts, it is 
necessary that an accurate surface survey should be made 
between the two ends. As the natural surface in a locality 
where a tunnel is necessary is almost invariably very steep and 
rough, it requires the employment of unusually refined methods 
of work to avoid inaccuracies. It is usual to run a line on the 
surface that will be at every point vertically over the center line 
of the tunnel. Tunnels are generally made straight unless 
curves are absolutely necessary, as curves add greatly to the 
cost. Fig. 87 represents roughly a longitudinal section of the 




-^00- *j— — 6000- — *i--— fOOO;^-— H- 6G0O ^— 5000— H 

Fig. 87. — Sketch of Section of the Hoosac TuNrrajL. 



Hoosac Tunnel. Permanent stations were located at A, B, C, 
D, Ej and Fj and stone houses were built at A, B, C, and D, 
These were located with ordinary field transits at first, and then 
all the points were placed as nearly as possible in one vertical 
plane by repeated trials and minute corrections, using a very 
large specially constructed transit. The stations D and F were 
necessary because E and A were invisible from C and B, The 
alinement at A and E having been determined with great accu- 
racy, the true alinement was easily carried into the tunnel. 

227 



228 RAILROAD CONSTRUCTION. § 195. 

The relative elevations of A and E were determined with 
great accuracy. Steep slopes render necessary many settings 
of the level per unit of horizontal distance and require that the 
work be unusually accurate to obtain even fair accuracy per 
unit of distance. The levels are usually re -run many times 
until the probable error is a very small quantity 

The exact horizontal distance between the two ends of the 
tunnel must also be known, especially if the tunnel is on a 
grade. The usual steep slopes and rough topography likewise 
lender accurate horizontal measurements very difficult. Fre- 
quently when the slope is steep the measurement is best ob- 
tained by measuring along the slope and allowing for grade. 
This may be very accurately done by employing two tripods 
(level or transit tripods serve the purpose very well), setting 
them up slightly less than one tape-length apart and measuring 
between horizontal needles set in wooden blocks inserted in the 
top of each tripod. The elevation of each needle is also observed. 
The true ' horizontal distance. between two successive positions 
of the needles then equals the square root of the difference of 
the squares of the inclined distance and the difference of eleva- 
tion. Such measurements will probably be more accurate than 
those made by attempting to hold the tape horizontal and 
plumbing down with plumb-bobs, because (1) it is practically 
difficult to hold both ends of the tape truly horizontal ; (2) on 
steep slopes it is impossible to hold the down-hill end of a 100- 
foot, tape (or even a 25-foot length) on a level with the other 
end, and the great increase in the number of applications of the 
unit of measurement very greatly increases the probable error 
of the whole measurement; (3) the vibrations of a plumb-bob 
introduce a large probability of error in transferring the meas- 
urement from the elevated end of the tape to the ground, and 
the increased number of such applications of the unit of meas- 
urement still further increases the probable error. 

195. Surveying down a shaft. If a shaft is sunk, as at Sj 
Fig. 87, and it is desired to dig out the tunnel in both directions 
from the foot of the shaft so as to meet the headings from the 
outside, it is necessary to know, when at the bottom of the 
shaft, the elevation, alinement, and horizontal distance from 
each end of the tunnel. 

The elevation is generally carried down a shaft by means of 
a steel tape, This method involves the least number of appli- 



§ 195. TUNNELS. 229 

cations of the unit of measurement and greatly increases the 
accuracy of the final result. 

The horizontal distance from each end may be easily trans- 
ferred down the shaft by means of a plumb -bob, using some of 
the precautions described in the next paragraph. 

To transfer the alinemeni from the surface to the bottom of 
a shaft requires the highest skill because the shaft is always 
small, and to produce a line perhaps several thousand feet long 
in a direction given by two points 6 or 8 feet apart requires 
that the two points must be determined with extreme accuracy. 
The eminently successful method adopted in the Hoosac Tunnel 
will be briefly described: Tw^o beams w^ere securely fastened 
across the top of the shaft (1030 feet deep), the beams being 
placed transversely to the direction of the tunnel and as far 
apart as possible and yet allow - plumb-lines, hung from the 
intersection of each beam wdth the tunnel center line, to swing 
freely at the bottom of the shaft. These intersections of the 
beams with the center line were determined by averaging the 
results of a large number of careful observations for alinement. 
Two fine parallel wires, spaced about -yq" apart, were then 
stretched betw^een the beams so that the center line of the 
tunnel bisected at all points the space betw^een the wires. 
Plumb-bobs, weighing 15 pounds, were suspended by fine wires 
beside each cross-beam, the wires passing between the two 
parallel alinement wires and bisecting the space. The plumb - 
bobs were allowed to swing in pails of water at the bottom. 
Drafts of air up the shaft required the construction of boxes 
surrounding the wires. Even these precautions did not suffice 
to absolutely prevent vibration of the wire at the bottom 
through a very small arc. The mean point of these vibrations 
in each case w^as then located on a rigid cross-beam suitably 
placed at the bottom of the shaft and at about the level of the 
roof of the tunnel. Short plumb-lines w^ere then suspended 
from these points w^henever desired; a transit w^as set (by trial) 
so that its line of collimation passed through both plumb-lines 
and the line at the bottom could thus be prolonged. 

Some recent experience in the ''Tamarack" shaft, 4250 feet 
deep, shows that the accuracy of the results may be affected by 
air-currents to an unsuspected extent. Tw^o 50-lb. cast-iron 
plumb-bobs w^ere suspended wdth No. 24 piano-wire in this 
shaft. The carefully measured distances between the wires 



230 



KAILROAD CONSTRUCTION. 



§19& 



at top and bottom were 16.32 and 16.43 feet respectively. 
After considerable experimenting to determine the cause of 
the variation, it was finally concluded that air-currents were 
alone responsible. The variation of the bobs from a true ver- 
tical plane passing through the wires at the top was of course 
an unknown quantity, but since the variation in one direction 
amounted to 0.11 foot, the accuracy in other directions was 
very questionable. This shows that a careful comparative 
measurement between the wires at top and bottom should 
always be made as a test of their parallelism. 

196. Underground surveys. Survey marks are frequently 
placed on the timbering, but they are apt to prove unreliable 
on account of the shifting of the timbering due to settlement 
of the surrounding material. They should never be placed at 
the bottom of the tunnel on account of the danger of being 
disturbed or covered up. Frequently holes are drilled in the 
roof and filled with wooden plugs in which a hook is screwed 
exactly on line Although this is probably the safest method, 
even these plugs are not always undisturbed, as the material, 
unless very hard, will often settle slightly as the excavation 
proceeds. When a tunnel is perfectly straight and not too long, 
alinement-points may be given as frequently as desired from 

permanent stations located outside 
the tunnel where they are not liable 
to disturbance. This has been ac- 
complished by running the aline- 
ment through the upper part of the 
cross-section, at one side of the cen- 
ter, where it is out of the way of 
the piles of masonry material, 
debris, etc., which are so apt to 
choke up the lower part of the 
cross-section. The position of this 
line relative to the cross-section 
being fixed, the alignment of any 
required point of the cross-section 
is readily found by means of a light 
frame or template with a fixed tar- 
get located where this line would intersect the frame when 
properly placed. A level-bubble on the frame will assist in 
setting the frr.me in its proper position. 




Fig. 88. 



§ 197. TUNNELS. 231 

In all tunnel surveying the cross-wires must be illuminated 
by a lantern, and the object sighted at must also be illuminated. 
A powerful dark-lantern with the opening covered with ground 
glass has been found useful. This may be used to illuminate a 
plumb-bob string or a very fine rod, or to place behind a brass 
plate having a narrow slit in it, the axis of the slit and plate 
being coincident with the plumb-bob string by which it is 
hung. 

On account of the interference to the surveying caused by 
the work of construction and also by the smoke and dust in the 
air resulting from the blasting, it is generally necessary to make 
the surveys at times when construction is temporarily suspended. 

197. Accuracy of tunnel surveying. Apart from the very 
natural desire to do surveying which shall check well, there is 
an important financial side to accurate tunnel surveying. If 
the survey lines do not meet as desired when the headings come 
together, it may be found necessary, if the error is of appreciable 
size, to introduce a slight curve, perhaps even a reversed curve, 
into the alinement, and it is even conceivable that the tunnel 
section would need to be enlarged somewhat to allow for these 
curves. The cost of these changes and the perpetual annoyance 
due to an enforced and undesirable alteration of the original 
design will justify a considerable increase in the expenses of the 
survey. Considering that the cost of surveys is usually but a 
small fraction of the total cost of the work, an increase of 10 or 
even 20% in the cost of the surveys will mean an insignificant 
addition to the total cost and frequenf/, if not generally, it will 
result in a saving of many times the increased cost. The 
accuracy actually attained in two noted American tunnels is 
given as follows: The Musconetcong tunnel is about 5000 feet 
long, bored through a mountain 400 feet high. The error of 
alinement at the meeting of the headings was 0'.04, error of 
levels 0'.015, error of distance 0'.52. The Hoosac tunnel is 
over 25000 feet long. The heading from the east end met the 
heading from the central shaft at a point 11274 feet from the 
east end and 1563 feet from the shaft. The error in alinement 
was Y6 oi an inch, that of levels ^' a few hundredths/' error of 
distance ** trifling.'' The alinement, corrected at the shaft, was 
carried on through and met the heading from the west end at a 
point 10138 feet from the west end and 2056 feet from the shaft. 
Here the error of alinement was ■^" and that of levels 0.134 foot. 



232 RAILROAD CONSTRUCTION. ' § 198. 

DESIGN 

198. Cross-section. Nearly all tunnels have cross-sections 
peculiar to themselves — all varying at least in the details. The 
general form of a great many tunnels is that of a rectangle sur- 
mounted by a semi-circle or semi-ellipse. In very soft material 
an inverted arch is necessary along the bottom. In such cases 
the sides will generally be arched instead of vertical. The sides 
are frequently battered. In very long tunnels, several forms 
of cross-section will often be used in the same tunnel, owing to 
differences in the material encountered. In solid rock, which 
will not disintegrate upon exposure, no lining is required, and 
the cross-section will be the irregular section left by the blasting, 
the only requirement being that no rock shall be left within the 
required cross-sectional figure. Farther on, in the same tunnel, 
when passing through some very soft treacherous material, it 
may be necessary to put in a full arch lining — top, sides, and 
bottom — which will be nearly circular in cross-section. For 
an illustration of this see Figs. 89 and 90. 

The cross-section recommended by the A. R. E. A. for single 
track is a rectangle 16 feet wide by 16 feet 6 inches high, sur- 
mounted by a semi-circle with a radius of 8 feet. The top of the 
tie is to be 2 feet above the bottom which is at sub-grade. If 
the surrounding material is yielding and exerts great pressure, 
the sides should be battered inward 1 foot at the bottom. For 
a double track tunnel the design is similar, except that the width 
is increased by the standard spacing between double tracks and 
the top is a compound curve made up of two 8-foot-radius 
curves at the sides which compound into a curve over the center 
which will give a clear height of 22 feet 6 inches over the center 
of each tie. The base of the roof curve is 13 feet 6 inches above 
the top of the ties. The bottom slopes to a central gutter which 
is 6 inches below the side corners, which are at sub-grade. Six- 
inch cast-iron pipes should be spaced as needed and run from 
each side to the central gutter. The width of both single and 
double track tunnels should be increased, if the tunnel is on a 
curve, and the track centers should also be displaced, so that 
the clearance on each side is as great as on a tangent. Figs. 
89, 90 and 91,* show some typical cross-sections. 

199. Grade. A grade of at least 0.2% is needed for drainage. 
If the tunnel is at the summit of two grades, the tunnel grade 

* Drinker's "Tunneling." 



§198. 



TUNNELS, 



233 




FiQ. 89. — HoosAc Tunnel. Section through Solid Rock, 




Fig. 90. — Hoosac TunneSc Section through Soft Ground. 



234 



RAILROAD CONSTRUCTION. 



200. 



should be practically level, with an allowance for drainage, the 
actual summit being at either end but not in the center. When 
the tunnel forms part of a long ascending grade, it is advisable 
to reduce the grade through the tunnel unless the tunnel is 
very short. The additional atmospheric resistance and the 
decreased adhesion of the driver wheels on the damp rails in 
a tunnel will cause an engine to work very hard and still more 
rapidly vitiate the atmosphere until the accumulation of poison- 
ous gases becomes a source of actual danger to the engineer and 




1 



Fig. 91. ^St. Cloud Tunnel. 



fireman of the locomotive and of extreme discomfort to the 
passengers. If the nominal ruling grade of the road were 
maintained through a tunnel, the maximum resistance would be 
found in the tunnel!. This would probably cause trains to stall 
there, which would be objectionable and perhaps dangerous. 

200. Lining. It is a characteristic of many kinds of rock 
and of all earthy material that, although they may be self- 
sustaining when first exposed to the atmosphere, they rapidly 
disintegrate and require that the top and perhaps the sides and 
even the bottom shall be lined to prevent caving in. In this 
country, when timber was cheap, it was formerly framed as an 
arch and used as the ^permanent lining, but masonry is always 



§201 



TUNNELS. 



235 



to be preferred. Frequently the cross-section is made extra 

large so that a masonry lining may subsequently be placed inside 
the wooden lining and thus postpone a large expense until the 
road is better able to pay for the work. In very soft unstable 
material, like quicksand, an arch of cut stone voussoirs may be 
necessary to withstand the pressure. A good quality of brick is 
occasionally used for lining, as they are easily handled and make 
good masonry if the pressure is not excessive. Only the best 
of cement mortar should be used, econom^^ in this feature being 
the worst of folly. Of course the excavation must include the 
outside line of the lining. Any excavation which is made out- 
side of this line (by the fall of earth or loose rock or by excessive 
blasting) must be refilled with stone well packed in. Occasionally 
it is necessary to fill these spaces with concrete. Of course it is 
not necessary that the lining be uniform throughout the tunnel. 
201. Shafts. Shafts are variously made with square, rectan- 
gular, elliptical, and circular cross-sections. The rectangular 




Fig. 92. —Connection with Shaft, Church Hill Tunnel. 



cross-section, with the longer axis parallel with the tunnel, is 
most usually employed. Generally the shaft is directly over the 
center of the tunnel, but that always impHes a comphcated con- 
neciion between the Hnings of the tunnel a^d shaft, provided 



236 



RAILROAD CONSTRUCTION. 



§202. 



such linings are necessary. It is easier to sink a shaft near to 
one side of the tunnel and make an opening through the nearly 
vertical side of the tunnel. Such a method was employed in the 
Church Hill Tunnel, illustrated in Fig. 92.* Fig. 93 f shows 
a cross-section for a large main shaft. Many shafts have been 
built with the idea of being left open permanently for ventila- 
tion and have therefore been elaborately lined with masonry. 




X. 



Fig. 93. — Cross-section, Large Main Shaft. 

The general consensus of opinion now appears to be that shafts 
are worse than useless for ventilation ; that the quick passage of 
a train through the tunnel is the most effective ventilator; and 
that shafts only tend to produce cross-currents and are ineffective 
to clear the air. In consequence, many of these elaborately 
lined shafts have been permanently closed, and the more recent 
practice is to close up a shaft as soon as the tunnel is completed. 
Shafts always form drainage -w^ells for the material they pass 
through, and sometimes to such an extent that it is a serious 
matter to dispose of the w^ater that collects at the bottom, 
requiring the construction of large and expensive drains. 

202. Drains. A tunnel will almost invariably strike veins of 
water which will promptly begin to drain into the tunnel and 
not only cause considerable trouble and expense during construc- 
tion, but necessitate the provision of permanent drains for its 
perpetual disposal. These drains must frequently be so large as 



* Drinker's '' Tunnel irag." 

t Rziha, "Lehrbuch der Gesammten Tunnelbaukunst." 



§203. TUNNELS. 237 

to appreciably increase the required cross-section of the tunnel. 
Generally a small open gutter on each side will suffice for this 
purpose, but in double-track tunnels a large covered drain is 
often built between the tracks. It is sometimes necessary to 
thoroughly grout the outside of the lining so that water will not 
force its way through the masonry and perhaps injure it, but 
may freely drain do\\Ti the sides and pass through openings in 
the side walls near their base into the gutters. 

CONSTRUCTION. 

203. Headings. The methods of all tunnel excavation de- 
pend on -the general principle that all earthy material, except 
the softest of liquid mud and quicksand, will be self-sustaining 
over a greater or less area and for a greater or less time after 
excavation is made, and the work consists in excavating some 
material and immediately propping up the exposed surface by 
timbering and poling-boards. The excavation of the cross- 
section begins with cutting out a ^'heading," which is a small 
horizontal drift whose breast is constantly kept 15 feet or more 
in advance of the full cross-sectional excavation. In solid 
self-sustaining rock, which will not decompose upon exposure 
to air, it becomes simply a matter of excavating the rock with 
the least possible expenditure of time and energy. In soft 
ground the heading must be heavily timbered, and as the heading 
is gradually enlarged the timbering must be gradually extended 
and perhaps replaced, according to some regular system, so that 
when the full cross-section has been ex- 
cavated it is supported by such timbering 
as is intended for it. The heading is 
sometimes made on the center line near 
the top; with other plans, on the center 
line near the bottom; and sometimes two 
simultaneous headings are run in the two 
lower corners. Headings near the bot- 
tom serve the purpose of draining the 
material above it and facilitating the 
excavation. The simplest case of head- 
ing timbering is that sho^Ti in Fig. 94, 
in which cross-timbers are placed at in- yiq. 94. 

tervals just under the roof, set in notches 
cut in the side walls and supporting poling-boards which sus- 




238 



RAILROAD CONSTRUCTION. 



§204. 



tain whatever pressure may come on them. Cross -timbers 
near the bottom support a flooring on which vehicles for trans- 
porting material may be run and under which the drainage 
may freely escape. As the necessity for timbering becomes 
greater, side timbers and even bottom timbers must be added, 
these timbers supporting poling-boards, and even the breast 
of the heading must be protected by boards suitably braced. 




Fig. 95. — Timbering for Tunnel Heading. 

as shown in Fig. 95. The supporting timbers are framed into 
collars in such a manner that added pressure only increases 
their rigidity. 

204. Enlargement. Enlargement is accomplished by remov- 
ing the poling-boards, one at a time, excavating a greater or less 
amount of material, and immediately supporting the exposed 
material with poling-boards suitably braced. (See Figs. 95 and 
96.) This work being systematically done, space is thereby 
obtained in which the framing for the full cross-section may be 
gradually introduced. The framing is constructed with a cross- 



§205. 



TUNNELS. 



239 



section so large that the masonry lining may be constructed 
within it. 
205. Distinctive features of various methods of construction. 

There are six general systems, known as the English, German, 
Belgian, French, Austrian, and American. They are so named 




Fig. 96. 



from the origin of the methods, although their use is not con- 
fined to the countries named. Fig. 97 shows by numbers (1 to 5) 
the order of the excavation within the cross-sections. The Eng- 
lish, Austrian, and American systems are alike in excavating the 
entire cross-section before beginning the construction of the 
masonry lining. The German method leaves a solid core (5) 
until practically the whole of the lining is complete. This has 
the disadvantage of extremely cramped quarters for work, poor 
ventilation, etc. The Belgian and French methods agree in 
excavating the upper part of the section, building the arch at 
once, and supporting it temporarily until the side walls are 
built. The Belgian method then takes out the core (3), removes 
very short sections of the sides (4) immediately underpinning 
the arch with short sections of the side walls and thus gradually 
constructing the whole side wall. The French method digs out 
the sides (3), supporting the arch temporarily with timbers and 
then replacing the timbers with masonry; the core (4) is taken 
out last. The French method has the same disadvantage as the 
German— -working in a cramped space. The Belgian and French 
systems have the disadvantage that the arch, supported tempo- 
rarily on timber, is very apt to be strained and cracked by the 
slight settlement that so frequently occurs in soft material. The 
English, Austrian, and American methods differ mainly in the 



240 



RAILROAD CONSTRUCTION. 



§205, 



design of the timbering. The English support the roof by lines 
of very heavy longitudinal timbers which are supported at com- 
paratively wide intervals by a heavy framework occupying the 



4 i 3 

1 
4- 

5 1 1 


4 



5 




ENGLISH 



AUSTRIAN 



/f 


3 1 


^ 




^ 


j 1 I 


^ 


, 


1 




1 . 


2 ! 




2 










! 


5 







4 


3 


4 


1 j 




1 











GERMAN 



BELGIAN 





FRENCH AMERICAN 

Fig. 97. —Order of Working by the Various Systems. 



whole cross-section. The Austrian system uses such frequent 
cross-frames of timber-work that poling-boards will suffice to 
support the material between the frames. The American sys- 
tem agrees with the Austrian in using frequent cross-frames 



§ 206. TUNNELS. 241 

supporting poling-boards, but differs from it in that the " cross- 
frames" consist simply of arches of 3 to 15 wooden voussoirs, 
the voussoirs being blocks of 12"Xl2" timber about 2 to 8 feet 
long and cut with joints normal to the arch. These arches are 
put together on a centering which is removed as soon as the arch 
is keyed up and thus immediately opens up the full cross-section, 
so that the center core (4) may be immediately dug out and the 
masonry constructed in a large open space. The American sys- 
tem has been used successfully in very soft ground, but its ad- 
vantages are greater in loose rock, when it is much cheaper than 
the other methods which employ more timber. Fig. 92 and 
Plate III illustrate the use of the American system. Fig. 92 
shows the wooden arch in place. The masonry arch may be 
placed when convenient, since it is possible to lay the track and 
commence traffic as soon as the wooden arch is in place. The 
student is referred to Drinker's ''Tunneling" and to Rziha's 
"Lehrbuch der Gesammten Tunnelbaukunst " for numerous 
illustrations of European methods of tunnel timbering. 

206. Ventilation during construction. Tunnels of any great 
length must be artificially ventilated during construction. If 
the excavated material is rock so that blasting is necessary, the 
need for ventilation becomes still more imperative. The inven- 
tion of compressed-air drills simultaneously solved two difficul- 
ties. It introduced a motive powder which is unobjectionable in 
its application (as gas would be), and it also furnished at the same 
time a supply of just w^hat is needed — pure air. If no blasting 
is done (and frequently even when there is blasting), air must be 
supplied by direct pumping. The cooling effect of the sudden 
expansion of compressed air only reduces the otherwise objection- 
ably high temperature sometimes found in tunnels. Since pure 
air is being continually pumped in, the foul air is thereby forced 
out. 

207. Excavation for the portals. Under normal conditions 
there is always a greater or less amount of open cut preceding 
and following a tunnel. Since all tunnel methods depend (to 
some slight degree at least) on the capacity of the exposed ma- 
terial to act as an arch, tjiere is implied a considerable thickness 
of material above the tunnel. This thickness is reduced to 
nearly zero over the tunnel portals and therefore requires special 
treatment, particularly when the material is very soft. Fig. 98 * 

* Rziha, "Lehrbuch der Gesammten Tiinnelbaukunst." 



242 



KAILROAD CONSTRUCTION, 



§208, 



illustrates one method of breaking into the ground at a portal. 
The loose stones are piled on the framing to give stability to the 
framing by their weight and also to retain the earth on the 




1 



Fig. 98. — Timbering for Tunnel Portal. 



,slope above. Another method is to sink a temporary shaft to 
the tunnel near the portal; immediately enlarge to the full size 
and build the masonry lining; then work back to the portal. 
This method is more costly, but is preferable in very treacherous 
ground, it being less liable to cause •landslides of the surface 
material, 

2o8. Tunnels vs. open cuts. In cases in which an open cut 
rather than a tunnel is a possibility the ultimate consideration 
is generally that of first cost combined with other financial con- 



IE 







Elevation op Portai,. 
Phcenixville Tunnel. P. S. V. R. R. 



LoNon-nDiNAL SBcnoK of Pobtai.. 



tLATl III. 




LONQITUDINAL SbCTIOH Of PoBTAIiu 



§209. 



TUNNELS. 



243 



siderations and annual maintenance charges directly or indirectly 
connected with it. Even when an open cut may be constructed 
at the same cost as a tunnel (or perhaps a little cheaper) the 
tunnel may be preferable under the following conditions: 

1. ^\Tien the soil indicates that the open cut would be liable 
to landslides. 

2. When the open cut would be subject to excessive snow- 
drifts or avalanches. 

3. AVhen land is especially costly or it is desired to run under 
existing costly or valuable buildings or monuments. When run- 
ning through cities, tunnels are sometimes constructed as open 
cuts and then arched over. 

These cases apply to tunnels vs. open cuts when the aline- 
ment is fixed by other considerations than the mere topography. 
The broader question of excavating tunnels to avoid excessive 
grades or to save distance or curvature, and similar problems, 
are hardly susceptible of general analysis except as questions of 
railway economics and must be treated individually. 

209, Cost of tunneling. The cost of any construction which 
involves such uncertainties as tunneling is very variable. It 
depends on the material encountered, the amount and kind of 
timbering required, on the size of the cross-section, on the price 
of labor, and especially on the reconstruction that may be neces- 
sary on account of mishaps. 

Headings generally cost $4 to $5 per cubic yard for excava- 
tion, while the remainder of the cross-section in the same tunnel 
may cost about half as much. The average cost of a large 
number of tunnels in this country may be seen from the follow- 
ing table:* 





( 


C^ost per cubic yard 




Cost per 
lineal foot. 




Excavation. 


Masonry. 




Material. 


Single. 






Single. 


Double. 


Single. 


Double. 


Double. 


Hard rock 

Loose rock 

Soft ground. . . . 


$5.89 
3.12 
3.62 


$5.45 
3.48 
4.64 


$12.00 

9.07 

15.00 


$8.25 
10.41 
10.50 


$69.76 

80.61 

135.31 


$142.82 
119.26 
174.42 



* Figures derived from Drinker's "Tunneling." 



244 RAILROAD CONSTRUCTION. § 209. 

A considerable variation from these figures may be foimd in 
individual cases, due sometimes to unusual skill (or the lack of 
it) in prosecuting the work, but the figures will generally be 
sufficiently accurate for preliminary estimates or for the com- 
parison of two proposed routes. 



CHAPTER VI. 
CITLVERTS AND MINOR BRIDGES. 

210. Definition and object. Although a variable percentage 
of the rain falHng on any section of country soaks into the 
ground and does not immediately reappear, yet a ver^^ large 
percentage flows over the surface, always seeking and following 
the lowest channels. The roadbed of a railroad is constantly 
intersecting these channels, which frequently are normally dry. 
In order to prevent injury to railroad embankments by the im- 
pounding of such rainfall, it is necessary to construct waterways 
through the embankment through which such rainflow may 
freely pass. Such waterways, called culverts, are also appli- 
cable for the bridging of very small although perennial streams, 
and therefore in this w^ork the term culvert will be applied to 
all water-channels passing through a railroad embankment 
which are not of sufficient magnitude to require a special struc- 
tural design, such as is necessary for a large masonry arch or a 
truss bridge. 

211. Elements of the design. A well-designed culvert must 
afford such free passage to the water that it will not ''back up" 
over the adjoining land nor cause any injury to the embankment 
or culvert. The ability of the culvert to discharge freely all the 
water that comes to it evidently depends chiefly on the area of 
the waterway, but also on the form, length, slope, and materials 
of construction of the culvert and the nature of the approach 
and outfall. When the embankment is very low and the amount 
of water to be discharged very great, it sometimes becomes 
necessary to allow the water to discharge ''under a head," i e., 
with the surface of the water above the top of the culvert. 
Safety then requires a much stronger construction than would 
otherwise be necessary to avoid injury to the culvert or embank- 
ment by washing. The necessity for such construction should 
be avoided if possible. 

245 



246 RAILROAD CONSTRUCTION. § 212. 



AREA OF THE WATERWAY. 

212. Elements involved. The determination of the required 
area of the waterway involves such a multiplicity of indeter- 
minate elements that any close determination of its value from 
purely theoretical considerations is a practical impossibilityo 
The principal elements involved are: 

a. Rainfall. The real test of the culvert is its capacity to 
discharge without injury the flow resulting from the extraordi- 
nary rainfalls and "cloud bursts" that may occur once in many« 
years. Therefore, while a knowledge of the average annual 
rainfall is of very little value, a record of the maximum rainfall 
during heavy storms for a long term of years may give a relative 
idea of the maximum demand on the culvert. 

b. Area of watershed. This signifies the total area of country 
draining into the channel considered. When the drainage area 
is very small it is sometimes included within the area surveyed 
by the preliminary survey. When larger it is frequently possi- 
ble to obtain its area from other maps with a percentage of 
accuracy sufficient for the purpose. Sometimes a special survey 
for the purpose is considered justifiable. 

c. Character of soil and vegetation. This has a large in- 
fluence on the rapidity with which the rainflow from a given 
area will reach the culvert. If the soil is hard and impermeable 
and the vegetation scant, a heavy rain will run off suddenly, 
taxing the capacity of the culvert for a short time, while a 
spongy soil and dense vegetation will retard the flow, making it 
more nearly uniform and the maximum flow at any one time 
much less. 

d. Shape and slope of watershed. If the watershed is very 
long and narrow (other things being equal), the water from the 
remoter parts will require so much longer time to reach the 
culvert that the flow will be comparatively uniform, especially 
when the slope of the whole watershed is very low. When the 
slope of the remoter portions is quite steep it may result in the 
nearly simultaneous arrival of a storm- flow from all parts of the 
watershed, thus taxing the capacity of the culvert. 

e. Effect of design of culvert. The principles of hydraulics 
show that the slope of the culvert, its length, the form of the 
cross-section, the nature of the surface, and the form of the 



§ 213. CULVERTS AND MINOR BRIDGES. 247 

approach and discharge all have a considerable influence on the 
area of cross-section required to discharge a given volume of 
water in a given time, but unfortunately the combined hy- 
draulic effect of these various details is still a very uncertain 
quantity. 

213. Methods of computation of area. There are three pos- 
sible methods of computation. 

(a) Theoretical. As shown above it is a practical impossi- 
bility to estimate correctly the combined effect of the great mul- 
tiplicity of elements which influence the final result. The nearest 
approach to it is to estimate b}^ the use of empirical formulae 
the amount of water which will be presented at the upper end 
of the culvert in a given time and then to compute, from the 
principles of hydraulics, the rate of flow through a culvert of 
given construction, but (as shown in § 212, e) such methods are 
still very unreliable, owing to lack of experimental knowledge. 
This method has apparently greater scientific accuracy than 
other methods, but a little study will show that the elements 
of uncertainty are as great and the final result no more reliable. 
The method is most reliable for streams of uniform flow, but 
it is under these conditions that method (c) is most useful. The 
theoretical method v*411 not therefore be considered further. 

(b) Empirical. As illustrated in § 214, some formulae make 
the area of waterway a function of the drainage area, the for- 
mula bemg affected by a coefficient the value of which is esti- 
mated between limits according to the judgment Assuming 
that the formulae are sound, their use only narroT\'s the limits of 
error, the final determination depending on experience and 
judgment. 

(c) From observation. This method, considered b}' far the 
best for permanent work, consists m observing the high- water 
marks on contracted channel-openings which are on the same 
stream and as near as possible to the proposed culvert. If the 
country is new and there are no such openings, the wisest plan 
is to bridge the opening by a temporary structure in wood which 
has an ample waterway (see § 158, 6, 4) and carefully observe 
all high-water marks on that opening during the 6 to 10 years 
which is ordinarily the minimum life of such a structure. As 
shown later, such observations may be utilized for a close com- 
putation of the required waterway. Method (b) may be utilized 
for an approximate calculation for the required area for the tem- 



l I. 



248 RAILROAD CONSTRUCTION. § 214. 

porary structure, using a value which is intentionally excessive, 
so that a permanent structure of sufficient capacity may subse- 
quently be constructed within the temporary structure. 

214. Empirical formulae. Two of the best known empirical 
formulae for area of the waterway are the following: 

(a) Myer's formula: 

Area of waterway in square feet = CX v^drainage area in acres, 
where (7 is a coefficient varying from 1 for flat country to 4 for 
mountainous country and rocky ground. As an illustration, if 
the drainage area is 100 acres, the waterway area should be from 
10 to 40 square feet, according to the value of the coefficient 
chosen. It should be noted that this formula does not regard 
the great variations in rainfall in various parts of the world nor 
the design of the culvert, and also that the final result depends 
largely on the choice of the coefficient. 

(b) Talbot's formula: . 

Area of waterway in square feet = (7X>y(drainage area in acres) ^ 
** For steep and rocky ground C varies from f to 1. For rolling 
agricultural country subject to floods at times of melting snow, 
and with the length of the valley three or four times its width, C 
is about i; and if the stream is longer in proportion to the area, 
decrease C, In districts not affected by accumulated snow, and 
Inhere the length of the valley is several times the width, J or I, 
or even less, may be used. C should be increased for steep side 
slopes, especially if the upper part of the valley has a much 
greater fall than the channel at the culvert.'' * As an illus- 
tration, if the drainage area is 100 acres the area of waterway 
should be (7X31.6. The area should then vary from 5 to 31 
square feet, according to the character of the country. Like 
the previous estimate, the result depends on the choice of a 
coefficient and disregards local variations in rainfall, except as 
they may be arbitrarily allowed for in choosing the coeffi- 
cient. 

215. Value of empirical formulae. The fact that these for- 
mulae, as well as many others of similar nature that have been 
suggested, depend so largely upon the choice of the coefficient 
shows that they are valuable ^'more as a guide to the judgment 
than as a working rule," as Prof. Talbot explicitly declares in 



* Prof. A. N. Talbot, "Selected Papers of the Civil Engineers' Club of 
the Univ. of Illinois." 



§ 216. CULVERTS AND MINOR BRIDGES. 249 

i 
commenting on his own formula. In short, they are chiefly valu- 
able in indicating a probable maximum and minimum between 
which the true result probably lies. 

216. Results based on observation. As already indicated in 
§ 213, observation of the stream in question gives the most 
reliable results. If the country is new and no records of the 
flow of the stream during heavy storms has been taken, even 
the life of a temporary wooden structure may not be long enough 
to include one of the unusually severe storms which must be 
allowed for, but there will usually be some high-water mark 
w^hich will indicate how much opening will be required. The 
following quotation illustrates this : '' A tidal estuary may gen- 
erally be safely narrowed considerably from the extreme water 
lines if stone revetments are used to protect the bank from 
wash. Above the true estuary, where the stream cuts through 
the marsh, we generally find nearly vertical banks, and we are 
safe if the faces of abutments are placed even with the banks. 
In level sections of the country, where the current is sluggish, 
it is usually safe to encroach somewhat on the general width 
of the stream, but in rapid streams among the hills the width 
that the stream has cut for itself through the soil should not h^ 
lessened, and in ravines carrying mountain torrents the open^ 
ings must be left very much larger than the ordinary appear- 
ance of the banks of the stream would seem to make neces- 
sary." * 

As an illustration of an observation of a storm-flow through 
a temporary trestle, the f ollow^ing is quoted : ^' Having the flood 
height and velocity, it is an easy matter to determine the vol- 
ume of w^ater to be taken care of. I have one ten-bent pile 
trestle 135 feet long and 24 feet high over a spring branch that 
ordinarily runs about six cubic inches per second. Last sum- 
mer during one of our heavy rainstorms (four inches in less 
than three hours) I visited this place and found by float obser- 
vations the surface velocity at the highest stage to be 1.9 feet 
per second. I made a high-water mark, and after the flood- 
water receded found the width of stream to be 12 feet and an 
average depth of 2} feet. This, mth a surface velocity of 1.9 
feet per second, would give approximately a discharge of 50 



* J. P. Snow, Boston & Maine Railway. From Report to Association of 
Railway Superintendents of Bridges and Buildings. 1897. 



250 RAILROAD CONSTRUCTION. § 217 

cubic feet, or 375 gallons, per second. Having this information 
it is easy to determine size of opening required." * 

217. Degree of accuracy required. The advantages result- 
ing from the use of standard designs for culverts (as well as 
other structures) have led to the adoption of a comparatively 
small number of designs. The practical use made of a compu- 
tation of required waterway area is to determine which one of 
several standard designs will most nearly fulfill the require- 
ments. For example, if a 24-inch iron pipe, having an area of 
3.14 square feet, is considered to be a little small, the next size 
(30-inch) would be adopted; but a 30-inch pipe has an area of 
4.92 square feet, which is 56% larger. A similar result, except 
that the percentage of difference might not be quite so marked, 
will be found by comparing the areas of consecutive standard 
designs for stone box culverts. 

The advisability of designing a culvert to withstand any 
storm-flow that may ever occur is considered doubtful. Several 
years ago a record-breaking storm in New England carried 
away a very large number of bridges, etc., hitherto supposed 
to be safe. It was not afterward considered that the design of 
those bridges was faulty, because the extra cost of constructing 
bridges capable of withstanding such a flood, added to interest 
for a long period of years, would be enormously greater than the 
cost of repairing the damages of such a storm once or twice in 
a century. Of course the element of danger has some weight, 
but not enough to justify a great additional expenditure, for 
common prudence would prompt unusual precautions during 
or immediately after such an extraordinary storm. 

PIPE CULVERTS. 

218. Advantages. Pipe culverts, made of cast iron or earthen- 
ware, are very durable, readily constructed, moderately cheap, 
will pass a larger volume of water in proportion to the area than 
many other designs on account of the smoothness of the sur- 
face, and (when using iron pipe) may be used very close to 
the track when a low opening of large capacity is required. 
Another advantage lies in the ease with which they may be 
inserted through a somewhat larger opening that has been 

* A. J. Kelley, Kansas City Belt Railway. From Report to Association 
of Railway Superintendents of Bridges and Buildings. 1897. 



§ 219. CULVERTS AND MINOR BRIDGES. 251 

temporarily lined with wood, without disturbing the roadbed 
or track 

219. Construction. Permanency requires that the founda- 
tion shall be firm and secure against being washed out. To 
accomplish this, the soil of the trench should be hollowed out to 
fit the lower half of the pipe, making suitable recesses for the 
bells. In very soft treacherous soil a foundation-block of con- 
crete is sometimes placed under each joint, or even throughout 
the whole length. When pipes are laid through a slightly 
larger timber culvert great care should be taken that the pipes 
are properly supported, so that there will be no settling nor 
development of unusual strains when the timber finally decays 
and gives way. To prevent the washing away of material 
around the pipe the ends should be protected by a bulkhead. 
This is best constructed of masonry (see Fig. 99), although wood 
is sometimes used for cheap and minor constructions. The joints 
should be calked, especially when the culvert is liable to run 
full or when the outflow is impeded and the culvert is liable to 
be partly or wholly filled during freezing weather. The cost of 
a calking of clay or even hydraulic cement is insignificant com- 
pared with the value of the additional safety afforded. When 
the grade of the pipe is perfectly uniform, a very low rate of 
grade wdll suffice to drain a pipe culvert, but since some uneven- 
ness of grade is inevitable through uneven settlement or im- 
perfect construction, a grade of 1 in 20 should preferably *be 
required, although much less is often used. The length of a 
pipe culvert is approximately determined as follows: 

Length = 2s {depth of embankment) + (width of roadbed), 

in which s is the slope ratio (horizontal to vertical) of the banks. 
In practice an even number of lengths should be used which will 
equal or exceed the length given by this formula. 

220. Iron-pipe culverts. Simple cast-iron pipes are used in 
sizes from 12" to 48" diameter. These are usually made in 
lengths of 12 feet with a few lengths of 6 feet, so that any required 
length may be more nearly obtained. The lightest pipes made 
are sufficiently strong for the purpose, and even those which 
would be rejected becauseof incapacity to withstand considerable 
internal pressure may be utilized for this work. In Fig. 99 are 
shown the standard plans used on the C. C. C. & St. L. Ry., 
which may be considered as typical plans. 




OtZ- 



-aJ 



0,2- 



t0;2- 



FiG. 99. — Standard Cast-iron 
Pipe Cut.vert. C. G. C. & 
St. L. Ry. (May 1893.) 



252 



§221. 



CULVERTS AND MINOR BRIDGES. 



253 



Pipes formed of cast-iron segments have been used up to 12 
feet diameter. The shell is then made comparatively thin, but is 
stiffened by ribs and flanges on the outside. The segments break 
joints and are bolted together through the flanges. The joints are 
made tight by the use of a tarred rope, together with neat cement. 
/ 221. Tile-pipe culverts. The pipes used for this purpose 
vary from 12" to 30" in diameter. When a larger capacity is 
required two or more pipes may be laid side by side, but in 
such a case another design might be preferable. It is frequently 
specified that " double-strength '' or "extra-heavy'' pipe shall 
be used, evidently with the idea that the stresses on a culvert- 
pipe are greater than on a sewer-pipe. But it has been con- 
clusively demonstrated that, no matter how deep the embank- 
ment, the pressure cannot exceed a somewhat uncertain maxi- 
mum, also that the greatest danger consists in placing the pipe 
so near the ties that shocks may be directly transferred to the 
pipe without the cushioning effect of the earth and ballast. 
When the pipes are well bedded in clear earth and there is a 




UP-STR-EAr/u.E3MD. DOWN-6TREAM E>lD. DOWN-STREA-M EMD. THREE PIPES'.^ 

Fig. 100. — Standard Vitrified-pipe Culvert. Plant System. (1891.)* 

sufficient depth of earth over them to avoid direct impact (at least 
three feet) the ordinary sewer-pipe will be sufficiently strong. 
" Double-strength " pipe is frequently less perfectly burned, and 
the supposed extra strength is not therefore obtained. In Fig. 100 
are shown the standard plans for vitrified-pipe culverts as used 



254 



RAILROAD CONSTRUCTION. 



§222. 



on the " Plant system. '^ Tile pipe is much cheaper than iron 
pipe, but is made in much shorter lengths and requires much 
more work in laying and especially to obtain a uniform grade. 

Concrete pipes, factory made, both plain and with metal rein- 
forcement, 12" to 48" in diameter, have come into use in recent 
years. They are stronger and more dependable than tile and 
there is no deterioration. 

BOX CULVERTS. 

222. Wooden box culverts. This form serves the purpose 
of a cheap temporary construction which allows the use of a 
ballasted roadbed. As in all temporary constructions, the area 
should be made considerably larger than the calculated area 
(§§213-216), not only for safety but also in order that, if the 
smaller area is demonstrated to be sufficiently large, the per- 
manent construction (probably pipe) may be placed inside with- 
out disturbing the embankment. All designs agree in using 
heavy timbers (12"X12", 10"X12", or 8"X12") for the side 
walls, cross-timbers for the roof, every fifth or sixth timber 
being notched down so as to take up the thrust of the side walls, 
and planks for the flooring. Fig. 101 shows some of the standard 
designs as used by the C, M. & St, P. Ry. 



I 




Fig. 101.— Standard Timber Box Culvert. C.,M.& St. P. Ry. (Feb. 1889.) 



223. Stone box culverts. In localities where a good quality 
of stone is cheap, stone box culverts are the cheapest form of 
permanent construction for culverts of medium capacity, but 
their use is decreasing owing to the frequent difficulty m obtain- 
ing really suitable stone within a reasonable distance of the 
culvert. The clear span of the cover-stones varies from 2 to 4 
feet. The required thickness of the cover-stones is sometimes 



1^ 



§223. 



CULVERTS AND MINOR BRIDGES. 



255 



calculated by the theory of transverse strains on the basis of 
certain assumptions of loading — as a function of the height of 
the embankment and the unit strength of the stone used. Such 
a method is simply another illustration of a class of calculations 
which look very precise and beautiful, but which are worse than 
useless (because misleading) on account of the hopeless imcer- 




PLAN 

Fig. 102. — Standard Single Stone Ctjlvert (3'X4'). 

(1S90.) 



N. & W. R.R. 



tainty as to the true value of certain quantities which must be 
used in the computations In the first place the true value of 
the unit tensile strength of stone is such an uncertain and variable 



256 



RAILROAD CONSTRUCTION. 



§223. 



quantity that calculations based on any assumed value for it are 
of small reliability. In the second place the weight of the prism 
of earth lying directly above the stone, plus an allowance for live 
load, is by no means a measure of the load on the stone nor of 
the forces that tend to fracture it. All earthwork will tend to 




PLAN 



Fig. 103. — Standard Double Stone Culvert (3'X40. N. & W. R. R. 

(1890.) 



form an arch above any cavity and thus relieve an uncertain 
and probably variable proportion of the pressure that might 
otherwise exist. The higher the embankment the less the pro- 



§224. 



CULVERTS AND MINOR BRIDGES, 



257 



portionate loading, until at some uncertain height an increase 
in height will not increase the load on the cover-stones. The 
effect of frost is likewise large, but uncertain and not computable. 
The usual practice is therefore to make the thickness such as 
experience has shown to be safe with a good quality of stone, 
i.e., about 10 or 12 inches for 2 feet span and up to 16 or 18 
inches for 4 feet span- The side w^alls should be carried down 
deep enough to prevent their being undermined by scour or 
heaved by frost. The use of cement mortar is also an important 
feature of first-class work, especially when there is a rapid scour- 
ing current or a liability that the culvert will run under a head. 
In Figs. 102 and 103 are shown standard plans for single and 
double stone box culverts as used on the Norfolk & Western R.R. 
224. Old-rail culverts. It sometimes happens (although very 
rarely) that it is necessary to bring the grade line within 3 or 4 
feet of the bottom of a stream and yet allow an area of 10 or 12 
square feet. A single large pipe of sufficient area could not be 
used in this case. The use of se^'eral smaller pipes side by side 
w^ould be both expensive and inefficient. For similar reasons 
neither wooden nor stone box culverts could be used. In such 
cases, as well as in many others where the head-room is not so 
limited, the plan illustrated in Fig. 104 is a very satisfactory 




Fig. 104. — Standard Old-rail Culvert. N. & W. R.R. (1895.) 



solution of the problem. The old rails, having a length of 8 or 
9 feet, are laid close together across a 6-foot opening. Some- 
times the rails are held together by long bolts passing through 



258 RAILROAD CONSTRUCTION. § 225. 

the webs of the rails. In the plan shown the rails are confined 
by low end walls on each abutment. This plan requires only 
15 inches between the base of the rail and the top of the culvert 
channel. It also gives a continuous ballasted roadbed. 

225. Reinforced Concrete Culverts. The development of 
reinforced concrete as a structural material is illustrated in its 
extensive adoption for arches and also for culverts. One of the 
special types which has been adopted is that of a box culvert 
which has a concrete bottom. Since this bottom can be made 
so that it will withstand an upward transverse stress, it furnishes 
a broad foundation for the whole culvert, and thus entirely 
eliminates the necessity for extensive footing to the side walls of 
the culvert, such as are necessary in soft ground with an ordinary 
stone culvert. Another advantage is that the inside of the cul- 
vert may be made perfectly smooth and thus offer less resistance 
to the passage of water through it. As may be noticed from 
Fig. 105, such a culvert is provided with flaring head walls, and 
sunken end walls, so that the water may not scour underneath 
the culvert, and other features common to other types. No 
attempt will here be made to discuss the design of reinforced 
concrete, except to say that all four sides of such a box culvert 
are designed to withstand a computed bursting pressure which 
tends to crush the flat sides inward. In Fig. 105 is shown one 
illustration of the many types of culverts which have been 
designed of reinforced concrete. 



ARCH CULVERTS. 

191. Influence of design on flow. The variations in the design 
of arch culverts have a very marked influence on the cost and 
efficiency. To combine the least cost with the greatest effi- 
ciency, due weight should be given to the following elements: 
(a) amount of masonry, (b) the simphcity of the constructive 
work, (c) the design of the wing walls, (d) the design of the 
junction of the wing walls with the barrel and faces of the arch, 
and (e) the safety and permanency of the construction. These 
elements are more or less antagonistic to each other, and the 
defects of most designs are due to a lack of proper proportion 
in the design of these opposing interests. The simplest con- 
struction (satisfying elements h and e) is the straight barrel arch 



^^ .fl.rt M«jTc3W a >IJ0 



STANDARD ARCH CULVERT 

8 FEET 

IJORFOLK & WESTERN R.R, 
(1891) 




^Vx.^ 



{Tu face paye 25'X) 



§226. 



CULVERTS AND MINOR BRIDGES. 



259 



/- 


i 

1 

-r* 
■<-i 

-U» 


< .A 




OT 

is 

»_■(- 
■*-;— 


^g LONGITUDINAL SECTION 






• / 












,_4__ + __a 1--A-. 

1 1 1 1 1 
4 — — .■.__. t— _4- — — -I'-- 


- 


■t- 
f- 
f- 
1— 

h- 
h- 




— > 


8 






—X- 


1 < 
1 ■ 




\-t~ 
\i 


1 ' 

-T- 
-T- 




1 
i-- 

1 — 

■1— 

J_- 
(-- 

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3 

— 4-- 

-4 • 




_ 


* C^I 




^0 


IV V 


^ /^T IV ^ 


1 










Ul- O '' 








260 



RAILROAD CONSTRUCTION. 



§227. 



between two parallel vertical head walls, as sketched in Fig. 
106, a. From a hydraulic standpoint the design is poor, as the 
water eddies around the corners, causing a great resistance 
which decreases the flow. Fig. 106, 6, shows a much better 




Fig. 106. — Types of Culverts. 

design in many respects, but much depends on the details of the 
design as indicated in elements (b) and (d). As a general thing 
a good hydraulic design requires complicated and expensive 
masonry construction, i.e., elements (6) and (d) are opposed. 
Design 106, c, is sometimes inapplicable because the water is 
liable to work in behind the masonry during floods and perhaps 
cause scour. This design uses less masonry than (a) or (6). 

227. Example of arch" culvert design. In Plate IV is shown 
the design for an 8-foot arch culvert according to the standard 
of the Norfolk and Western R. R. Note that the plan uses the 
flaring wing walls (Fig. 106, h) on the up-stream side (thus 
protecting the abutments from scour) and straight wing walls 
(similar to Fig, 106, c) on the down-stream end. This econo- 
mizes masonry and also simplifies the constructive w^ork. Note 
also the simplicity of the junction of the wing walls with the 
barrel of the arch, there being no re-entrant angles below the 
springing line of the arch. The design here shown is but one 
of a set of designs for arches varying in span from 6' to 30'. 



MINOR OPENINGS. 



228. Cattle-guards, (a) Pit guards. Cattle-guards will be 
considered under the he^d of minor openings, since the old- 
fashioned plan of pit guards, which are even now defended and 



§228. 



CULVERTS AND MINOR BRIDGES. 



261 



preferred by some railroad men, requires a break in the con- 
tinuity of the roadbed. A pit about three feet deep, five feet 




Fig. 107. — Cattle-guard with Wooden Slats. 



long, and as wide as the width of the roadbed, is walled up with 
stone (sometimes with wood), and the rails are supported on 
heavy timbers laid longitudinally with the rails. The break in 
the continuity of the roadbed produces a disturbance in the 
elastic wave running through the rails, the effect of which is 
noticeable at high velocities. The greatest objection, however, 
lies in the dangerous consequences of a derailment or a failure 
of the timbers owing to unobserved decay or destruction by 
fire— caused perhaps by sparks and cinders from passing loco- 
motives. The very insignificance of the structure often leads 
to careless inspection. But if a single pair of wheels gets off the 
rails and drops into the pit, a costly wreck is inevitable. 

(b) Surface cattle-guards. These are fastened on top of the 
ties; the continuity of the roadbed is absolutely unbroken and 
thus is avoided much of the danger of a bad wrecJ? owing to a 
possible derailment. The device consists essentially of overlay- 
ing the ties (both inside and outside the rails) with a surface on 



262 



RAILROAD CONSTRUCTION. 



§228. 



which cattle will not walk. The multitudinous designs for such 
a surface are variously effective in this respect. An objection, 




Fig. 108. — Sheffield Cattle-qxtard.. 



CENTER SECTION 



r:'^ 



"END SECTION 




Illustrating how 
.Sections should be 
placed on ties and 
manner of fastening 



FiQ. 109. — Climax Cattle-guard (tile). 



which is often urged indiscriminately against all such designs, is 
the liability that a brake-chain which may happen to be drag- 
ging may catch in the rough bars which are used. The bars 



§ 229. CULVERTS AND MINOR BRIDGES. 263 

are sometimes " home-made/* of wood, as shown in Fig. 107. 
Steel guards may be made as shown in Fig. 108. The general 
construction is the same as for the wooden bars. The metal 
bars have far greater durability, and it is claimed that they are 
more effective in discouraging cattle from attempting to cross. 
229. Cattle-passes. Frequently when a railroad crosses a 
farm on an embankment, cutting the farm into two parts, the 
railroad company is obliged to agree to make a passageway 
through the embankment sufficient for the passage of cattle and 
perhaps even farm-wagons. If the embankment is high enough 
so that a stone arch is practicable, the initial cost is the only 
great objection to such a construction; but if an open w^ooden 
structure is necessary, all the objections against the old-fashioned 
cattle-guards apply with equal force here. The avoidance of a 
grade crossing which would otherwise be necessary is one of the 
great compensations for the expense of the construction and 
maintenance of these structures. The construction is some- 
times made by placing two pile trestle bents about 6 to 8 feet 
apart, supporting the rails by stringers in the usual way, the 
special feature of this construction being that the embankments 
are filled in behind the trestle bents, and the thrust of the em- 
bankments is mutually taken up through the stringers, which 
are notched at the ends or other\^dse constructed so that the}^ 
may take up such a thrust. The designs for old-rail culverts 
and arch culverts are also utilized for cattle-passes when suitable 
and convenient, as well as the designs illustrated in the following 
section, and the reinforced concrete design of § 225. 

230. Standard stringer and I-beam bridges. The advantages 
of standard designs apply even to the covering of short spans 
with wooden stringers or with I beams — especially since the 
methods do not require much vertical space between the rails 
and the upper side of the clear opening, a feature which is often 
of prime importance. These designs are chiefly used for cul- 
verts or cattle-passes and for crossing over highwaj^s — providing 
such a narrow opening would be tolerated. The plans all imply 
stone abutments, or at least abutments of sufficient stability to 
withstand all thrust of the embankments. Some of the designs 
are illustrated in Plate V. The preparation of these standard 
designs should be attacked by the same general methods as 
already illustrated in § 190. When computing the required 



264 RAILROAD CONSTRUCTION. § 230. 

transverse strength, due allowance should be made for lateral 
bracing, which should be amply provided for. Note particu- 
larly the methods of bracing illustrated in Plate V. The designs 
calling for iron (or steel) stringers may be classed as permanent 
constructions, which are cheap, safe, easily inspected and main- 
tained, and therefore a desirable method of construction. 



1^'boLT EV! 




8 X 10 TIES, 14. 



f- 10"c-20*6'0 





(i'o face page 264.) 






ffi 


Y 






T m 


.NCHO 


1 I 




: b"x , 


o'ties, u 


"lONO, 


.■C.TOO.T 


1 




J 


E= 


o'C-20*6 


oXou.. 


ZffiE-- 


TS,.«l«C,ER 



„0» JOLT, ,-„,.. [T^ 




eL 



STANDARD I-BRIDGES-14-FT. SPAN. 

NORFOLK AND WESTERN R.R. . 
(1891.) 




(i'o jace pays 264.) 



CHAPTER VII. 

BALLAST. 

231. Purpose and requirements. ^' The object of the ballast 
is to transfer the applied load over a large surface; to hold the 
timber work in place horizontally; to carry off the rain-water 
from the superstructure and to prevent freezing up in winter; 
to afford means of keeping the ties truly up to the grade line; 
and to give elasticity to the roadbed.'* This extremely con- 
densed statement is a description of an ideally perfect ballast. 
The value of any given kind of ballast is proportional to the 
extent to which it fulfills these requirements. The ideally 
perfect ballast is not necessarily the most economical ballast 
for all roads. Light traffic generally justifies something cheaper, 
but a very common error is to use a very cheap ballast when a 
small additional expenditure would procure a much better 
ballast, which would be much more economical in the long run. 

232. Materials. The materials most commonly employed are 
gravel and broken stone. In many sections of the country 
other materials which more or less perfectly fulfill the require- 
ments as given above, are used. The various materials includ- 
ing some of these special types have been defined by the American 
Railway Engineering Association as follows: 

DEFINITIONS. 

Ballast. Selected material placed on the roadbed for the 
purpose of holding the track in line and surface. 

Stone ballast. Stone broken by artificial means into small 
fragments of specified sizes. 

Burnt clay. A clay or gumbo which has been burned into 
material for ballast. 

Chats. Tailings from mills in which zinc, lead, silver and other 
ores are separated from the rocks in which they occur. 

265 



266 KAILROAD CONSTRUCTION. § 232. 

Chert. An impure flint or hornstone occurring in beds. 

Cinders. The residue from the coal used in locomotives and 
other furnaces. 

Gravel. Worn fragments of rock, occurring in natural 
deposits, that will pass through a 2i-inch ring and be retained 
upon a No. 10 screen. 

Gumbo. A term commonly used for a peculiarly tenacious 
clay, containing no sand. 

Sand. Any hard, granular, comminuted rock which will 
pass through a No. 10 screen and be retained upon a No. 50 
screen. 

Slag. The waste product, in a more or less vitrified form, o! 
furnaces for reduction of ore. Usually the product of a blast- 
furnace. 

There is still another classification which may or may not be 
considered as ballast. It is perhaps hardly correct to speak 
of the natural soils as ballast, yet many miles of cheap rail- 
ways are "ballasted'' with the natural soil, which is then called 
Mud ballast. 

Broken or crushed stone. Rock ballast is generally specified 
to be that which may all be passed through a 1^^ inch (or 2 inch) 
ring, but which cannot pass through a |-inch mesh. It is most 
easily handled with forks. This method also has the advantage 
that when it is being rehandled the fine chips which would 
interefere with effectual drainage will be screened out. Rock 
ballast is more expensive in first cost and is also more trouble- 
some to handle, but in heavy traffic especially, the track will be 
kept in better surface and will require less work for maintenance 
after the ties have become thoroughly bedded. 

Burnt clay. Th2s material has been used in many sections of 
the country where broken stone or gravel are unobtainable 
except at a prohibitive cost, and where a suitable quality of 
clay is readily obtained. This clay should be of " gumbo '* 
variety and contain no gravel. It is sometimes burnt in a 
kiln, or it is sometimes burnt by piling the clay in long heaps 
over a mass of fuel, the pile being formed in such a way that 
a temporary but effectual kiln is made. It is necessary that 
a clear, clean fuel shall be used and that the firing shall be 
done by a man who is experienced in maintaining such a fire 
until the burning is completed. Such ballast may be burned 
very hard and it will last from four to six years. The cost of 



§232. . BALLAST. 267 

burning varies from 30 to 60 cents per cubic yard, according 
to the circumstances. 

Chats. This is a form of ballast which is peculiar to South- 
western Missouri and Southeastern Kansas. When this mate- 
rial was first used it was obtained from the refuse piles of the 
mills which treated the zinc and lead ores mined in those regions. 
With the processes then employed the material was obtained 
in lumps as large as broken stone, and they were considered to 
be as valuable as broken stone for ballast. Improvements in 
the processes of treating the ores have resulted in making this 
by-product very much smaller grained and of less value as bal- 
last, although it is Mill considered a desirable form of ballast 
where it may readily be obtained. It should be noted that it 
is classed with gravel and cinders in the forms of cross-section 
shown later. 

Chert, This is a form of flint or hornstone which occurs in 
nodules of a size that is suitable for ballast, and is a very good 
type of ballast wherever it is found, but its occurrence is com- 
paratively infrequent. It is classed with cemented gravel in 
the design of cross-sections of ballast. 

Cinders. This is one of the most universal forms of ballast, 
since it is a by-product of every road which uses coal as fuel. 
The advantages consist in the fairly good drainage, the ease of 
handling and the cheapness — after the road is in operation. 
One of the greatest disadvantages is the fact that the cinders are 
readily reduced to dust, which in dry weather becomes very 
objectionable. Cinders are usually considered preferable to 
gravel in yards. 

Gravel. This is one of the most common forms of good 
ballast. There are comparatively few railroads which cannot 
find, at some place along their line, a gravel pit which will 
afford a suitable supply of gravel for ballast. Sometimes it is 
unnecessary to screen it, but usually it is better to screen the 
gravel over a screen having a J-inch mesh so as to screen 
out all the dirt and the finer stones. 

Sand. Railroads which run along the coast are frequently 
ballasted merely with the sand obtained in the immediate 
neighborhood. One great advantage lies in the almost perfect 
drainage which is obtained. 

Slag. When slag is readily obtainable it furnishes an ex- 
cellent ballast which is free from dust and perfect in drainage 



268 RAILROAD CONSTRUCTION. § 233. 

qualities. Slag is classified with Clashed rock in the cross- 
sections shown below, but it should be noted that this only 
applies to the best qualities of slag, since its quality is quite 
variable. 

Mud ballast. When the natural soil is gravelly so that rain 
will drain through it quickly, it will make a fair roadbed for 
light traffic, but for heavy traffic, and for the greater part of 
the length of most roads, the natural soil is a very poor material 
for ballast; for, no matter how suitable the soil might be along 
limited sections of the road, it would practically never happen 
that the soil would be uniformly good throughout the whole 
length of the road. Considering that a heavy rain will in one 
day spoil the results of weeks of patient ** surfacing'' with mud 
ballast, it is seldom economical to use "mud" if there is a 
gravel-bed or other source of ballast anywhere on the line of 
the road. 

233. Cross-sections. The required depth of the cross-section 
to thQ sub-soil depends largely on the weight of the rolling 
stock which is to pass over the track. A careful examination 
of a roadbed to determine the changes which take place under 
th© tiea and also an examination of the track and ties during 
the passage of a heavy train shows that the heavy loads which 
are now common on railroad tracks force the tie into the bal- 
last with the passage of every wheel load. The effect on the 
ballast is a greater or less amount of crushing of the ballast. 
Even thft very hardest grades of broken stone are more or less 
crushed by grinding against each other during the passage of a 
train. The softer and weaker forms of ballast are ground up 
much more quickly. One result is the formation of a fine dust 
which interferes with the proper drainage of water through the 
[ballast. A second result is the compression of the ballast imme- 
diately under the tie into the sub-soil. In a comparatively 
short time a hole is formed under the tie which acts virtually . 
like a pump. With every rise and fall of the tie under each 
wheel load, the tie actually pumps the water from the surround- 
ing ballast and sub-soil into these various holes. When the 
ballast is of such a character that the water does not drain 
through it easily, the water will settle in these holes long tmough 
to seriously deteriorate the ties. When the track beco^Jies so 
much out of line or level, or so loose that it needs to be tamped 
up, the process of tamping has practically the effect of deepen- 



§ 234. BALLAST, 269 

ing the amount of ballast immediately under the tie, while the 
sub-soil is forced up between the ties. A longitudinal section 
of the sub-soil of a track which has been frequently tamped 
generally has a saw-tooth appearance, and the sub-soil, instead 
of being a uniform line, has a high spot between each tie, while 
the ballast is considerably below its normal level immediately 
under the tie. 

234. Classification of Railroads. The American Railway En- 
gineering Association has divided railroads into three classes 
with respect to the standards of construction which should be 
adopted for ballasting, as well as other details of construction. 
The three classes are as follows (quoted from the Association 
Manual): 

'^ Class 'A' shall include all districts of a railway having more 
than one main track, or those districts of a railway having a 
single main track with a traffic that equals or exceeds the follow- 
ing: 

Freight-car mileage passing over districts per year per 

mile 150000 

or, 
Passenger-car mileage per annum per mile of district. . . 10000 

with maximum speed of passenger-trains of 50 miles per hour. 

*^ Class *B' shall include all districts of a railway having a 
single main track with a traffic that is less than the minimum 
prescribed for Class ^ A' and that equals or exceeds the following : 

Freight-car mileage passing over districts per year per 

mile 50000 

or, 
Passenger-car mileage per annum per mile of district. . . 5000 

with maximum speed of passenger-trains of 40 miles per hour. 

*^ Class ^C shall include all districts of a railway not meeting 
the traffic requirements of Classes ^A^ or ^B.' " 

The classification was adopted on the consideration that 
quality of traffic as well as mere tonnage should determine 
the classification of a railroad. For example, it is considered 
that a road which operates a train at a speed of 50 miles an 

(hour should adopt the first class or Class '^A'' standards, even 
though there is but one train per day on that railroad. It 

llikewise means that any road whose traffic makes necessary the 



270 RAILROAD CONSTRUCTION. § 235. 

construction of a regular double track should adopt the first - 
class specifications. ^ 

235. Recommended sections for the several classifications. 
In Fig. 110 are shown a series of cross-sections which were 



_>|.124«-3'3"-^' 



-10 0" 



I SLOPE 2 TO 



_'- --^^^ 



SLOPE Vn TO THE FOOT 






CRUSHED ROCK AND SLAG 




CRUSHED ROCK AND SLAG 




PROVIDE DRAINS WHERE NEEDED 



GRAVEL, CINDERS, CHATS, ETC., 



SELECT COARSE STONE 
FOR END OF DRAIN 




GRAVEL, CINDERS, CHATS, ETC., 
Fig. 110. — Cross-sections of Ballast for Class **A" Roads. 

recommended by that association for Class "A*^ traffic. It 
should be noticed that in each case the cross-section of the 
roadbed from shoulder to shoulder of the roadbed is 20 feet 
plus the space between track centers for double track if any. 



§ 235. BALLAST. 271 

The width of side ditches is merely added to that of the roadbed. 
The clear thickness of the ballast underneath the ties is made 
12 inches, but even this should be considered as the minimum 
depth and is recommended for use only on the firmest, most 
substantial and well-drained subgrades. The slope of |- inch 
to the foot from the center of the track to the end of the tie, 
which is common to all the cross-sections, is designed with the 
idea of allowing a clear space of 1 inch underneath the rail. 
The ballast is then rounded off on a curve of 4 feet radius and 
finally reaches the subsoil on a slope which is 2 : 1 for broken 
stone, and 3:1 for all other materials. The flat slope adopted 
for gravel, etc., which adds considerably to the required width 
of roadbed, has been so designed in order that the considerable 
mass of material at the ends of the ties shall be better able to 
hold the track in place laterally. The sod on the embank- 
ment over the shoulder of the roadbed up to within 12 inches 
of the edge of the ballast is strongly recommended on account 
of the protection it affords to the shoulder of the roadbed. 
It should be noticed that the latest decision of that associa- 
tion regarding the form of subgrade is that the subgrade should 
be made level and not crowned, as suggested and discussed in 
§93. 

In Fig. Ill are shown a series of cross-sections for vanous 
classes of ballast for railroads that belong to Class ^'B." It 
may be noted that the thickness of the ballast under the tie 
is 9 inches for this class. The width of roadbed between the 
shoulders, recommended for Class ''B" is 16 feet. As before, 
the width of the ditches is supposed to be added to this width. 
It should be noted that when using cementing gravel and chert 
the slope of 3 : 1 is made to begin at the bottom of the tie in- 
stead of at a point about 2 inches below the top of the tie. 
This is done in order to prevent water from accumulating 
around the end of the tie in a material which is less permeable 
than the other forms of ballast. 

In Fig. 112 are shown two cross-sections for ballast for roads 
belonging to Class ''C." On roads of this class it is assumed 
that crushed rock will not be used for ballast. The width of 
roadbed between shoulders is 14 feet, while the depth of ballast 
underneath the tie is 6 inches. 

It should be noticed that the above sections issued by the 
associatioa do not include any cross-section which is recom- 



J 



272 



RAILROAD CONSTRUCTION, 



§235. 



mended when no special ballast is used other than the natural 
soil. In such a case a cross-section very similar to the sec- 
tions shown for cementing gravel and chert should be used. The 
essential feature of such a section is that the soil, which is 
probably not readily permeable, should be kept away from 



SLDPEj^'Vo THE FOOT 




GRAVEL, CINDERS, CHATS, ETC. 




CEMENTING. GRAVEL A^ID CHERT. 
Fig. 111. — Cross-sections of Ballast for Class **B** Roads. 



the ends of the ties. Specifications for the placing of mud 
ballast, as well as other forms of ballast, have frequently speci- 
fied that the ballast should be crowned about 1 inch above the 
level of the tops of the ties in the center of the track. This 
feature of any cross-section, although proposed, was rejected 
by the association, in spite of the fact that when a tie is so 
imbedded it certainly will have a somewhat greater holding 
nower in the ballast. 



§236. 



BALLAST. 



273 



236. Proper depth of ballast. The depth of ballast is officially 
defined by the A. R. E. A. as "the distance from the bottom of 
the tie to the top of the subgrade." In the recommended sec- 
tions (Figs. 110 to 112) the depth shown varies from 6 inches 
to 12 inches. But the Ballast Committee reported in 1915 as 
a recommended conclusion that ''From the data available, it 
is concluded that with ties 7 in. by 9 in. by 8^ ft., spaced approx- 
imately 24 in. to 25.5 ins., center to center, a depth of 24 inches 
of stone ballast is necessary to produce uniform pressure on the 
subgradCj^'and a combination of a lower layer of gravel or cinder 

I 



-ro'- 



1 



.SLOPEHTO THE. FOOT 

^ ^SLOPES TO 1 




y 



GRAVEL, CINDERS^ CHATS. ETC. 




CEMENTING GRAVEL AND CHERT. 
Fig. 112. — Cross-sections of Ballast for Class "C* Roads. 



ballast, 18 inches to 14 inches, and an upper layer of stone ballast, 
6 inches to 10 inches, approximately 24 inches deep in the ag- 
gregate, with the same spacing of the ties, will produce nearly 
the same results." New sections for Class ^^A'^ roads which 
would conform with the above were also recommended. These 
were not adopted, but the adoption (substantially) was prob- 
ably only postponed. Future specifications will probably re- 
quire that a sub-ballast of less expensive material shall be laid 
under the ballast which immediately supports the rails for 
all Class ^^A," and perhaps Class "B," roads. As previously 
stated, old track generally has a depth of ballast under the tie 
which is greater than the 2 foet recommended — often 3 or 4 feet. 



274 RAILROAD CONSTRUCTION. § 237. 

237. Methods of laying ballast. The cheapest method of 
laying ballast on new roads is to lay ties and rails directly, on 
the prepared subgrade and run a construction train over the 
track to distribute the ballast. Then the track is lifted up until 
sufficient ballast is worked under the ties and the track is prop- 
erly surfaced. This method, although cheap, is apt to injure 
the rails by causing bends and kinks, due to the passage of 
loaded construction trains when the ties are very unevenly and 
roughly supported, and the method is therefore condemned and 
prohibited in some specifications. The best method is to draw 
in carts (or on a contractor's temporary track) the ballast that 
is required under the level of the bottom of the ties. Spread 
this ballast carefully to the required surface. Then lay the ties 
and rails, which will then have a very fair surface and uniform 
support. A construction train can then be run on the rails and 
distribute sufficient additional ballast to pack around and 
between the ties and make the required cross-section. 

The necessity for constructing some lines at an absolute 
minimum of cost and of opening them for traffic as soon aa 
possible has often led to the policy of starting traffic when 
there is little or no ballast — perhaps nothing more than a mere 
tamping of the natural soil under the ties. When this is done 
ballast may subsequently be drawn where required by the train- 
load on flat cars and unloaded at a minimum of cost by means 
of a "plough.'' The plough has the same width as the cars and 
is guided either by a ridge along the center of each car or by 
short posts set up at the sides of the cars. It is drawn from one 
end of the train to the other by means of a cable. The cable is 
sometimes operated by means of a small hoisting-engine car- 
ried on a car at one end of the train. Sometimes the locomo- 
tive is detached temporarily from the train and is run ahead 
with the cable attached to it. 

238. Cost. The cost of ballast in the track is quite a variable 
item for different roads, since it depends (a) on the first cost of 
the material as it comes to the road, (6) on the distance from 
the source of supply to the place where it is used, and (c) on 
the method of handling. The first cost of cinder or slag is 
frequently insignificant. A gravel-pit may cost nothing except 
the price of a little additional land beyond the usual limits of 
the right of way. Broken stone will usually cost $1 or more 
per cubic yard. If suitable stone is obtainable on the com- 



§ 238. BALLAST. 275 

pany's land, the cost of blasting and breaking should be some- 
what less than this. The cost of hauling will depend on the 
distance hauled, and also, to a considerable extent, on the limi- 
tations on the operation of the train due to the necessity of keep- 
ing out of the way of regular trains. There is often a needless 
waste in this way. The '^mud train*' is considered a pariah and 
entitled to no rights whatever, regardless of the large daily cost 
of such a train and of the necessary gang of men. The cost of 
broken-stone ballast in the track is estimated at $1.25 per cubic 
yard. The cost of gravel ballast is estimated at 60 c. per cubic 
yard in the track. The cost of placing and tamping gravel 
ballast is estimated at 20 c. to 24 c. per cubic yard, for cinders 
12 c. to 15 c. per cubic yard. The cost of loading gravel on 
cars, using a steam-shovel, is estimated at 6 c. to 10 c. per 
cubic yard.* 

• Report Roadmasters' Association. 1885, 



CHAPTER VIII. 

TIES. 
AND OTHER FORMS OF RAIL SUPPORT. 

239, Various methods of supporting rails. It is necessary 

that the rails shall be sufficiently supported and braced, so that 
the gauge shall be kept constant and that the rails shall not be 
subjected to excessive transverse stress. It is also preferable 
that the rail support shall be neither rigid (as if on solid rock) 
nor too yielding, but shall have a uniform elasticity throughout. 
These requirements are more or less fulfilled by the following 
methods. 

(a) Longitudinals. Supporting the rails throughout their 
entire length. This method is very seldom used in this country 
except occasionally on bridges and in terminals when the 
longitudinals are supported on cross-ties. In § 264 will be 
described a system of rails,- used to some extent in Europe, 
having such broad bases that they are self-supporting on the 
ballast and are only connected by tie-rods to maintain the gauge. 

(b) Cast-iron "bowls" or "pots." These are castings resem- 
bling large inverted bowls or pots, having suitable chairs on 
top for holding and supporting the rails, and tied together with 
tie-rods. They will be described more fully later (§ 263). 

(c) Cross- ties of metal or wood. These will be discussed in 
the following sections. 

240. Economics of ties. The true cost of ties depends on the 
relative total cost of maintenance for long periods of time. The 
first cost of the ties delivered to the road is but one item in the 
economics of the question. Cheap ties require frequent renew- 
als, which cost for the labor of each renewal practically the 
same whether the tie is of oak or of hemlock. Cheap ties make 
a poor roadbed which will require more track labor to keep even 
in tolerable condition. The roadbed will require to be disturbed 
so frequently on account of renewals that the ties never get an 
Opportunity to get settled and to form a smooth roadbed for any 
length of time. Irregularity in width, thickness, or length of 
ties is especially detrimental in causing the ballast to act and 
wear unevenly. The life of ties has thus a more or less direct 
influence on the life of the rails, on the wear of rolling stock, and 
on the speed of trains. These last items are not so readily 
reducible to dollars and cents, but when it can be shown that 

276 



!k^ 



§241. 



TIES. 



277 



,the total cost, for a long period of time, of several renewals of 
cheap ties, with all the extra track labor involved, is as great as 
or greater than that of a few renewals of durable ties, then there 
is no question as to the real economy. In the following dis- 
cussions of the merits of untreated ties (either cheap or costly), 
chemically treated ties, or metal ties, the true question is there- 
fore of the ultimate cost of maintaining any particular kind of 
ties for an indefinite period, the cost including the first cost of 
the ties, the labor of placing them and maintaining them to 
surface, and the somewhat uncertain (but not therefore non- 
existent) effect of frequent renewals on repairs of rolling stock, 
on possible speed, etc. 

WOODEN TIES. 

241. Choice of wood. This naturally depends, for any partic- 
ular section of country, on the supply of wood which is most 
readily available. The woods most commonly used, especially 
in this country, are oak and pine, oak being the most durable 
and generally the most expensive. Redwood is used very ex- 
tensively in California and proves to be extremely durable, so 
far as decay is concerned, but it is very soft and is much injured 
by '' rail-cutting." This defect is being partly remedied by the 
use of tie-plates, a^ will be explained later. Cedar, chestnut, 
hemlock, and tamarack are frequently used in this country. In 
tropical countries very durable ties are frequently obtained from 
the hard woods pecuhar to those countries. 



Table XXII. — number and value of cross-ties used on 

STEAM AND STREET RAILWAYS IN UNITED STATES IN 1906. 

(U. S. Dept. Agric. — Forestry Service, No. 124.) 



Kind of wood. 


Number of 
ties. 


Per cent. 


Total value. 


Aver, value 


Oaks 


45,357,874 

18,841,210 

7,248,562 

7,083.442 

6,588,975 

5,104,496 

3,969,605 

2,576,859 

2,058,198 

1,248,629 

554,738 

373,387 

1,828,067 


44.1 
18.3 
7.1 
6.9 
6.4 
5.0 
3.9 
2.5 
2.0 
1.2 
0.5 
0.3 
1.8 


$23,278,052 

9,567,745 

3,010,392 

3.310,116 

2,995,942 

1,862,135 

1,698,027 

889,561 

582,968 

536,172 

210,818 

151,052 

726,144 


$0.51 


Southern pines 

Douglas fir 


.51 

.42 


Cedar 


.47 


Chestnut 


.49 


Cypress 


.36 


Western pine 

Tamarack 


.43 
.35 


Hemlock 


.28 


Redwood 


.43 


Lodgepole pine 

White pine 


.38 
.40 


All others 


.40 






Total 


102,834,042 


100.0 ' $48,819,124 


$0.47 



278 RAILROAD CONSTRUCTION. § 242. 

The limitations of timber supply have somewhat diminished 
the use of oak and increased the use of the softer woods in recent 
years. 

242. Durability. The durabihty of ties depends on the cli- 
mate; the drainage of the ballast; the volume, weight, and 
speed of the traffic ; the curvature, if any ; the use of tie-plates ; 
the time of year of cutting the timber; the age of the timber 
and the degree of its seasoning before placing in the track; the 
nature of the soil in which the timber is grown; and, chiefly, 
on the species of wood employed. The variability in these 
items will account for the discrepancies in the reports on the life 
of various woods used for ties. 

White oak is credited with a life of 5 to 12 years, depending 
principally on the traffic. It is both hard and durable, the 
hardness enabling it to withstand the cutting tendency of the 
rail-flanges, and the durability enabling it to resist decay. Pine 
and redwood resist decay very well, but are so soft that they are 
badly cut by the rail-flanges and do not hold the spikes very 
well, necessitating frequent respiking. Since the spikes must 
be driven within certain very limited areas on the face of each 
tie, it does not require many spike-holes to "spike-kill" the 
tie. On sharp curves, especially with heavy traffic, the wheel- 
flange pressure produces a side pressure on the rail tending to 
overturn it, which tendency is resisted by the spike, aided some- 
times by rail-braces. Whenever the pressure becomes too great 
the spike will yield somewhat and will be slightly withdrawn. 
The resistance is then somewhat less and the spike is soon so 
loose that it must be redriven in a new hole. If this occurs 
very often, the tie may need to be replaced long before any decay 
has set in. When the traffic is very light, the wood very dura- 
ble, and the climate favorable, ties have been known to last 
25 years. 

243. Dimensions. The usual dimensions for the best roads 
(standard gauge) are 8' to 9' long, 6" to 7" thick, and 8'' to 
10'' wide on top and bottom (if they are hewed) or 8'' to 9" 
wide if they are sawed. For cheap roads and light traffic the 
length is shortened sometimes to 7' and the cross-section also 
reduced. On the other hand a very few roads use ties 9' 6" long. 

Two objections are urged against sawed ties: first, that the 
grain is torn by the saw, leaving a woolly surface which induces 
decay; and secondly, that, since timber is not perfectly straight- 



^ 



§ 244. TIES. 279 

grained, some of the fibers are cut obliquely, exposing their ends, 
which are thus Hable to decay. The use of a ^'planer-saw'' ob- 
viates the first difficulty. Chemical treatment of ties obviates 
both of these difficulties. Sawed ties are more convenient to 
handle, are a necessity on bridges and trestles, and it is even 
claimed, although against commonly received opinion, that 
actual trial has demonstrated that they are more durable than 
hewed ties. 

244. Spacing. The spacing is usually 14 to 16 ties to a 30- 
foot rail. This number is sometimes reduced to 12 and even 
10, and on the other hand occasionally increased to 18 or 20 by 
employing narrower ties. There is no economy in reducing the 
number of ties very much, since for any required stiffness of 
track it is more economical to increase the number of supports 
than to increase the weight of the rail. The decreasing cost of 
rails and the increasing cost of ties have materially changed the 
relation between number of ties and weight of rail to produce a 
given stiffness at minimum cost, but many roads have found it 
economical to employ a large number of ties rather than increase 
the weight of the rail. On the other hand there is a practical 
limit to the number that may be employed, on account of the 
necessary space between the ties that is required for proper 
tamping. This width is ordinarily about twice the width of the 
tie. At this rate, with light ties 6'' wide and with 12" clear 
space, there would be 20 ties per 30-foot rail, or 3520 per mile. 
The smaller ties can generally be bought much cheaper (propor- 
tionately) than the larger sizes, and hence the economy. 

Track instructions to foremen generally require that the 
spacing of ties shall not be uniform along the length of any 
rail. Since the joint is generally the weakest part of the rail 
structure, the joint requires more support than the center of the 
rail. Therefore the ties are placed with but 8" or 10" clear 
space between them at the joints, this applying to 3 or 4 ties at 
each joint; the remaining ties, required for each rail length, are 
^equally spaced along the remaining distance. 

245. Specifications. The specifications for ties are apt to 
include the items of size, kind of wood, and method of construc- 
.tion, besides other minor directions about time of cutting, sea- 
:Soning, delivery, quality of timber, etc. 

(a) Size. The particular size or sizes required w^U be some- 
what as indicated in § 243. 



280 RAILROAD CONSTRUCTION. § 246. 

(b) Kind of wood. When the kind or kinds of wood are 
specified^ the most suitable kinds that are available in that 
section of country are usually required. 

(c) Method of construction. It is generally specified that the 
ties shall be hewed on two sides; that the two faces thus made 
shall be parallel planes and that the bark shall be removed. It 
is sometimes required that the ends shall be sawed off square; 
that the timber shall be cut in the winter (when the sap is down) ; 
and that the ties shall be seasoned for six months These last 
specifications are not required or lived up to as much as their 
importance deserves. It is sometimes required that the ties shall 
be delivered on the right of way, neatly piled in rows, the alter- 
nate rows at right angles, piled if possible on ground not lower 
than the rails and at least ten feet away from the nearest rail, 
the lower row of ties resting on two ties which are themselves 
supported so as to be clear of the ground. 

(d) Quality of timl^er. The usual specifications for sound 
timber are required, except that they are not so rigid as for a 
better class of timber work The ties must be sound, reason- 
ably straight-grained, and not very crooked — one test being that 
a line joining the center of one end with the center of the middle 
shall not pass outside of the other end. Splits or shakes, espe- 
cially if severe, should cause rejection. 

Specifications sometimes require that the ties shall be cut 

from small trees, making 
what is known as ^'pole 
ties" and definitely con- 
demning those which are 

POLE TIE. SLAB TIE. QUARTER TIE. i • , r- i 

cut or split from larger 
Fig. 113. — Methods op cutting Ties. , , . . , a i u 

trunks, givmg two 'slab 

ties" or four " quarter ties" for each cross-section, as is illustrated 
in Fig. 113. Even if pole ties are better, their exclusive use 
means the rapid destruction of forests of young trees. 

246. Regulations for laying and renewing ties. The regula- 
tions issued by railroad companies to their track foremen will 
generally include the following, in addition to directions regard- 
ing dimensions, spacing, and specifications given in §§ 242-245. 
When hewn ties of somewhat variable size are used, as is fre- 
quently the case, the largest and best are to be selected for use 
as joint ties. If the upper surface of a tie is found to be warped 
(contrary to the usual specifications) so that one or both rails do 




^ 



§247. TIES. 281 

not get a full bearing across the whole width of the tie, it must 
be adzed to a true surface along its whole length and not mereJs^ 
notched for a rail-seat. When respiking is necessary and spikes 
have been pulled out, the holes should be immediately plugged 
with "wooden spikes," which are supplied to the foreman for 
that express purpose, so as to fill up the holes and prevent the 
decay which would otherwise take place when the hole becomes 
fiUed with rain-water. Ties should always be laid at right angles 
to the rails and never obliquely Minute regulations to prevent 
premature rejection and renewal of ties are frequently made. It 
is generally required that the requisitions for renewals shall be 
made by the actual count of the individual ties to be rencAved 
instead of by any wholesale estimates. It is unwise to have ties 
of widely variable size, hardness, or durability adjacent to each 
other in the track, for the uniform elasticity, so necessary for 
smooth riding, will be unobtainable under those circumstances. 
After a considerable discussion of the two policies of tie 
renewals over long continuous stretches of track or of single tie 
renewals where individually needed, the A. R. E. A. has decided 
in favor of single tie renewals, as being most economical and 
producing least track disturbance. 

247. Dating nails. These are made of iron or steel, galvanized 
with zinc. They should be 2 J inches long, J inch in diameter, 
with f -inch head, which has two figures Ye "^^h high, denoting 
the year, which are stamped, by depression, into the head. 
They should be driven into the upper side of all treated ties, 
10 inches inside the rail, on the line side of the track. The use 
of such dates gives definite knowledge of the life of the tie when 
it is renewed and a means of studying the effectiveness of the tie 
treatment. 

248. Cost of ties. When railroads can obtain ties cut by 
farmers from woodlands in the immediate neighborhood, the 
price will frequently be as low as 35 cents for the smaller sizes, 
running up to 60 cents for the larger sizes and better quahties, 
especially when the timber is not very plentiful. Sometimes if a 
railroad cannot procure suitable ties from its immediate neigh- 
borhood, it will find that adjacent railroads control all adjacent 
sources of supply for their own use and that ties can only be 
procured from a considerable distance, with a considerable 
added cost for transportation. First-class oak ties cost about 80 
to 90 cents and frequently much more. 



282 • RAILROAD CONSTRUCTION. § 249. 



PRESERVATIVE PROCESSES FOR WOODEN TIES. 

249. General principles. Wood has a fibrous cellular struc- 
ture, the cells being filled with sap or air. The woody fiber is 
but little subject to decay unless the sap undergoes fermentation. 
Preservative processes generally aim at removing as much of the 
water and sap as possible and filling up the pores of the wood 
with an antiseptic compound. The most common methods 
all agree in this general process and only differ in the method 
employed to get rid of the sap and in the antiseptic chemical 
with which the fibers are filled. One valuable feature of these 
processes lies in the fact that the softer cheaper woods are more 
readily treated than are the harder woods and from them a tie 
can be made which will be as durable as the best (from the stand- 
point of decay), and, if protected from mechanical wear by tie- 
plates, will have a very long life. The following woods may be 
used without preservative treatment: White oak family, long- 
leaf strict heart yellow pine, cypress, excepting the white cypress, 
redwood, white cedar, chestnut, catalpa, locust, except the 
honey locust, walnut and black cherry. The following woods 
should preferably not be used without preservative treatment: 
Red oak family, beech, elm, maple, gum, loblolly, short-leaf. 
Western yellow pine, Norway, North Carolina pine and other 
sap pines, red fir, spruce, hemlock, and tamarack. It is better 
to use an excess of chemical rather than not enough. Ties 
should be grouped before treatment; for example, green ties 
should not be mixed with seasoned ties, since the treatment 
should be different. Ties should be air-seasoned before being 
treated. When there is time to air-season them at the plant 
before treatment, they should be piled in groups having the same 
degree of seasoning, so that they rest on seasoned stringers, the 
lowest ties at least 6 inches from the ground, which should be 
thoroughly drained and cleared from weeds, high grass and 
decaying matter. The ties should not be allowed to over- 
season or deteriorate. Ties which show signs of checking should 
be secured with S-irons or bolts to prevent further checking. 
When ties are to be adzed or bored for the use of tie-plates or 
screw spikes, the adzing or boring should be done before chemical 
treatment. When it is necessary to treat unseasoned or only 
partially seasoned ties, they should be steamed to remove the 



i^ 



§ 250. TIES. 283 

To do the work, long cylinders, which may be opened at the 
ends, are necessary. Usually the timbers are run in and out on 
iron carriages running on rails fastened to braces on the inside of 
the cylinder. When the load has been run in, the ends of the 
cylinder are fastened on. The water and air in the pores of the 
wood are drawn out by subjecting the wood alternately to steam- 
pressure and to the action of a vacuum-pump. Live steam 
should be admitted so that a pressure of 20 lbs. is produced 
within 30 to 50 minutes. This pressure may be maintained from 
1 to 5 hours, depending on the condition of the wood, but the 
pressure should never exceed 20 lbs. A vent should be provided 
to allow the escape of air and condensed water. After steaming, 
a vacuum of not less than 24 inches of mercury at sea-level (or 
correspondingly less for higher altitudes), shall be produced 
and maintained for half an hour. Then, without breaking the 
vacuum, the chemical shall be admitted. 

250. Creosoting. This process consists in impregnating the 
wood with creosote oil, a product obtained from coal-gas tar 
or coke oven tar which shall be free from any tar, including coal- 
gas tar, oil or residue obtained from petroleum or any other 
source. The pure creosote oil is strongly recommended by the 
A.R. E. A., but they recognize that the practice of using other coal 
tar distillates, when the available supply of creosote is inadequate, 
is firmly established, and have made specifications accordingly. 

It would require about 35 to 50 lbs of creosote to completely 
fill the pores of a cubic foot of wood. But it would be impossible 
to force such an amount into the wood, nor is it necessary or 
desirable. About 10 lbs. per cubic foot, or about 35 lbs. per tie, 
is all that is necessary. For piling placed in salt water about 
18 to 20 lbs. per cubic foot is used, and the timber is then per- 
fectly protected against the ravages of the teredo navalis. 
After one of the vacuum periods, the cylinder is filled with 
creosote oil at a temperature of about 170° F. The pumps 
are kept at work until the pressure is about 80 to 100 lbs. per 
square inch, and is maintained at this pressure from one to two 
hours according to the size of the timber. The oil is then 
withdrawn, the cylinders opened, the train pulled out and an- 
other load made up in 40 to 60 minutes. The average time 
required for treating a load is about 18 or 20 hours, the absorption 
about 10 or 11 lbs. of oil per cubic foot, and the cost (1894) 
from $12.50 to $14.50 per thousand feet B. M. 



284 RAILROAD CONSTRUCTION. § 251. 

251. Buraettizing (chloride-of-zinc process). This process is 
very similar to the creosoting process except that the chemical is 
chloride of zinc. The chemical is heated to 140° F. before using. 
The preliminary treatment of the wood to alternate vacuum and 
pressure is not continued for quite so long a period as in the 
creosoting process. Care must be taken, in using this process, 
that the ties are of as uniform quality as possible, for seasoned 
ties will absorb much more zinc chloride than unseasoned (in the 
same time)^ and the product will lack uniformity unless the sea- 
soning is imiform. The amount of solution injected shall be 
equivalent to § lb. of dry soluble zinc-chloride per cubic foot of 
timber. The solution shall be as weak as can be used and still 
obtain the desired absorption of zinc-chloride, and shall not be 
stronger than 5%. If the cylinders are provided with steam 
coils, steam pressure shall be maintained in these coils during 
treatment. One great objection to burnettized ties is the fact 
that the chemical is somewhat easily washed out, when the wood 
again becomes subject to decay. Another objection, which is 
more forcible with respect to timber subject to great stresses 
as in trestles, than to ties, is the fact that when the solution 
of zinc chloride is made strong (over 3%) the timber is made 
very brittle and its strength is reduced. The reduction in 
strength has been shown by tests to amount to ^ to yV of the 
ultimate strength, and that the elastic limit has been reduced 
by about \. 

252. Kyanizing (bichloride-of -mercury or corrosive-sublimate 
process). This is a process of '' steeping.'^ It requires a much 
longer time than the previously described processes, but does not 
require such an expensive plant. Wooden tanks of sufficient 
size for the timber are all that is necessary. The corrosive subli- 
mate is first made into a concentrated solution of one part of 
chemical to six parts of hot water. When used in the tanks this 
solution is weakened to 1 part in 100 or 150. The wood will 
absorb about 5 to 6.5 pounds of the bichloride per 100 cubic 
feet, or about one pound for each 4 to 6 ties. The timber is 
allowed to soak in the tanks for several days, the general rule 
being about one day for each inch of least thickness and one day 
over — ^which means seven days for 6-inch ties, or thirteen (to 
fifteen) days for 12-inch timber (least dimension). The process is 
somewhat objectionable on account of the chemical being such a 
virulent poison, workmen sometimes being sickened by the fumes 



§253. TIES. 285 

arising from the tanks. On the Baden Railway (Germany) 
kyanized ties last 20 to 30 years. On this railway the wood is 
always air-dried for two weeks after impregnation and before 
being used, which is thought to have an important effect on its 
durability. The solubility of the chemical and the liability of 
the chemical washing out and leaving the wood unprotected is 
an element of weakness in the method. 

253. Zinc-tanning process. The last two methods described 
(as well as some others employing similar chemicals) are open 
to the objection that since the wood is impregnated with an 
aqueous solution, it is liable to be washed out very rapidly if the 
wood is placed under water, and will even disappear, although 
more slowly, under the action of moisture and rain. Several 
processes have been proposed or patented to prevent this. By 
one of these processes the timber is successively subjected to the 
action of chemicals, each individually soluble in water, and hence 
readily impregnating the timber, but the chemicals when brought 
in contact form insoluble compounds which cannot be washed 
out of the wood-cells. After injecting the zinc-chloride, as 
before described, the solution is run off and the ties drained for 
15 minutes. Then a 2% solution of tannic acid, made from 
6| lbs. of 30% extract of tannin and 100 lbs. of water is run 
in and maintained at 100 lbs. pressure for one-half hour. 
Then a solution of glue made by dissolving 2.1 lbs. of glue 
containing 50% gelatine in 100 lbs. of water is run in and 
maintained at 100 lbs. pressure for one-half hour. The glue 
and tannin combine to form an insoluble leathery compound in 
the cells, which will prevent the zinc chloride from being 
washed out. 

254. Zinc-creosote emulsion process. The chemical is an 
emulsion which will leave in the wood an equivalent of 0.4 lb. of 
dry, soluble zinc-chloride and from 1.25 to 1.5 lbs. of creosote per 
cubic foot. The zinc-chloride must not be stronger than 3.5%. 
The emulsion must be effectively mixed in a storage tank and 
heated to at least 140° F. before it enters the cylinder, where the 
pressure is raised to 100 lbs. per square inch and maintained 
there until the required amount of chemical has been absorbed 
by the wood. 

255. Two-injection zinc-creosote process. The zinc-chloride 
and creosote are injected separately. The zinc-chloride must be 
as weak as possible (not more than 5%), and yet strong 



286 RAILROAD CONSTRUCTION. § 256. 

enough so that the equivalent of 0.3 lb. can be injected per 
cubic foot. After impregnation, the remaining zinc-chloride 
is run out and the creosote is forced in and maintained at 100 
lbs. pressure until the wood has absorbed about 3 lbs. of oil per 
cubic foot. 

256. Cost of Treating. The cost of treating ties by the vari- 
ous methods has been estimated as follows* — assuming that 
the plant was of sufficient capacity to do the work economic- 
ally; creosoting, 25 cents per tie; vulcanizing, 25 cents per tie; 
burnettizing (chloride of zinc), 8.25 cents per tie; kyanizing (steep- 
ing in corrosive sublimate), 14.6 cents per tie; WeUhouse process 
(chloride of zinc and tannin), 11.25 c. per tie. These estimates 
are only for the net cost at the works and do not include the 
cost of hauling the ties to and from the works, which may mean 
5 to 10 c. per tie. Some of these processes have been installed 
on cars which are transported over the road and operated where 
most convenient. An estimate made in 1907 by Prof. Gellert 
AUeman on the cost of treating ties, each assumed to have a 
volume of 3 cubic feet, the cost "not including royalty on pa- 
tents, profit, interest, or depreciation, all of which vary widely 
at the various plants," is as follows: 

Zinc chloride ..,.,.... 16 cents 

'* *' and creosote ...27 ** 

Creosote, 10 pounds to the cubic foot ... 55 " 

The very grert, increase in these prices, especially for creosot- 
ing, is due to the enormous increase in late years in the con- 
sumption and in the price of creosote. 

257. Economics of treated ties. The fact that treated ties are 
not universally adopted is due to the argument that the added 
life of the tie is not worth the extra cost. If ties can be bought 
for 25 c, and cost 25 c. for treatment, and the treatment only 
doubles their life, there is apparently but little gained except 
the work of placing the extra tie in the track, which is more 
or less offset by the interest on 25 c. for the life of the untreated 
tie, and the larger initial outlay makes a stronger impression on 
the mind than the computed ultimate economy. But when 
(utilizing some statistics from the Pittsburg, Ft. Wayne & 

* Bull. No. 9, U. S. Dept. of Agric, Div. of Forestry. App. No. 1, by 
Henry Flad. 



§ 257. TIES. 287 

Chicago Railroad) it is found that white oak ties laid in rock 
ballast had a life of 10.17 years, and that hemlock ties treated 
with the zinc-tannin process and laid in the same kind of ballast 
lasted 10.71 years, then the economy is far more apparent. 
Unfortunately no figures were given for the cost of these ties 
nor for the cost of the treatment; but if we assume that the 
white oak ties cost 75 c. and the hemlock ties 35 c. plus 20 c. 
for treatment, there is not only a saving of 20 c. on each tie, 
but also the advantage of the slightly longer life of the treated 
tie. In the above case the total life of the two kinds of ties 
is so nearly the same that we may make an approximation of 
their relative worth by merely comparing the initial cost; but 
usually it is necessary to compare the value of two ties one 
of which may cost more than the other, but will last considerably 
longer. The mathematical comparison of the real value of 
two ties under such conditions may be developed as follows: 
The real cost of a tie, or any other similar item of constructive 
twork, is measured by the cost of perpetually maintaining that 
item in proper condition in the structure. It will be here 
assumed that the annual cost of the trackwork, which is assign- 
able to the tie, is the same for all kinds of ties, although the 
difference probably lies in favor of the more expensive and 
most durable ties. By assuming this expense as constant, the 
remaining expense may be considered as that due to the cost 
of the new ties whenever necessary, plus the cost of placing 
them in the track. We also may combine these two items 
in one, and consider that the cost of placing a tie in the track, 
which we will assume at the constant value of 20 c. per tie, 
regardless of the kind of tie, is merely an item of 20 e. in the 
total cost of the tie. We will assmne that T^ is the present 
cost of a tie, the cost including the preservative treatment if 
any, and the cost of placing in the track. The tie is assumed 
to last n years. At the end of n years another tie is placed 
in the track, and, for lack of more precise knowledge, we will 
assume that this cost T2 equals T^. The "present worth'' 
of T2 is the sum which, placed at compound interest, would 
equal T2 at the end of n years, and is expressed by the quantity 

T 

' nj^ \n ^ ^ which r equals the rate of interest. Similarly at 

the end of 2n years we must expend a sum T^ to put in the third 
^ tie, and the present worth of the cost of that third tie is ex- 



288 RAILROAD COHS1RI5CTION. §257. 

T 
pressed by the fraction ttx^Vo"* ^® i^^y similarly express 

the present worths of the cost of ties for that particular spot 
for an indefinite period. The sum -of all these present worths ♦ 
is given by the sum of a converging series and equals (assuming ' 

that all the T's are equal) . , . --. But instead of laying 

aside a sum of money which will maintain a tie in that par- 
ticular place in perpetuity, we may compute the annual sum 
which must be paid at the end of each year, which would be 
the equivalent. We will call that annual payment Aj and 
then the present worths of all these items are as follows: 

For the first payment ... . » 

For the second payment . *, 

A 
For the third payment , .^ ; 

A 
For the nth payment r— — r-. 

^ (l+r)n 

After the next tie is put in place we have the present worths 
of the annual payments on the second tie, of which the first 
one would be 

For the (n + 1) payment (i^^)(n+i r 

Similarly after x ties have been put in place the last pay- 

A 
ment for the x tie would have a present worth -— . The 

{l-{-r\nx 

sum of all these present worths is represented by the sum of 

• A 

a converging series and equals the very simple expression —• 

But since the sum of the present worths of these annual pay- 
ments must equal the sum of the present worths of the payments 
made at intervals of n years, we may place these two summa- 
tions equal to each other, and say that 

(1+rr-l 



§ 257. TIES. 289 

Values of A for various costs of a tie T on the basis that r 
equals 5% have been computed and placed in Table XVIII. 
To illustrate the use of this table, assume that we are comparing 
the relative values of two ties, both untreated, one of them 
a white oak tie which will cost, say 75 c, and will last twelve 
years, the other a yellow pine tie which will cost, say 35 c, 
and will last six years. Assuming a charge for each case of 
20 c. for placing the tie in the track, we have as the annual 
charge against the white oak tie, which costs 95 c. in the track, 
10.72 c. The pine tie, costing 55 c. in the track and lasting 
six years, will be charged with an annual cost of 10.48 c, which 
shows that the costs are practically equal. It is probably 
true that the track work for maintaining the white oak would 
be less than that for the pine tie, but since the initial cost of 
the pine tie is less than that of the oak tie, it would probably 
be preferred in this case, especially if money was difficult to 
obtain. It may be interesting to note that if a comparison is 
made from a similar table which is computed on the basis of 
compounding the money at 4% instead of 5%, the annual 
charges would be 10.13 and 10.49 c. for the oak and pine ties 
respectively, fhus showing that when money is "easier'* the 
higher priced tie has the greater advantage. 

Example 2. Considering again the comparison previously 
made of a white oak untreated tie which was assumed to cost 
75 c, and a hemlock treated tie, which cost 35 c. for the tie 
and 20 c. for the treatment, the total costs of these ties laid 
in the track would therefore be 95 c. and 75 c. respectively. 
These ties had practically the same life (10.17 and 10.71 years), 
but in order to use the table, we will call it ten years for each 
tie. The annual charge against the oak tie would therefore 
be 12.30 c, while that against the hemlock tie would be 9.72 c. 
This gives an advantage in the use of the treated tie of 2.58 c. 
per year, which capitalized at 5% would have a capitalized 
value of 51.6 c. 

The Atchison, Topeka and Santa Fc R. R. has compiled a 
record of treated pine ties removed in 1897, '98, '99, and 1900, 
showing that the average life of the ties removed had been about 
11 years. On the Chicago, Rock Island and Pacific R. R., the 
average life of a very large number of treated hemlocK and 
tamarack ties was found to be 10.57 years. Of one lot of zl,850 
ties, 12% still remained in the track after 15 years' exposure. 

.It has been demonstrated that much depends oa the minor 



290 RAILROAD CONSTRUCTION. § 258. 

details of the process — ^whatever it may be. As an illustra- 
tion, an examination of a batch of ties, treated by the zinc- 
creosote process, showed 84% in service after 13 years' expo- 
sure; another batch, treated by another contractor by the same 
process (nominally), showed 50% worthless after a service of six 
years. 

METAL TIES. 

258. Extent of use. In 1894 * there were nearly 35000 miles 
of '^ metal track '' in various parts of the world. Of this total, 
there were 3645 miles of ^' longitudinals " (see § 264), found exclu- 
sively in Europe, nearly all of it being in Germany. There 
were over 12000 miles of '' bowls and plates '^ (see § 263), found 
almost entirely in British India and in the Argentine Republic. 
The remainder, over 18000 miles, was laid with metal cross-ties 
of various designs. There were over 8000 miles of metal cross- 
ties in Germany alone, about 1500 miles in the rest of Europe, 
over 6000 miles in British India, nearly 1000 miles in the rest 
of Asia, and about 1500 miles more in various other parts of the 
world. Several railroads in this country have tried various de- 
signs of these ties, but their use has never passed the experi- 
mental stage. These 35000 miles represent about 9% of the 
total railroad mileage of the world — nearly 400000 miles. They 
represent about 17.6% of the total railroad mileage, exclusive of 
the United States and Canada, where they are not used at all, 
except experimentally, t In the four years from 1890 to 1894 the 
use of metal track increased from less than 25000 mile3 to nearly 
35000 miles. This increase was practically equal to the total in- 
crease in railroad mileage during that time, exclusive of the 
increase in the Unite'd States and Canada. This indicates a 
large growth in the percentage of metal track to total mileage, 
and therefore an increased appreciation of the advantages to be 
derived from their use. 

259. Durability. The durability of metal ties is still far 
from being a settled question, due lar/^ely to the fcict that the 
best form for such ties is not yet determined, and that a large 
part of the apparent failures in metal ties hr.vo been evidently 
due to defective design. Those in favor of tlici.i estimate the 
life as from 30 to 50 years. The opponentrj i^Iace it at not more 

* Bulletin No. 9, U. S. Dept. of Agriculture, Div. of Forestry, 
t See § 260 for a later development. 



I 



§259. TIES. 291 

than 20 years, or perhaps as long as the best of wooden ties. 
UnUke the wooden tie, however, which deteriorates as much 
with time as with usage, the Hfe of a metal tie is more largely a 
function of the traffic. The life of a well-designed metal tie has 
been estimated at 150000 to 200000 trains; for 20 trains per 
day, or say 6000 per year, this w;ould mean from 25 to 33 years. 
20 trains per day on a single track is a much larger number than 
will be found on the majority of railroads. Metal ties are found 
to be subject to rust, especially when in damp localities, such as 
tunnels; but on the other hand it is in such confined localities, 
where renewals are troublesome, that it is especially desirable to 
employ the best and longest-lived ties. Paint, tar, etc., have 
been tried as a protection against rust, but the efficacy of such 
protection is as yet uncertain, the conditions preventing any re- 
newal of the protection — such as may be done by repainting a 
bridge, for example. Failures in metal cross-ties have been 
largely due to cracks which begin at a corner of one of the square 
holes which are generally 'punched through the tie, the holes 
being made for the bolts by which the rails are fastened to the 
tie. The holes are generally punched because it is cheaper. 
Reaming the holes after punching is thought to be a safeguard 
against this frequent cause of failure. Another method is to 
round the corners of the square punch with a radius of about 
\". If a crack has already started, the spread of the crack may 
be prevented by drilling a small hole at the end of it. 

260. Form and dimensions of metal cross- ties. Since stability 
in the ballast is an essential quality for a tie, this must be accom- 
plished either by turning do^vn the end of the tie or by having 
some form of lug extending downward from one or more points 
of the tie. The ties are sometimes depressed in the center (see 
Plate VI, N. Y. C. & H. R. R. R. tie) to allow for a thick cover- 
ing of ballast on top in order to increase its stability in the 
ballast. This form requires that the ties should be sufficiently 
well tamped to prevent a tendency to bend out straight, thus 
widening the gauge. Many designs of ties are objectionable 
because they cannot be placed in the track without disturbing 
adjacent ties. The failure of many metal cross-ties, otherwise 
of good design, may be ascribed to too light weight. Those 
weighing much less than 100 pounds have proved too light. 
From 100 to 130 pounds weight is being used satisfactorily on 
German railroads. The general outside dimensions are about 



292 RAILROAD CONSTRUCTION. § 260. 

the same as for wooden ties, except as to thickness. The metal 
is generally from J'' to f " thick. They are; of course, only made 
of wrought iron or steel, cast iron being used only for " bowls " 
or " plates " (see § 263). The details of construction for some 
of the most commonly used ties may be seen by a study of 
Plate VI. 

The Carnegie tie is perhaps the only tie whose use on steam 
railroads in this country has passed the experimental stage. 
The Bessemer and Lake Erie R. R. in 1910 had 188 miles of 
track laid with these ties, and other roads are making extensive 
experiments. One practical difficulty, which is not of course 
insuperable, arises from the common practice of using the rails 
as parts of an electrical circuit for a block-signal system, which 
requires that the rails shall be insulated from each other. This 
requires that these metal ties shall be insulated from the rails. 
A method of insulation which is altogether satisfactory and 
inexpensive is yet to be determined. It is claimed that, on 
account of the better connection between the rail and the tie, 
there is less wear and more uniform wear to the rail. It is 
also claimed that there is greater lateral rigidity in the rails 
and ties (considered as a structure) and that this decreases the 
trackwork necessary to maintain alinement. These ties weigh 
19.7 pounds per linear foot, or about 167 pounds for an 8 foot 
6 inch tie. Even at the lowest possible price per pound the 
cost of the tie and its fastenings must be two or three times 
that of the best oak tie with spikes and even tie plates. It 
has been impossible to estimate the probable life of these ties. 
Until a reasonably close estimate of the life of steel ties can 
be determined, no proper comparison can be made of their 
economy relative to that of wooden ties. A study of Table 
XVIII will show that a tie which costs, say three times as much 
as a cheap tie, must last more than three times as long in order 
that the annual charge against the tie shall be as low as that of 
the cheaper tie. For example, let us assume that the cost of a 
metal tie, laid in the track, is $2.55 and that it will last 20 years. 
From Table XVIII we may find that the annual charge against 
$2.55 at 5% for 20 years = (2 X8.02) +4.41 =20.45 cents. Com- 
pared with a tie costing 65 cents, plus 20 cents for track laying, 
we find that the cheaper tie will only cost 19.63 cents per year 
even if it only lasts 5 years. Of course the claimed advantage 
of better track and less cost for track maintenance, using steel 
ties, will tend to offset, so far as it is true, the disadvantage of 



I 




L1VE5EY BOWL, OaflO 



Plate VI.— Some Forms of Metal Ties. 
iBelween pp. 292 and 293.) 



§ 261. TIES. 293 

the extra cost of the metal tie. Even if the extra work per tie 
amounts to only one-half hour for one man in a year, the cost of it, 
say 6 cents, will utterly change the relative economics of the two 
ties. 

261. Fastenings, The devices for fastening the rails to the 
ties should be such that the gauge may be widened if desired on 
curves, also that the gauge can be made true regardless of slight 
inaccuracies in the manufacture of the ties, and also that shims 
may be placed under the rail if necessary during cold weather 
when the tie is frozen into the ballast and cannot be easily 
disturbed. Some methods of fastening require that the base of 
the rail be placed against a lug which is riveted to the tie or 
which forms a part of it. This has the advantage of reducing 
the number of pieces, but is apt to have one or more of the 
disadvantages named above. Metal keys or wooden wedges are 
sometimes used, but the majority of designs employ some form 
of bolted clamp. The form adopted for the experimental ties 
used by the N. Y. C. & H. R. R. R. (see Plate VI) is especially 
ingenious in the method used to vary the gauge or allow for 
inaccuracies of manufacture. Plate VI shows some of the 
methods of fastening adopted on the principal types of ties. 

262 Cost. The cost of metal cross-ties in Germany averages 
about 1.6 c. per pound or about $1.60 for a 100-lb. tie. The ties 
manufactured for the N. Y. C. & H. R. R. R. in 1892 weighed 
about 100 lbs. and cost $2.50 per tie, but if they had been made 
in larger quantities and with the present price of steel the cost 
would possibly have been much lower. The item of freight 
from the place of manufacture to the place where used is no 
inconsiderable item of cost with some roads. Metal cross-ties 
have been used by some street railroads in this country. Those 
used on the Terre Haute Street Railway weigh 60 pounds and 
cost about 66 c. for the tie, or 74 c. per tie with the fastenings. 



263. Bowls or plates. 'As mentioned before, over 12000 miles 
of railway, chiefly in British India and in the Argentine Repub- 
lic, are laid with this form of track. It consists essentially of 
large cast-iron inverted ''bowls" laid at intervals under each 
rail and opposite each other, the opposite bowls being tied 
together with tie-rods. A suitable chair is riveted or bolted on 
to the top of each bowl so as to properly hold the rail. Being 



294 EAILROAD CONSTRUCTION. § 264. 

made of cast iron, they are not so subject to corrosion as steel 
or wrought iron. They have the advantage that when old and 
worn out their scrap value is from 60% to 80% of their initial 
cost, while the scrap value of a steel or wrought-iron tie is prac- 
tically nothing. Failure generally occurs from breakage, the 
failures from this cause in India being about 0.4% per annum. 
They weigh about 250 lbs. apiece and are therefore quite expen- 
sive in first cost and transportation charges. There are miles 
of them in India which have already lasted 25 years and are 
still in a serviceable condition. Some illustrations of this form 
of tie are shown in Plate VI. 

264. Longitudinals.* This form, the use of which is con- 
fined almost exclusively to Germany, is being gradually replaced 
on many lines by metal cross- ties. The system generally con- 
sists of a compound rail of several parts, the upper bearing rail 
being very light and supported throughout its length by other 
rails, which are suitably tied together with tie-rods so as to 
maintain the proper gauge, and which have a sufficiently broad 
base to be properly supported in the ballast. One great objec- 
tion to this method of construction is the 
difficulty of obtaining proper drainage espe- 
cially on grades, the drainage having a ten- 
dency to follow along the lines of the rails. 
^222^^ The construction is much more complicated 

^ ,, on sharp curves and at frogs and switches. 

Fig. 114. 

Another fundamentally different form of 

longitudinal is . the Haarman compound "self -bearing" rail, 

having a base 12" wide and a height of 8", the alternate sections 

breaking joints so as to form a practically continuous rail. 

Some of the other forms of longitudinals are illustrated in 
Plate VI. 

For a very complete discussion of the subject of metal ties, 
see the "Report on the Substitution - of Metal for Wood in 
Railroad Ties" by E. E. Russell Tratman, it being Bulletin 
No. 4, Forestry Division of the U. S. Dept. of Agriculture. 

♦Although the discussion of longitudinals might be considered to be 
long more properly to the subject of Rails, yet the essential idea of all de- 
signs must necessarily be the support of a rail-head on which the rolling 
stock may run, and therefore this form, unused in this country, will be 
briefly described here. 



§ 265. TIES. 29S 

265. Reinforced Concrete Ties. The wide application of 
reinforced concrete to various structural purposes, combined 
with its freedom from decay, has led to its attempted adop- 
tion for ties. In the annual Proceedings of the American 
Railway Engineering Association for 1907 is a report on over a 
dozen different designs, the most of which were shown to be 
incapable of enduring traffic except on sidings. The ties are 
particularly subject to fracture if struck by a derailed car. A 
similar progress report, made in 1911, again indicated that a 
practicable concrete tie for general use has not yet been invented. 

The annual report for 1915 again contained a review of all such 
ties then in service. In no case was there any considerable 
stretch of track laid with concrete ties — merely a few used experi- 
mentally in scattered places. The reports are full of instances 
of ties being fractured by a derailment after short service. 



CHAPTER IX. 
RAILS. 

266. Early forms. The first rails ever laid wei:e wooden 
stringers which were used on very short tram-roads around coal- 
mines. As the necessity for a more durable rail increased, 
owing chiefly to the invention of the locomotive as a motive 
power, there were invented successively the cast-iron "fish- 
belly'' rail and various forms of wrought-iron strap rails which 
finally developed into the T rail used in this country and the 
double-headed rail, supported by chairs, used so extensively in 
England. The cast-iron rails were cast in lengths of about 3 
feet and were supported in iron chairs which were sometimes 
set upon stone piers. A great deal of the first railroad track 
of this country was laid with longitudinal stringers of v/ood 
placed upon cross-ties, the inner edge of the stringers being 
protected by wrought- iron straps. The "bridge" rails were 
first rolled in this country in 1844. The "pear" section was 
an approach to the present form, but was very defective on 
account of the difficulty of designing a good form, of joint. The 
"Stevens" section was designed in 1830 by Col. Robert L. 
Stevens, Chief Engineer of the Camden and Amboy Railroad; 
although quite defective in its proportions, according to the 
present knowledge of the requirements, it is essentially the pres- 
ent form. In 1836, Charles Vignoles invented essentially the 
same form in England; this form is therefore known throughout 
England and Europe as the Vignoles rail. 

267. Present standard forms. The larger part of modern 
railroad track is laid with rails v/hich are either '^T" rails or 
the double-headed or " bull-headed " railc which are carried in 
chairs. The double-headed rail was designed with r. symmetri- 
cal form with the idea that after one head had been worn out 
b}^ traffic the rail could be reversed, and that its life would be 
practically doubled. Experience has showm that the wear of the 

296 



§267. 



RAILS. 



297 



rail in the chairs is very great; so much so that when one head 
has been worn out by traflSc the whole rail is generally useless. 




BALT. & OHIO R. R. 
QUINCYR.R. 1843. "BULL-HEAD.' 



1826. 





VIGNOLES. 1836. 




CAMDEN & AMBOY. 8TEPHENJS01<. "PEAR." 

1832. 1838. 




"FISH-BELLY"— CAST IRON. 



] 



CAST IRON. 




REYNOJ-DS— 176^. 

Fia. 115. — Early Forms of Rails. 

If the rail is turned over, the worn places, caused by the chairs, 
make a rough track and the rail appears to be more brittle and 
subject to fracture, possibly due to the crystallization that may 
have occurred during the previous usage and to the reversal of 
stresses in the fibers. Whatever the explanation, experience has 
demonstrated the ]act. The ^^bull-headed" 
rail has the lower head only large enough to 
properly hold the wooden k^ys with which 
the rail is secured to the chairs (see Fig. 116) 
and furnish the necessary strength. The use 
^HEADED rIi^In^^" ^^ ^"^^^^ ^^^^^ rcquircs the use of two cast- 
Chair. iron chairs for each tie. It is claimed that 

such track is better for heavy and fast traffic, but it is more 




298 



RAILROAD CONSTRUCIION. 



§267. 



expensive to build and maintain. It is the standard form of 
track in England and some parts of Europe. 

Until after 1893 there was a very great multiplicity in the 
designs of " T " rails as used in this country, nearly every 
prominent railroad having its own special design, which perhaps 
differed from that of some other road by only a very minute and 
insignificant detail, but which nevertheless would require a 
complete new set of rolls for rolling. This had a very appreciable 
effect on the cost of rails. In 1893, the American Society of 
Civil Engineers, after a very exhaustive investigation of the 




Fig. 117. — Standard Rail Sections. 



subject, extending over several years, having obtained the opin- 
ions of the best experts of the country, adopted a series of sec- 
tions which have been very extensively adopted by the railroads 
of this country. 

In 1909 the American Railway Association and the American 
Railway Engineering Association, by combined action, developed 
a series of sections. Fig. 117 shows diagrammatically all of 
these sections and their variations with different weights and 
systems are shown by the tabular values for the lettered dimen- 
sions. It may be noted that the radii of the upper and lower 
corners of the flanges and of the lower corners of the head are 
constant ire") for all weights of rail and for all systems. 



§267. 



RAILS. 



299 






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American Railway 
Association 

and 


American Railway 

Engineering 

Association 





300 



EAILKOAD CONSTRUCTION. 



§268. 




Fig, 118. — Relation 
OF Rail to Wheel 

TREAD. 



The chief features of disagreement among railroad men relate 
to the radius of the upper corner of the head and the slope of the 
side of the head. The radius (j^") adopted by the A. S. C. E. 
for the upper corner (constant for all weights) is a little more 
than is advocated by those in favor of " sharp corners " who 
prefer a radius of J". On the other hand it is much less than 
is advocated by those who consider that it 
should be nearly equal to (or even greater 
than) the larger radius universally adopted 
for the corner of the wheel-flange. The 
discussion turns on the relative rapidity of 
rail wear and the wear of the wheel-flanges 
as affected by the relation of the form of the 
wheel-tread to that of the rail. It is argued 
that sharp rail corners wear the wheel- 
flanges so as to produce sharp flanges, 
which are liable to cause derailment at 
switches and also to require that the tires of 
engine-drivers must be more frequently 
turned down to their true form. On the 
other hand it is generally believed that rail wear is much less 
rapid when the area of contact between the rail and wheel- 
flange is small, and that when the rail has worn down, as it inva- 
riably does, to nearly the same form as the wheel-flange, the rail 
wears away very quickly. The A. R. E. A. system uses f " radius 
^br all rail weights. The ^' B '' sections were proposed to satisfy 
^hose that desired that the head should be narrower and deeper 
than as found in the ^^ A '' sections. The A. R. E. A. Manual 
(1915), suggests that if a section is found to be inadequate because 
of lack of depth of head, the next heavier section will be found 
more desirable and economical. 

268. Weight for various kinds of traffic. The heaviest rails 
in regular use weigh 120 lbs. per yard, and even these are only 
used on some of the heaviest traffic sections of such roads as the 
N. Y. Central, the Pennsylvania, the N. Y., N. H. & H., and 
a few others. Probably the larger part of the mileage of the 
country is laid with 70- to 80-lb. rails — considering the fact that 
'' the larger part of the mileage " consists of comparatively light- 
traffic roads and may exclude all the heavy trunk lines. Very 
light-traffic roads are sometimes laid with 56-lb. rails. Roads 
with fairly heavy traffic generally use 85- to 95-ib. rails, espe- 



§268. RAILS. 301 

cially when grades are heavy and there Is much and sharp curva- 
ture. The tendency on all roads is toward an increase in the 
weight, rendered necessary on account of the increase in the 
weight and capacity of rolling stock, and due also to the fact that 
the price of rails has been so reduced that it is both better and 
cheaper to obtain a more solid and durable track by increasing 
the weight of the rail rather than by attempting to support a 
weak rail by an excessive number of ties or by excessive track 
labor in tamping. It should be remembered that in buying rails 
the mere weight is, in one sense, of no importance. The im- 
portant thing to consider is the strength and the stiffness. If 
we assume that all weights of rails have similar cross-sections 
(which is nearly although not exactly true), then, since for beams 
of similar cross-sections the strength varies as the cube of the 
homologous dimensions and the stiffness as the fourth power^ 
while the area (and therefore the weight per unit of length) 
only varies as the square, it follows that the stiffness varies as 
the square of the weight, and the strength as the f power of the 
weight. Since for ordinary variations of weight the price per 
ton is the same, adding (say) 10% to the weight (and cost) adds 
21% to the stiffness and over 15% to the strength. As another 
illustration, using an 80-lb. rail instead of a 75-lb. rail adds only 
6|% to the cost, but adds about 14% to the stiffness and nearly 
11% to the strength. This shows why heavier rails are more 
economical and are being adopted even when they are not abso- 
lutely needed on account of heavier rolling stock. The stiffness, 
strength, and consequent diurability are increased in a much 
greater ratio than the cost. 

The relation between weight of rail and the weight on the 
drivers of the locomotives which are to run on it has been briefly 
expressed by the Baldwin Locomotive Works as ^' 300 pounds of 
wheel per pound of rail per yard.'' This rule may be utilized 
by making a diagram as shown in Fig. 119. For example, if it is 
desired to use a type of locomotive with 170,000 lbs. on the 
drivers and also 75-lb. rails, four pairs of drivers will be needed 
and such a type of locomotive should be used. By using 95-lb. 
rails the same weight on the drivers could be placed on three axles. 
As another example, a Pacific-type locomotive, with 150,000 lbs. 
on its six drivers, should have a rail with a minimum weight of 83 
lbs., or say an 85-lb. rail. Whatever elements are given, the cor- 
responding proper value for the other element may be derived. 



302 



RAILROAD CONSTRUCTION. 



§269. 



269. Effect of stiffness on traction. A very important but 
generally unconsidered feature of a stiff rail is its effect on trac- 
tive force. An extreme illustration of this principle is seen 
when a vehicle is drawn over a soft sandy road. The constant 
compression of the sand in front of the wheel has virtually the 
same effect on traction as drawing the wheel up a grade whose 



500,000 



450.000 



400,000 



350,000 



2 300,000 
^ 250,000 



150,000 



100,000 



60,000 









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60 



60 70 80 90 

Weight of Rail -pounds per yard 



100 



Fig. 110 — Curves for Finding the Number of Drivers Needed for 
Given Weight on Driving Wheels and Weight of Rails. 

steepness depends on the radius of the Avheel and the depth of 
the rut. On the other hand, if a wheel, made of perfectly 
elastic material, is rolled over a surface which, while supported 
with absolute rigidity, is also perfectly elastic, there would be a 
forward component, caused by the expanding of the compressed 
metal just behind the center of contact, w^hich would just bal- 
ance the backward component. If the rail was supported 
throughout its length by an absolutely rigid support, the high 
elasticity of the wheel-tires and rails would reduce this form of 



i 



§270. RAILS. 303 

resistance to an insignificant quantity, but the ballast and even 
the ties are comparatively inelastic. When a weak rail yields, 
the ballast is more or less compressed or displaced, and even 
though the elasticity of the rail brings it back to nearly its 
former place, the work done in compressing an inelastic material 
is wholly lost. The effect of this on the fuel account is certainly 
very considerable and yet is frequently entirely overlooked. It 
is practically impossible to compute the saving in tractive power, 
and therefore in cost of fuel, resulting from a given increase in 
the w^eight and stiffness of the rail, since the yielding of the rail 
is so dependent on the spacing of the ties, the tamping, etc. But 
it is not difficult to perceive in a general way that such an econ- 
omy is possible and that it should not be neglected in considering 
the value of stiffness in rails. 

270. Length of rails. The recommended standard minimum 
length of rails is 33 feet. In recent years many roads have been 
trying 45-foot and even 60-foot rails. The argument in favor of 
longer rails is chiefly that of the reduction in track-joints, which 
are costly to construct and to maintain and are a fruitful source 
of accidents. Mr. Morrison of the Lehigh Valley R. R.* declares 
that, as a result of extensive experience with 45-foot rails on 
that road, he finds that they are much less expensive to handle, 
and that, being so long, they can be laid around sharp curves 
without being curved in a machine, as is necessary with the 
shorter rails. The great objection to longer rails lies in the 
difficulty in allowing for the expansion, which will require, in 
the coldest weather, an opening at the joint of nearly f for a 
60-foot rail. The Pennsylvania R. R. and the Norfolk and 
Western R. R. each have a considerable mileage laid with 60-foot 
rails. 

271. Expansion of rails. Steel expands at the rate of .0000065 
of its length per degree Fahrenheit. The extreme range of tem- 
perature to which any rail will be subjected will be about 160°, 
or say from -20° F. to +140° F. With the above coefficient 
and a rail length of 60 feet the expansion would be 0.0624 foot, 
or about J inch. But it is doubtful whether there would ever 
be such a range of motion even if there were such a range of 
temperature. Mr. A. Torrey, chief engineer of the Mich. Cent. 
R. R., experimented wdth a section over 500 feet long, which, 

* Report, R )admasters Association, 1895. 



304 EAILROAD CONSTRUCTION. §272. 

although not a single rail, was made "continuous'^ by rigid 
splicing, and he found that there was no appreciable additional 
contraction of the rail at any temperature below +20° F. The 
reason is not clear, but the fact is undeniable. 

The heavy girder rails, used by the street railroads of the 
country, are bonded together with perfectly tight rigid joints 
which do not permit expansion. If the rails are laid at a tem- 
perature of 60° F. and the temperature sinks to 0°, the rails 
haA^e a tendency to contract .00039 of their length. If this 
tendency is resisted by the friction of the pavement in which the 
rails are buried, it only results in a tension amounting to .00039 
of the modulus of elasticity, or say 10920 pounds per square 
inch, assuming 28 000000 as the modulus of elasticity. This 
stress is not dangerous and may be permitted. If the tempera- 
ture rises to 120° F., a tendency to expansion and buckling will 
take place, which will be resisted as before by the pavement, 
and a compression of 10920 pounds per square inch will be in- 
duced, which will likewise be harmless. The range of tempera- 
ture of rails which are buried in pavement is much less than 
when they are entirely above the ground and will probably 
never reach the above extremes. Rails supported on ties which 
are only held in place by ballast must be allowed to expand and 
contract almost freely, as the ballast cannot be depended on to 
resist the distortion induced by any considerable range of tem- 
perature, especially on curves. 

272. Rules for allowing for temperature. Track regulations 
generally require that the track foremen shall use iron (not 
wooden) shims for placing between the ends of the rails while 
splicing them. The thickness of these shims should vary with 
the temperature. Some roads use such approximate rules as the 
following : " The proper thickness for coldest weather is -f-Q of an 
inch; during spring and fall use \ of an inch, and in the very 
hottest weather re of an inch should be allowed.'' This is on 
the basis of a 30-foot rail. When a more accurate adjustment 
than this is desired, it may be done by assuming some very high 
temperature (100° to 125° F.) as a maximum, when the joints 
should be tight; then compute in tabular form the spacing for 
each temperature, varying by 25°, allowing 0".0643 (very 
nearly re'O for each 25° change. Such a tabular form would 
be about as follows (rail length 33 feet): 



§273, 



RAILS. 



305 



Temperature . . 


Over 100° 


100°-75° 

_ 


75°-50° 


50°-25° 


35°-0° 


Below 0° 


Rail opening . . . 


Close 


fe'' 


¥' 


1^" 


¥' 


^"" 



One practical difficulty in the way of great refinement in this 
work is the determination of the real temperature of the rail 
when it is laid. A rail lying in the hot sun has a very much 
higher temperature than the air. The temperature of the rail 
cannot be obtained even by exposing a thermometer directly to 
the sun, although such a result might be the best that is easily 
obtainable. On a cloudy or rainy day the rail has practically 
the same temperature as the air; therefore on such days there 
need be no such trouble. 

273. Chemical composition. About 98 to 99.5% of the com- 
position of steel rails is iron, but the value of the rail, as a rail, 
is almost wholly dependent upon the large number of other 
chemical elements which are, or may be, present in very small 
amounts. The manager of a steel-rail mill once declared that 
their aim was to produce rails ha\dng in them — 

Carbon 0.32 to 0.40% 

Silicon 0.04 to 0.06% 

Phosphorus 0.09 to 0.105% 

Manganese 1.00 to 1.50% 

The analysis of 32 specimens of rails on the Chic, Mil. & St 
Paul R. R. showed variations as follows: 

Carbon 0.211 to 0.52% 

Silicon. , , 0.013 to 0.256% 

Phosphorus : 0.055 to 0. 181% 

Manganese... 0.35 to 1.63% 



These quantities have the same general relative proportions 
as the rail-mill standard given above, the differences lying 
mainly in the broadening of the limits. Increasing the per- 
centage of carbon by even a few hundredths of one per cent 
makes the rail harder, but likewise more brittle. If a track is 
well ballasted and not subject to heaving by frost, a harder and 
more brittle rail may be used without excessive danger of break- 
age, and such a rail will wear much longer than a softer tougher 



306 



KAILROAD CONSTRUCTION. 



§274. 



rail, although the softer tougher rail may be the better rail for 
a road having a less perfect roadbed. 

A small but objectionable percentage of sulphur is some- 
times found in rails, and very delicate analysis will often show 
the presence, in very minute quantities, of several other chem- 
ical elements. The use of a very small quantity of nickel or 
aluminum has often been suggested as a means of producing 
a more durable rail. The added cost and the uncertainty of 
the amount of advantage to be gained has hitherto prevented 
the practical use or manufacture of such rails. 

274. Proposed standard specifications for steel rails. The 
following specifications for steel rails are those proposed by a 
committee of the American Railway Engineering Association in 
March, 1910: 

PROCESS OF MANUFACTURE. 

1. The entire process of manufacture shall be in accordance 
with the best current state of the art. 

(a) Ingots shall be kept in a vertical position until ready to 
be rolled, oruntil the metal in the interior has had time to solidify. 
(6) Bled ingojs shall not be used. 

CHEMICAL COMPOSITION. 

2. The chemical composition of the steel from which the rails 
are rolled shall be within the following limits: 



Carbon 

Manganese 

Silicon 

Phosphorus not to ex- 
ceed 

Sulphur not to exceed . 



Bessemer. 



80 lbs. and 
under. 



0.40 to 0.50 
0.80 to 1.10 
0.10 to 0.20 

0.10 
0.075 



85 to 100 lb£ 
inclusive. 



0.45 to 0.55 
0.85 to 1.15 
0.10 to 0.20 

■ 0.10 
0:075 



Open-hearth. 



80 lbs. and 
under. 



0.53 to 0.66 
0.75 to 1.00 
0.10 to 0.20 

0.04 
0.06 



85 to 100 lbs. 
inclusive. 



0.63 to 0.76 
0.75 to 1.00 
0.10 to 0.20 

0.04 
0.06 



3. When lower phosphorus can be secured in Bessemer or 
open-hearth steel, the carbon shall be increased at the rate of 
0.035 for each 0.01 reduction in phosphorus. 

The percentages of carbon, manganese, and siHcon in an entire 
order of rails shall average as high as the mean percentages be- 
tween the upper and lower limits. 



§ 274. RAILS. 307 



SHEARING. 

4. There shall be sheared from the end of the bloom formed 
from the top of the ingot, sufficient discard to insure sound rails. 
All metal from the top of the ingot, whether cut from the bloom 
or the rail, is the top discard. 

SHRINKAGE. 

5. The number and passes and speed of train shall be so regu- 
lated that, on leaving the rolls at the final pass, the temperature 
of the rails will not exceed that which requires a shrinkage allow- 
ance at the hot saws, for a 33-ft. rail of 100 lbs. section of 6J ins., 
and J in. less for each 10 lbs. decrease of section, these allow- 
ances to be decreased at the rate of x^o iii- for each second, 
of time elapsed between the rail leaving the finishing rolls and 
being sawed. The bars shall not be held for the purpose of 
reducing their temperature, nor shall any artificial means of 
cooling them be used between the leading and finishing passes, 
nor after they leave the finishing pass. 

SECTION. 

6. The section of rail shall conform as accurately as possible 
to the templet furnished by the railroad company. A variation 
in height of ^ in. less or ^ in. greater than the specified height, 
and ^ in. in width of flange, will be permitted; but no variations 
shall be allowed in the dimensions affecting the fit of spHce bars. 

WEIGHT. 

7. The weight of the rail shall be maintained as nearly as pos- 
sible, after complying with the preceding paragraph, to that 
specified in the contract. 

A variation of one-half of one per cent from the calculated 
weight of section, as appHed to an entire order, will be 
allowed. 

Rails will be accepted and paid for according to actual weight. 

LENGTH. 

8. The standard length of rail shall be 33 ft. Ten per 
cent of the entire order will be accepted in shorter lengths varying 



308 RAILROAD CONSTRUCTION. § 274. 

as follows: 30 ft., 28 ft., and 26 ft. A variation of I in. from the 
specified length will be allowed. 

All No. 1 rails less than 33 ft. shall be painted green on both 
ends. 

STRAIGHTENING. 

9. Care shall be taken in hot-straightening rails, and it shall 
result in their being left in such condition that they will not 
vary throughout their entire length more than four (4) ins. from 
a straight line in any direction when dehvered to the cold- 
straightening presses. Those which vary beyond that amount, 
or have short kinks, shall be classed as second quality rails and 
be so marked. The distance between supports of rails in the 
straightening press shall not be less than forty-two (42) ins.; 
supports to have flat surfaces and out of wind. Rails shall be 
straight in line and surface and smooth on head when finished, 
final straightening being done while cold. They shall be sawed 
square at ends, variations to be not more than yj in., and prior 
to shipment shall have the burr caused by the saw cutting 
removed and the ends made clean. 

DRILLING. 

10. Circular holes for joint bolts shall be drilled in accordance 
with specifications of the purchaser. They shall in every respect 
conform accurately to drawing and dimensions furnished and 
shall be free from burrs. 

BRANDING. 

11. The name of the maker, the weight of the rail, and the 
month and year of manufacture shall be rolled in raised letters 
and figures on the side of the web. The number of the heat and 
a letter indicating the portion of the ingot from which the rail 
was made shall be plainly stamped on the web of each rail, where 
it will not be covered by the splice bars. Rails to be lettered 
consecutively A, B, C, etc., the rail from the top of the ingot 
being A. In case of a top discard of twenty or more per cent 
the letter A will be omitted. Open-hearth rails to be branded 

DROP TESTS. 

12. Drop tests shall be made on pieces of rail rolled from the 
top of the ingot, not less than four (4) ft. and not more than six 
(6) ft. long, from each heat of steel. These test pieces shaU be 



§274. EAiLS. 309 

cut from the rail bar next to either end of the top rail, as selected 
by the inspector. 

The temperature of the test piece shall be between forty (40) 
and one hundred (100) degrees Fahrenheit. 

The test pieces shall be placed head upward on soHd supports, 
five (5) ins. top radius, three (3) ft. between centers, and. sub- 
jected to impact tests, the tup falling free from the following 
heights: 

60- and 70-lb. rail 16 ft. 

80-, 85-, and 90-lb. rail 18 ft. 

100-lb. rail 20 ft. 

The test pieces which do not break under the first drop shall 
be nicked and tested to destruction. ' 

DEFLECTION. 

13. It is proposed to prescribe, under this head, the require- 
ments in regard to deflection, fixing maximum and minimum 
limits, as soon as proper deflection limits have been decided on. 

(a) Two pieces shall be tested from each heat of steel. If 
either of these test pieces breaks, a third piece shall be tested. 
If two of the test pieces break without showing ph3^sical defect, 
all rails of the heat will be rejected absolutely. If two of the 
test pieces do not break, all rails of the heat will be accepted 
as No. 1 or No. 2 classification, according as the deflection is 
less or more, respectively, than the prescribed Hmit.* 

(6) If, however, any test piece broken under test ^^A" shows 
physical defect, the top rail from each ingot of that heat shall be 
rejected. 

(c) Additional tests shall then be made of test pieces selected 
by the inspector from the top end of any second rails of the same 
heat. If two of out three of these second test pieces break, the 
remainder of the rails of the heat will also be rejected. If two out 
of three of these second test pieces do not break, the remainder 
of the rails of the heat will be accepted, provided they conform 
to the other requirements of these specifications, as No. 1 or 
No. 2 classification, according as the deflection is less or more, 
respectively, than the prescribed limit.* 

(d) If any test piece, test ^ ^A," does not break, but when nicked 

* This clause to be added when the deflection limits are specified. 



310 RAILROAD CONSTRUCTION. § 274. 

and tested to destruction shows interior defect, the top rails from 
each ingot of that heat shall be rejected. 

DROP-TESTING MACHINE. 

14. The drop-testing machine shall be the standard of the 
American Railway Engineering and Maintenance of Way Asso- 
ciation, and have a tup of 2000 lbs. weight, the striking face 
of which shall have a radius of five (5) ins. 

The anvil block shall be adequately supported and shall weigh 
20 000 lbs. 

The supports shall be a part of or firmly secured to the anvil. 

NO. 1 RAILS. 

15. No. l^rails shall be free from injurious defects and flaws 
of all kinds. 

NO. 2 RAILS, 

16. Rails which, by reason of surface imperfections, are not 
accepted as No. 1 rails, will be classed as No. 2 rails, but rails 
which in the judgment of the inspector contain physical defects 
which impair their strength, shall be rejected. 

No. 2 rails to the extent of five (5) per cent of the whole order 
will be received. All rails accepted as No. 2 rails must have the 
ends painted white, and shall have two prick punch marks on 
the side of the web near the heat number near the end of the 
rail, so placed as not to be covered by the splice bars. 

Rails improperly drilled or straightened, or from which the 
burrs have not been properly removed, shall be rejected, but 
may be accepted after being properly finished. 

All classes of rails must be kept separate from each other and 
shipped in separate cars. 

All rails must be loaded in the presence of the inspector. 

INSPECTION. 

17. (a) Inspectors representing the purchaser shall have free 
entry to the works of the manufacturer at all times while the 
contract is being executed, and shall have all reasonable facilities 
afforded them by the manufacturer to satisfy them that the rails 
have been made in accordance with the terms of the specifica- 
tions* 

(6) For Bessemer steel the manufacturer shall, before the rails 



§275. 



KAILS. 



311 



are shipped, furnish the inspector daily with carbon determina- 
tions for each heat, and two complete chemical analyses every 
twenty-fom* hours representing the average of the other elements 
contained in the steel, for each day and night turn. These analy- 
ses shall be made on drilhngs taken from small test ingots. 
The drillings for analyses shall be taken from the ladle test 
ingot at a distance of J in. beneath the surface. 

For open-hearth steel, the makers shall furnish the inspectors 
with the complete chemical analysis for each melt. 

(c) On request of the inspector, the manufacturer shall furnish 
a portion of the test ingot for check analysis. 

(d) All tests and inspections shall be made at the place of 
manufacture, prior to shipment, and shall be so conducted as not 
to unnecessarily interfere with the operation of the mill. 

(e) Rails to be accepted must meet all of the requirements 
of the specifications. 



275. Rail wear on tangents. When the wheel loads on a rail 
are abnormally heavy, and particularly when the rail has but 
httle carbon and is unusually soft, the concentrated pressure 
on the rail is frequently greater than the elastic limit, and the 
metal " flows " so that the head, although greatly abraded, will 
spread somewhat outside of its original Hues, as shown in Fig. 120. 
The rail wear that occurs oa tangents is almost exclusively on top. 





Fig. 120. 



Fig. 121. 



276. Rail wear on curves. On curves the maximum rail wear 
occurs on the inner side of the head of the outer rail, giving a 
worn form somewhat as shown in Fig. 121. The dotted Hne 
shows the naturfe and progress of the rail wear on the inner rail 
of a curve. Since the pressure on the outer rail is somewhat 
lateral rather than vertical, the ^' flow " does not take place to 
the same extent, if at all, on the outside, and whatever flow would 
take place on the inside is immediately worn off by the wheel- 



312 RAILROAD CONSTRUCTION. § 277. 

flange. Unlike the wear on tangents, the wear on curves is at a 
greater rate as the rail becomes more worn. 

The inside rail on curves wears chiefly on top, the same as 
on a tangent, except that the wear is much greater owing to the 
longitudinal slipping of the wheels on the rail, and the lateral 
slipping that must occur when a rigid four-wheeled truck is 
guided around a curve. The outside rail is subjected to a 
greater or less proportion of the longitudinal slipping, likewise 
to the lateral slipping, and, worst of all, to the grinding action 
of the flange of the wheel, which grinds off the side of the head. 

277. Experimental determination of rail wear. Several years 
ago a series of tests for rail wear were made on the Northern 
Pacific R. R. by taking up, weighing, and replacing, each year, 
the several groups of rails under test. Some of these rails were 
on tangents, the others on curves of various curvature. Some 
of the rails of each group were made of Bessemer steel, the others 
of open-hearth steel. No tests T^ere made to determine the loss 
of weight through mere oxidation. All of the rails were in 
service for five years and some lasted for six years or more, but 
the loss in weight during the sixth year was nearly always equal 
to, and in some cases twice as much as, the loss during the pre- 
ceding five years. Some of the rails lost over 10% of their 
weight, or about one-fourth the weight of the head, before being 
removed. Although the tests were too few to establish any 
positive laws, some tendencies which may be observed will give 
at least an approximate idea of the laws of rail wear. 

1. The average loss of weight during the first five years on 
20 rails on tangents was 0.412 lb. per yard per 10,000,000 tons 
of traffic. 

2. Ten of these same rails were kept in place at least one year 
longer and during the sixth year lost almost twice as much metal 
as during the previous five years; in other words, about two- 
thirds of the entire loss occurred during the sixth year. 

3. The average loss of weight during the first five years from 
20 rails on a tangent was 0.463 lb. per yard per 10,000 trains. 
The relation between mere tonnage and number of trains could 
not be even indicated by so few tests. There is reason to believe 
that engine drivers are more responsible for rail wear than mere 
car-wheel tonnage. This practically means that one effect of 
grade is to increase rail wear, since more (or heavier) engines 
are needed to haul a given car tonnage. 



§ 278. RAILS. 313 

4. The wear of the outer rail of curves is, of course, far greater 
than that of the inner rail, but the figures obtained did not seem 
to follow any rational law, the ratio of outer to inner rail wear 
varying from 144 to 244%, with an average of 182%. 

5. The average rail wear on curves, averaging inner and outer 
rails, per yard, per degree of curve, per 10,000,000 tons traffic, 
varied from 0.145 lb. for a 4° .04' curve down to 0.102 lb. per 
degree for a 10° 13' curve. Based on the four curves tested, the 
results seemed to point to the law that rail wear on curves does 
not increase as fast as the degree of the curve. 

6. Although the tests were too few to establish any law, the 
increase of the mean rail wear on curves with increase in degree 
of curve was very regular and indicated that the average rail 
wear on a curve of about 6° 40' is about twice as great as that on 
a tangent. 

7. The wear on open-hearth rails was almost invariably less 
than that on Bessemer rails, under identical conditions. 

2'/8. Cost of rails. In 1873 the cost of steel rails was about 
$120 per ton, and the cost of iron rails about $70 per ton. 
Although the steel rails were at once recognized as superior to 
iron rails on accoimt of more uniform wear, they were an expen- 
sive luxury. The manufacture of steel rails by the Bessemer 
process created a revolution in prices, and they steadily dropped 
in price until, many years ago, steel rails were manufactured 
and sold for $22 per ton. For several years since then the price 
was very imiform at $28 per ton at the mill. But now (1916) 
the advantages of open-hearth steel are better appreciated and 
a large proportion of rails are being rolled from open-hearth 
steel, which commands about $2 per ton more. At present 
(1916) the current prices at Pittsburgh mills run at about $33 
per ton for Bessemer and $35 for open-hearth. 

At such prices there is no longer any demand for iron rails, 
since the cost of manufacturing them is substantially the same 
as that of steel rails, while their durabihty is unquestionably 
inferior to that of steel rails. Rail quotations are generally on 
the basis of '' long tons " of 2240 lbs. 

The freight charge for transporting rails from the mill to the 
place where used is usually so large that it adds a very appreciable 
amount to the cost per ton. A^ an approximation, the freight 
may be estimated as 0.6 cent per ton-mile, or $3.00 per ton for 
a haul of 500 miles. 



CHAPTER X. 

RAIL-FASTENINGS. 

RAIL-JOINTS 

279. Theoretical requirements for a perfect joint. A perfect 
rail-joint is one that has the same strength and stiffness — no 
more and no less — as the rails which it joins, and which will 
not interfere with the regular and uniform spacing of ties. It 
should also be reasonably cheap both in first cost and in cost of 
maintenance. Since the act J on of heavy loads on an elastic rail 
is to cause a wave of translation in front of each wheel, any 
change in the stiffness or elasticity of the rail structure will 
cause more or less of a shock, which must be taken up and 
resisted by the joint. The greater the change in stiffness the 
greater the shock, and the greater the destructive action of the 
shock. The perfect rail-joint must keep both rail-ends truly in 
line both laterally and vertically, so that the flange or tread of 
the wheel need not jump or change its direction of motion sud- 
denly in passing from one rail to the other. A consideration of 
all the above requirements will show that only a perfect welding 
of rail-ends would produce a joint of uniform strength and stiff- 
ness which would give a uniform elastic w^ave ahead of each 
wheel. As welding is impracticable for ordinary railroad work 
(see § 271), some other contrivance is necessary which will 
approach this ideal as closely as may be. 

280. Efficiency of the ordinary angle-bar. Throughout the 
middle portion of a rail the rail acts as a continuous girder. If 
we consider for simplicity that the ties are unyielding, the deflec- 
tion of such a continuous girder between the ties will be but 
one-fourth of the deflection that would be found if the rail were 
cut half-way betw^een the ties and an equal concentrated load 
were divided equally between the two unconnected ends. The 
maximum stress for the continuous girder would be but one-half 
of that in the cantilevers. Joining these ends with rail- joints 
will give the ordinary "suspended'^ joint. In order to main- 

314 



§281. RAIL-FASTENINGS. 315 

tain uniform strength and stiffness the angle-bars must supply 
the deficiency. These theoretical relations are modified to an 
unknown extent by the unknown and variable yielding of the 
ties From some experiments made by the Association of 
Engineers of Maintenance of Way of the P. R. R.* the following 
deductions were made: 

1. The capacity of a " suspended '^ joint is greater than that 
of a "supported" joint — whether supported on one or three 
ties. (See §282.) 

2. That (with the particular patterns tested) the angle-bars 
alone can carry only 53 to 56% of a concentrated load placed 
on a joint. 

3. That the capacity of the whole joint (angle-bars and rail) 
is only 52.4% of the strength of the unbroken rail. 

4. That the ineffectiveness of the angle-bar is due chiefly to 
a deficiency in compressive resistance. 

Although it has been universally recognized that the angle- 
bar is not a perfect form of joint, its simplicity, cheapness, and 
reliability have caused its almost universal adoption. Within a 
very few years other forms (to be described later) have been 
adopted on trial sections and have been more and more extended, 
until their present use is very large. These designs all agree in 
using metal below the base of the rail, as is shown in the several 
designs on Plate VII, but the general type shown in Fig. 119 
is still (1916) in most common use. 

281. Effect of rail gap at joints. It has been found that the 
jar at a joint is due almost entirely to the deflection of the joint 
and scarcely at all to the small gap required for expansion. 
This gap causes a drop equal to the versed sine of the arc ha\dng 
a chord equal to the gap and a radius equal to the radius of 
the wheel. Taking the extreme case (for a 30-foot rail) of a f " 
gap and a 33'' freight-car wheel, the drop is about toW"- ^^ 
order to test how much the jarring at a joint is due to a gap be- 
tween the rails, the experiment was tried of cutting shallow 
notches in the top of an otherwise solid rail and running a loco- 
motive and an inspection car over them. The resulting jarring 
was practically imperceptible and not comparable to the jar pro- 
duced at joints. Notwithstanding this fact, many plans have 

* Roadmasters Association of America — Reports for 1897. 



316 



RAILROAD CONSTRUCTION. 



§282. 



been tried for avoiding this gap. The most of these plans con- 
sist essentially of some form of compound rail, the sections 
breaking joints. (Of course the design of the compound rail 
has also several other objects in view.) In Fig. 122 are shown a 





Fig. 122. — Compound Rail Sections. 



few of the very many designs which have been proposed. These 
designs have invariably been abandoned after trial. Another 
plan, which has been extensively tried on the Lehigh Valley 
R. R., is the use of mitered joints. The advantages gained by 
their use are as yet doubtful, while the added expense is unques- 
tionable. The ^' Roadmasters Association of America" in 1895 
adopted a resolution recommending mitered joints for double 
track, but their use has been abandoned. 

282. " Supported," " suspended," and " bridge " joints. In a 
supported joint the ends of the rails are on a tie. If the angle- 
plates are short, the joint is entirely supported on one tie; if 
very long, it may be possible to place three ties under one angle- 
bar and thus the joint is virtually supported on three ties rather 
than one. In a suspended joint the ends of the rails are midway 
between two ties and the joint is supported by the two. There 
have alw^ays been advocates of both methods, but suspended 
joints are more generally used than supported joints. The 
opponents of three-tie joints claim that either the middle tie will 
be too strongly tamped, thus making it a supported joint, or 
that, if the middle tie is weakest, the joint becomes a very long 
(and therefore weak) suspended joint between the outer joint- 
ties, or that possibly one of the outer joint-ties gives way, thus 
breaking the angle-plate at the joint. Another objection which 
is urged is that unless the bars are very long (say 44 inches, as 
used on the Mich. Cent. R. R.) the ties are too close for proper 
tamping. The best answer to these objections is the successful 
use of these joints on several heavy-traffic roads. 

"Bridget-joints are similar to suspended joints in that the 
joint is supported on two ties, but there is the important differ- 
ence that the bridge joint supports the rail from underneath and 



§ 283. RAIL-FASTENINGS. 317 

there is no transverse stress in the rail, whereas the suspended 
joint requires the combined transverse strength of both angle- 
bars and rail. A serious objection to bridge-joints hes in the 
fact of their considerable thickness between the rail base and the 
tie. When joints are placed ^^ staggered '^ (as is now the invariable 
standard practice), rather than ^'opposite," the ties support- 
ing a bridge-joint must either be notched down, thus weak- 
ening the tie and promoting decay at the cut, or else the tie 
must be laid on a slope and the joint and the opposite rail do not 
get a fair bearing. 

283. Failures of rail-joints. It has been observed on double- 
track roads that the maximum rail wear occurs a few inches 
beyond the rail gap at the joint in the direction of the traffic. 
On single-track roads the maximum rail wear is found a few 
inches each side of the joint rather than at the extreme ends of 
the rail, thus showing that the rail end deflects down under the 
wheel until (wdth fast trains especially) the wheel actually jumps 
the space and strikes the rail a few inches beyond the joint, the 
impact producing excessive wear. This action, which is called 
the ''drop,'' is apt to cause the first tie beyond the joint to 
become depressed, and unless this tie is carefully watched and 
maintained at its proper level, the stresses in the angle-bar may 
actually become reversed and the bar may break at the top. The 
angle-bars of a suspended joint are normally in compression at 
the top. The mere reversal of the stresses would cause the bars 



Fig. 123. — Effect of "Wheel Dkop" (Exaggerated), 

to give way with a less stress than if the stress were always the 
same in kind. A supported joint, and especially a three-tie 
joint (see § 282), is apt to be broken in the same manner. 

284. Standard angle-bars. An angle-bar must be so made 
as to closely fit the rails. The great multipHcity in the designs 
of rails (referred to in Chapter IX) results in a corresponding 
variety in the detailed dimensions of the angle-bars. The 
absolutely essential features required for a fit are (1) the angles 
of the upper and lower surfaces of the bar where they fit against 



318 



RAILROAD CONSTRUCTION. 



§284. 



the rail, and (2) the height of the bar. The bolt-holes in the 
bar and rail must also correspond. The holes in the angle-plates 
are elongated or made oval, so that the track-bolts, which are 
made of corresponding shape immediately under the head, will 
not be turned by jarring or vibration. The holes in the rails 
are made of larger diameter (by about i^") than the bolts, so as 
to allow the rail to expand with temperature. 

In Table XXIV and in Fig. 124 are shown the angles and 
dimensions for angle-plates to fit the standard rail sections 

shown in §267. Note that the 
^[^ y J dimension a for the splice-bar 

corresponds with dimension F for 
the rail and that 7^4 and the 
angle a. are the same for both for 

f^_ each type of rail. These dimensions 

were copied from the 1916 Hand- 
book of the Carnegie Steel Co. 
Although they correspond perfectly 
with the rail standards of the A. 
R. E. A., that association has not 
yet adopted any such definite 
standard dimensions for a rail-joint. 
The standard driUing for bolt- 
holes in splice-bar, as adopted by 
the A R. E. A. in 1914, is as follows : 

For 6-bolt splices, 5 spaces of 5J inches. 
For 4-bolt splices, 3 spaces of 5J inches. 

No definite recommendation was made by the Association 
as to the total length of angle bars, but the committee recom- 
mended that, on the basis of the above spacing of holes, 24 inches 
is a satisfactory length for a 4-bolt splice and 32 inches for a 
6-bolt sphce, in both cases using suspended joints. On this 
basis, the spacing from the center of the last hole to the end of 
the bar would be 3f inches for the 4-bolt splice and 2| inches for 
the 6-bolt splice. 

In Plate VII are shown some of the many designs which have 
been competing for favor and which have been more or less 
extensively tried out for both steam and electric railroad work. 
While many thousands in the aggregate have been placed on 
various roads, no one design has succeeded in displacing the 




Fig. 



124. — Standard 
Bar. 



Angle 



§284, 



RAIL-PASTENINGS, 



319 



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320 RAILROAD CONSTRUCTION. § 285. 

angle-bar. There are necessarily as many variations in the 
details of the angle-bars as there are variations in the sizes of 
rails, beside other shght variations, but all cross-sections are 
similar to that shown in Fig. 124. This general design prob- 
ably represents the majority of all the spHce-plates in the 
country. 

285. Specifications for steel splice-bars. Formerly these were 
made of either Bessemer or open-hearth steel. Now (1916), the 
specifications of the A. E. R. A. require open-hearth steel exclu- 
sively. Two grades^ are used. The special requirements of the 
*' high-carbon steel joint bars '' are that the phosphorus shall 
not exceed 0.04%; that the tensik strength of a J-inch test 
specimen shall be at least 85,000 lbs. per square inch, the elonga- 
tion at least 16% in 2 inches and that it shall bend 90° without 
fracture on the outside around an arc, the diameter of which is 
three times the thickness of the test piece. Also, they must be 
punched, slotted and shaped at a temperature of not less than 
800° C. or 1470° F. The other grade is "heat-treated, oil- 
quenched steel joint bars," These must have a tensile strength 
of at least 100,000 pounds per square inch, a yield point of at least 
70,000, an elongation in 2 inches of not less than (1,500,000 -r- 
tensile strength) per cent, which must not be less than 12, and 
also that it shall bend 90° without fracture on the outside around 
an arc, the diameter of which is 1 J times the thickness of the test 
piece. The joint bars shall be heated and quenched in an oil 
bath from a temperature of about 810° C. (1490° F.) and shall 
be kept in the oil bath until cold enough to be handled. As 
before, they must be punched, slotted and shaped at a tempera- 
ture of not less than 800° C. or 1470° F. There are the usual 
specifications about accuracy of workmanship, marks rolled in 
the steel, inspection, etc. 

TIE-PLATES. 

286. Advantages, (a) As already indicated in § 242, the 
life of a soft-wood tie is very much reduced by " rail-cutting " 
and " spike-killing," such ties frequently requiring renewal long 
before any serious decay has set in. It has been practically 
demonstrated that the " rail-cutting " is not due to the mere 
pressure of the rail on the tie, even with a maximum load on 
the rail, but is due to the impact resulting from vibration and 



1 




WOLHAUPTER JOINT 



ft.ATE VII.— Some forms op Rail Joints 
^Between pp. 320 and 321 ) 




BONZANO RAIL JOINT. 



§ 287. RAIL-FASTENINGS. 321 

to the longitudinal working of the rail. It has been proved 
that this rail-cutting is practically prevented by the use of tie- 
plates, (h) On curves there is a tendency to overturn the outer 
rail due to the lateral pressure on the side of the head. 
This produces a concentrated pressure of the outer edge of the 
base on the tie which produces rail-cutting and also draws the 
inner spikes. Formerly the only method of guarding against 
this was by the use of " rail-braces," one pattern of which is 
shown in Fig. 125. But shoulder tie-plates serve the purpose 
even better and rail-braces are fCr'^%^ 

now only used for guard rails and ^^^.^^V 

stock rails at switches, (c) Driv- f^^^^J^^V. 

ing spikes through holes in the ^X ^^^^^^ ^^ 
plate enables the spikes on each ^ ^^ks ^^^^^^^^^ 
side of the rail to mutually sup- ^k ^^^ /\^^^j 
port each other, no matter in nScn/ / li^& / 

which (lateral) direction the rail ^^^^?w^ / m ^^^^ 
may tend to move, and this prob- [^"^n^;' / ■ ^^j^^^ Ik^^ 
ably accounts in large measure /^ ^■'\' ,^^^v '^^^^ 
for the added stabihty obtained i^^^ ^ "^!^^%;^^'^'^^ 
by the use of tie-plates, id) The ^^^^^r;^^^^^ ^ 

wear in spikes, called '^ necking," Fig. 125. — ^Atlas Brace K. ' 
caused by the vertical vibration 

of the rail against them, is very greatly reduced, (e) The cost is 
very small compared with the value of the added hfe of the tie, 
the large reduction in the work of track maintenance, and the 
smoother running on the better track vvhich is obtained. It has 
been estimated that by the use of tie-plates the life of hard- 
wood ties is increased from one to three years and the life of 
soft-wood ties is increased from three to six years. From the 
very nature of the case, the value of tie-plates is greater when 
they are used to protect soft ties. 

287. Elements of the design. The Am. Rwy. Eng. Assoc, has 
stated these principles in its Manual, as follows : 

1. '^ Plates shall not be less than 6 inches in width, and as 
much wider as consistent with the class of ties to be used." The 
use of a wide tie presumes heavy traffic and heavy wheel loads 
and, therefore, the area of the plate should be increased by widen- 
ing the plate. 

2. '^ The length of the plates [parallel with the length of the 
tie] shall not be less than the safe-bearing area of the ties divided 



322 RAILROAD CONSTRUCTION. § 287. 

by the width of the plate, and, when made for screw spikes, shall 
be so shaped as to provide proper support for the screw spikes." 
335 lbs. per square inch is declared to be, by test, the minimum 
safe-bearing load. Tie-plates sometimes sink quickly and deeply 
into the tie, thus proving that the area is inadequate for the wheel 
loads and traffic on them. 

3. " The thickness of the plate shall be properly proportioned 
to the length.'' Tie-plates have been used as thin as ^ inch, 
but it is now being reahzed that the real function of the plate 
is to be a hearing plate which shall distribute the load, rather 
than a mere surface plate which shall protect the tie from abra- 
sion. The Track Committee of the A. R. E. A. recommended 
that the plates should be at least f inch thick under either edge 
of the rail. Although the Association refused to concur, the 
discussion developed the fact that the thin plates formerly used 
have been found to be too thin and that thicker plates are more 
satisfactory. 

4. " Plates shall have a shoulder at least i of an inch high. 
The distance from the edge of rail base to the end of the tie-plate 
on the outer side must be uniform, and in excess of the projection 
inside of the rail base.'' 

5. " Where treated ties are used or where plates are for screw 
spikes, a flat-bottom plate is preferable. Where ribs of any kind 
are used on base of plate, these shall be few in number and not to 
exceed J inch in depth." This specification is in direct contrast 
to the older designs which had been corrugations and even 
" claws " which were forced deeply into the tie, in order to anchor 
the plate immovably to the tie. But experience has proved that 
these corrugations hasten deterioration. In spite pf this, the 
type using claws (see Fig. 126) is still the standard on some roads. 

6. *' Punching must correspond to the slotting in the sphce- 
bars and, where advisable, may be so arranged that the plates 
may be used for joints. Spike holes may be punched for varying 
widths of rail base where the slotting will permit such punching 
without the holes interfering with each other and when the plate 
is of such design that the additional holes will not impair the 
strength of the plate." 

Tie-plates are variously made of steel, wrought iron and malle- 
able iron. Tie-plates are peculiarly subject to rust, especially 
as an effect of brine drippings from refrigerator cars. The 
comparative immunity from rust of malleable iron explains its 



§287. 



RAIL-FASTENINGS, 



323 



use for this purpose. The specifications for steel and wrought 
iron are similar to other physical tests for such a metal when 
toughness rather than high ultimate strength is desired. The 
malleable iron tie-plates have lugs cast on them for testing pur- 
poses. When this lug is broken off, it must not break easily, as 
cast iron, but must show toughness. The fracture must show a 
narrow band of white metal on the surface, the center portion 




Wolhaupter 





Hound grooved-tapered-flat 
bottom-shoiilder tie plate 



f% 



-11- 



^ 



P.B.R. flat bottom tie filate 



Claw and shoulder tie plate 

Fig. 126. — Various Forms of Tie-plates. 



being dark and fiberless. The plates must, when tested, bend 
sufficiently to prove thorough annealing. 

The holes in a tie-plate should be about x^" larger than the 
size of the intended spike. The length of the plate, perpendicular 
to the rail, should be such that there is a shoulder of 1| to 2 J in. 
on each side of the rail base, a little more on the outside than on 
the inside. For very heavy traffic the thickness should be §" to 
I"; for lighter traffic, they may be as thin as f ". Flat-bottom 
plates should be at least f thick; corrugated plates, being 
somewhat stiff er, may be thinner for the same, service. The tie- 
plates under the joint ties must be somewhat longer than the 



324 RAILROAD CONSTRUCTION. § 288. 

intermediates, in order to allow for the extra length from out to 
out of the angle-plates. 

288. Method of setting. A very important detail in the 
process of setting the tie-plates on the ties is that the plates 
should be rigidly attached to the ties in their intended position 
during the process of setting. If tie-plates with flat bottoms 
are used, the surface of the tie must be adzed, so that it is not 
only plane but level, so that there will be no danger that the 
plate will rock on the tie. When using tie-plates which are 
corrugated on the under surface, it is necessary to force them 
into the tie until the under side of the plate is flush with the 
surface of the tie. This requires a pressure of several thousand 
pounds. Sometimes trackmen have depended on the easy 
process of waiting for passing trains to force the corrugations 
into the tie until the plate is in its intended position. Until 
the plates are finally set the spikes cannot be driven home, 
and this apparently cheap and easy process generally results 
in loose spikes and rails. The best method for new work is 
to drive the plates into the tie before setting the tie in position. 
A tie-plate gauge holds both tie-plates in their proper relative 
position, and both plates may be driven by the use of heavy 
beetles. When it is necessary to place the plate under the rail 
and drive it in, it is somewhat difficult to drive it by striking 
the plate with a swage on each side of the rail alternately. 
When it is struck on one side, the other side flies up unless held 
down by a wedge driven between the plate and the rail on the 
other side of the rail. A straddler, which straddles the rail 
somewhat like an inverted U, is very useful for this purpose, 
since it makes it possible to strike the head of the straddler and 
force down both sides of the plate at once. The Southern 
Pacific Railroad Company has rigged up a small pile-driver on 
a hand-car, which is used in connection with a straddler to drive 
the tie-plates into position. Some western railroads have even 
adopted the process of rigging up a flat car with a machine 
which will press the tie-plates into place in the ties before the 
ties are placed in the track. 

SPIKES. 

289. Requirements. The rails must be held to the ties by a 
fastening which will not only give sufficient resistance, but which 



§289. 



RAIL-FASTENINGS . 



325 



will retain its capacity for resistance. It must also be cheap 
and easily applied. The ordinary' ti;ack-spike fulfills the last 
requirements, but has comparatively small resisting power, com- 
pared with screws or bolts. Worse than all, the tendency to 




^ 



Fig. 127. 





Fig. 128. 



vertical vibration in the rail produces a series of upward pulls on 
the spike that soon loosens it. When motion has once begun 
the capacity for resistance is greatly reduced, and but little more 
vibration is required to pull the spike out so much that redriving 
is necessary. Driving the spike to place again in the same hole 
is of small value except as a very temporary expedient, as its 
holding powder is then very small. Redriving the spikes in new 
holes very soon "spike-kills'' the tie. Many plans have been 
devised to increase the holding power of spikes, such as making 
them jagged, twisting the spike, swelling the spike at about the 
center of its length, etc. But it has been easily demonstrated 
that the fibers of the wood are generally so crushed and torn by 
driving such spikes that their holding power is less than that of 
the plain spike, and the durabihty is greatly diminished. 

The ordinary spike (see Fig. 127) is made with a square cross- 
section which is uniform through the middle of its length, the 
lower If in. tapering down to a chisel edge, the upper part swelling 
out to the head. The Goldie spike (see Fig. 128) aims to improve 
this form by reducing to a minimum the destruction of the 



F^ 



326 RAILROAD CONSTRUCTION. § 290. 

fibers. To this end, the sides are made smooth, the edges are 
clean-cut, and the point, instead of being chisel-shaped, is ground 
down to a p3n:amidal form. Such fiber-cutting as occurs is thus 
accomphshed without much crushing, and the fibers are thus 
pressed away from the spike and slightly downward. Any 
tendency to draw the spike will, therefore, cause the fibers to 
press still harder on the spike and thus increase the resistance. 
A series of tests made by a committee of the A. R. E. A. and 
reported to the 1914 Convention, established some very valuable 
conclusions with respect to the use of the ordinary cut spike. 
Spikes with sharp pyramidal points and with various degrees of 
bluntness, and also the ordinary chisel-pointed spike, were driven 
into ties and other timbers and were withdrawn by a testing 
machine. Then the timbers were cut so as to expose the holes 
to their full length, so that the crushing of the fibers by the spike 
driving could be observed. A series of photographs illustrated 
this feature. In some cases the spikes were driven into |-in. 
bored holes, some of which were 2i ins. deep, but the most of them 
were 4 ins. deep. In other cases, the spikes were driven without 
previous boring. The following conclusions were unmistakable. 

1. The spike with a pyramidal point about 1 in. long (vir- 
tually the '^ Goldie " design Fig. 128), has greater holding power, 
not only when it first begins to yield, but also afterward while the 
spike is being drawn out. 

2. The long-pointed spikes crushed the fiber far less than any 
other type. 

3. The chisel-pointed spike, virtually as shown in Fig. 127, and 
which is the type now in most common use, has the least holding 
power and is more destructive in crushing the fibers. 

4. Spikes driven into f-in. bored holes have greater holding 
power than when driven without boring, and the crushing of the 
fiber is much less. This indicates the very real economy in bor- 
ing holes where the life of the tie is an economical consideration. 

290. Driving. The holding power of a spike depends largely 
on how it is driven. If the blows are eccentric and irregular 
in direction, the hole will be somewhat enlarged and the hold- 
ing power largely decreased. The spikes on each side of the 
rail in any one tie should not be directly opposite, but should 
be staggered. Placing them directly opposite will tend to split 
the tie, or at least decrease the holding power of the spikes. 
The direction of staggering should be reversed in the two pairs 



§291. 



RAIL-FASTENINGS, 



327 



of spikes in any one tie (see Fig. 129). 
vent any twisting of the tie in 
the ballast, which would other- 
wise loosen the rail from the tie. 
291. Screw spikes. The D., 
L. & W. R. R. began the general 
use of screw spikes for all new 
work and for extensive track 
renewals in 1910. In five years 
they used over 12,000,000 screw 
spikes. The design is shown in 
Fig. 130. From a report made 



This will tend to pre- 




FiQ. 129. — Spike-dbivinq. 



Fig. 130. — Screw Spike, D. L. 
&. W. R. R. 



by Mr. G. .). Ray, Chief Engineer, to the A. R. E. A., the follow- 
ing facts and conclusions are deduced : 

1. The use of screw spikes, in conjunction with suitable tie- 
plates, is almost a necessity in order to fully utilize the durability 
of a treated tie. A treated tie is seldom removed on account of 
decay in the body of the tie. Its destruction is generally due to 
" spike-killing,'^ rail cutting, or to the decay which comes im- 
mediately after mechanical injury to the wood under the rail. 
Screw spikes and tie-plates largely prevent this mechanical injury. 

2. '^ As a rule, with woods which it will pay to treat, the poorer 
the quality of the timber the more elaborate and expensive the 
fastening must be if the mechanical life of the tie is made to 
approach the life of the treated timber.'' 

3. " Tie-plates should be used on all ties where screw spikes 
are used." 



r^ 



328 RAILROAD CONSTRUCTION. § 291. 

4. *' Four holes should be provided for screw spikes, so that 
two extra holes will be available if needed/' 

5. '' The size of screw spikes and the design of the thread should 
be carefully considered before a screw spike is adopted. There- 
after no changes should be made; otherwise the new screw spikes 
cannot be used in old holes without damaging the wood fiber/' 

6. " The screw-spike head should have tapering sides to pre- 
vent turning in the wrench socket after the size of the head has 
been diminished by rust/' 

7. '^ When screw spikes are fully seated, no further strain 
should be put on them, as this will tend to destroy the threads 
in the wood or injure the spikes." 

8. '^All ties should be bored at the treating plant before treat- 
ment. This can be done while the ties are being adzed, and not 
only insures that the holes are bored sufficiently deep, but provides 
for good treatment of all wood adjacent to the spike holes." 

9. " Where the ties are bored before treatment, the track must 
be to proper gauge before the ties can be placed." 

10. ^^ The holes for screw spikes should be of proper dimensions 
for the class of wood used, with due regard to the size of screw 
spike used." 

11. '^A limited number of holes can be bored with one bit, after 
which its size will diminish so as to make it unfit for a hole of a given 
size." [The paper nowhere makes any statement as to the size of 
the bored hole in comparison with the diameter of the screw. The 
bored hole should have about the same diameter as the diameter of 
the screw at the base of the screw thread, but the hardness of the 
wood requires some variation, since, if the hole is too small, it will 
be impossible to turn the screw. The exact diameter must be de- 
termined for each kind of wood and must be strictly maintained.] 

12. " Holes should be bored somewhat deeper than the length 
of the screw spike. There is no serious objection to boring the 
holes clear through the ties." 

13. '^ Not only is the lateral and vertical resistance of a screw 
spike greater than that of a cut spike when both are first applied, 
but the lateral and vertical resistance of a loose screw spike is 
considerably greater than the lateral and vertical resistance of a 
loose cut spike." 

14. " When the threads in the tie are entirely destroyed, a 
screw lining (any one of several different varieties) may be used 
with good results." 



§292. 



RAIL-FASTENINGS. 



329 



15. '^All ties should be bored and adzed before treatment. This 
insures good gauge, a perfect bearing for the tie-plates and good 
treatment under the rail seat and around the screw-spike holes." 

16. " In placing screw spikes, they should be driven by ham- 
mer only sufficient to make the threads take hold. If rigid in- 
structions are not carried out, laborers will continually overdrive 
spikes and thus destroy the wood fibers near the top of the holes." 

17. ^' The best results with the screw spikes can be expected in 
new construction, and where the number of screw spikes in tie 
renewals predominate over cut spikes." 

18. " The use of screw spikes for the past five years has not 
made it necessary to increase the number of sectionmen per mile 
of track." 

19. ^' Whether or not it will pay to use screw spikes will depend 
upon the cost of ties, their probable life and the amount of traffic." 

292. " Wooden spikes." Among the regulations for track- 
laying given in § 246, mention was made' of wooden ^' spikes," 
or plugs, which are used to fill up the holes when spikes are 
withdrawn. The value of the policy of filling up these holes is 
unquestionable, since the expense is insignificant compared with 
the loss due to the quick and certain decay of the tie if these 
holes are allowed to fill with water and remain so. But the 
method of making these plugs is variable. On some roads they 
are "hand-made" by the trackmen out of otherwise use- 
less scraps of lumber, the work being done at odd mo- 
ments. This policy, while apparently cheap, is not 
necessarily so, for the hand-made plugs are irregular 
in size and therefore more or less inefficient. It is 
also quite probable that if the trackmen are required to 
make their own glugs, they would spend time on these 
very cheap articles which could be more profitably em- 
ployed otherwise. Since the holes made by the spikes 
are larger at the top than they are near the bottom, the 
plugs should 7iot be of uniform cross-section but should 
be shghtly wedge-shaped. .The ''Goldie tie-plug" 
(see Fig. 131) has been designed to fill these require- 
ments. Being machine-made, they are uniform in 
size; the}^ are of a shape which will best fit the hole; 
they can be furnished of any desired wood, and at a 
cost which makes it a wasteful economy to attempt j^jq^ 131 
to cut them bv hand. 



330 RAILROAD CONSTRUCTION. § 293. 



TRACK-BOLTS. 

293. Essential requirements. The track-bolts must have 
sufficient strength and must be screwed up tight enough to hold 
the angle-plates against the rail with sufficient force to develop 
the full transverse strength of the angle-bars. On the other 
hand the bolts should not be screwed so tight that slipping may 
not take place when the rail expands or contracts with tempera- 
ture. It would be impossible to screw the bolts tight enough to 
prevent slipping during the contraction due to a considerable fall 
of temperature on a straight tracks but when the track is curved, 
or when expansion takes place, it is conceivable that the resist- 
ance of the ties in the ballast to lateral motion may be less than 
the resistance at the joint. A test to determine this resistance 
was made by Mr. A. Torrey, chief engineer of the Mich. Cent. 
R. R., using 80-lb. rails and ordinary angle-bars, the bolts being 
screwed up as usual. If required a force of about 31000 to 
35000 lbs. to start the joint, which would be equivalent to the 
stress induced by a change of temperature of about 22°. But 
if the central angle of any given curve is small, a comparatively 
small lateral component w411 be sufficient to resist a compression 
of even 35000 lbs. in the rails. Therefore there will ordinarily 
be no trouble about having the joints screwed too tight. The 
vibration caused by the passage of a train reduces the resistance 
to slipping. This vibration also facilitates an objectionable 
feature, viz., loosening of the nuts of the track-bolts. The bolt 
is readily prevented from turning by giving it a form which is 
not circular immediately under the head and making corre- 
sponding holes in the angle-plate. Square holes would answer 
the purpose, except that the square corners in the holes in the 
angle-plates would increase the danger of fracture of the plates. 
Therefore the holes (and also the bolts, under the head) are 
made of an oval form, or perhaps a square form with rounded 
corners, avoiding angles in the outline. 

" As a rule, as large track-bolts should be used as the rail and 
splice-bars will permit." [From 1915 Manual, A. R. E. A.] 
There is always some danger that a trackman may stretch a bolt 
beyond its elastic limit. A pull of 100 lbs. on a 33-inch track 
wrench will induce a stress of about 45000 lbs. per square inch 
in a |-inch track bolt. The same work on a 1-inch bolt would 
produce a stress of about 35000 lbs. per square inch. In order to 



§294. 



KAIL-FASTENINGS. 



331 



obtain the necessary toughness, bolts must be made of low-carbon 
steel or of nickel-steel, untreated or heat-treated. When made 
of carbon steel, specifications require an elastic limit of at least 
35,000 lbs. per square inch but at the same time an elongation of 
25% in 2 inches and a reduction of area of at least 50%. A 
harder steel would have a higher elastic Hmit, but would not 
be sufficienlty ductile. Higher elastic hmits, with suflficient 
ductility, may be obtained by .using untreated nickel or other 
alloy steel (at least 45,000 lbs. per square inch), or heat- 
treated nickel or other alloy steel (at least 75,000 lbs. per square 
inch). The elastic Hmit shall not be less than 50% of the ulti- 
mate. Added strength can only be obtained by using larger 
bolts or a more expensive metal. 

294. Design of track-bolts. In Fig. 132 is shown a common 
design of track-bolt. In its general form this represents the 
bolt used on nearly all roads, 
being used not only with the 
common angle-plates, but also 
with many of the improved de- 
signs of rail-joints. The varia- 
tions are chiefly a general in- 
crease in size to correspond with 
the increased weight of rails, 
besides variations in detail di- 
mensions which are frequently 
unimportant. The diameter is 
usuaUy f" to F'; 1" bolts are 
used -for 100-lb. rails. As to 
length, the bolt should not ex- 
tend more than |" outside of 
the nut when it is screwed up. 

If it extends farther than this it is Hable to be broken off by a 
possible derailment at that point. The lengths used vary from 
3i", which may be used with 60-lb. rails, to 5'', which is required 
with 100-lb. rails. The length required depends somewhat on 
the type of nut-lock used. 




Fig. 132.— Tbace-bolt. 



NUT-LOCKS. 



295. Design of nut-locks. The designs for nut-locks may be 
divided into three classes: (a) those depending entirely on an 



^ 



332 



RAILROAD CONSTRUCTION, 



§295. 




VERONA 




VULCANIZED FIBRE 





IMPROVED VERONA 



v^^ NATI0N.A1. 




Columbia.Nut Lock 



tsxsnsraS^^ \sssnnssar 




JONES 



Fig. 133. — ^Types of Nut-locks. 



§ 295. KAIL-FASTENINGS. 333 

elastic washer which absorbs the vibration which might other- 
wdse induce turning; (h) those which jam the threads of the 
bolt and nut so that, when screwed up, the frictional resistance 
is too great to be overcome by vibration; (c) the "positive'* 
nut-locks — those which mechanically hold the nut from turning. 
Some of the designs combine these principles to some extent. 
The "vulcanized fiber'' nut-lock is an example of the first class. 
It consists essentially of a rubber washer which is protected by 
an iron ring. When first placed this lock is effective, but the 
rubber soon hardens and loses its elasticity and it is then ineffec- 
tive and worthless. Another illustration of class (a) is the use 
of wooden blocks, generally 1'' to 2'' oak, which extend the 
entire length of the angle-bar, a single piece forming the washer 
for the four or six bolts of a joint. This form is cheap, but the 
wood soon shrinks, loses its elasticity, or decays so that it soon 
becomes worthless, and it requires constant adjustment to keep 
it in even tolerable condition. The "Verona" nut-lock is 
another illustration of class (a) which also combines some of the 
positive elements of class (c). It is made of tempered steel and, 
as shown in Fig. 133, is warped and has sharp edges or points. 
The w^arped form furnishes the element of elastic pressure when 
the nut is screwed up. The steel being harder than the iron of 
the angle-bar or of the nut, it bites into them, owing to the 
great pressure that must exist when the washer is squeezed 
nearly flat, and thus prevents any backward movement, although 
forward movement (or tightening the bolt) is not interfered 
with. The " National" nut-lock is a type of the second class (h), 
in which, like the " Harvey" nut-lock, the nut and lock are com- 
bined in one piece. With six-bolt angle-bars and 30-foot rails, 
this means a saving of 2112 pieces on each mile of single track. 
The "National" nuts are open on one side. The hole is drilled 
and the thread is cut slightly smaller than the bolt, so that when 
the nut is screwed up it is forced slightly open and therefore 
presses on the threads of the bolt with such force that vibration 
cannot jar it loose. Unlike the " National" nut, the ^' Harvev " 
nut is solid, but the form of the thread is progressively varied so 
that the thread pinches the thread of the bolt and the frictional 
resistance to turning is too great to be affected by vibration. 

The "Columbia" nut-lock is a two-piece nut, both parts of 
which must turn simultaneously. As shown in the figure, one 



334 RAILROAD CONSTRUCTION. § 295. 

section wedges into the other. The greater the tension in the 
bolt, the greater the wedging action and the greater the friction 
to prevent turning. 

The '^ Jones'' nut-lock, belonging to class (c), is a type of a 
nut-lock that does not depend on elasticity or jamming of screw- 
threads. It is made of a thin flexible plate, the square part of 
which is so large that it will not turn after being placed on the 
bolt. After the nut is screwed up, the thin plate is bent over so 
that the re-entrant angle of the plate engages the corner of the 
nut and thus mechanically prevents any turning. The metal 
is supposed to be sufficiently tough to endure without fracture 
as many bendings of the plate as will ever be desired. Nut- 
locks of class (c) are not in common use. 

The above types have been discussed in order to show the 
development of the various devices. With but few exceptions, 
the standard nut-lock is a steel spring ring of the same general 
class as the Verona. The A. R. E. A. have prepared specifica- 
tions for such nut-locks which include the following: 

" After the finished nut-lock has been subjected for one hour to 
pressure sufficient to compress it flat and has been released, its 
reaction shall be not less than two-thirds its height or thickness 
of section, provided thickness is less than width of section. If 
the section is square, the reaction must be not less than one-half 
its thickness. If height or thickness of section is more than 
width, the reaction shall be not less than the width of the section. 
The internal diameters naturally affect the percentage of reac- 
tion, and the above specifications apply to nut-locks of internal 
diameters from y|- in. to 1 3^ ins. Owing to the difficulty of 
establishing a common rate of percentage that shaU be uniformly 
applicable to any internal diameter of any nut-lock of any secti6n 
it has been sought to cover the matter as above. Amount and 
durability of reactionary power under constant pressure is the 
true test of any spiral spring nut-lock. The percentage of reac- 
tion increases proportionately with the increased internal diam- 
eter of any given section." 

'^ With one end of the finished nut-lock secured in a vise, and 
the opposite end twisted to 45 degrees, there must be no sign of 
fracture. When further twisted until broken, the fracture must 
show a good quahty of steel." 



CHAPTER XI. 

SWITCHES AND CROSSINGS. 

SWITCH CONSTRUCTION. 

296. Essential elements of a switch. Flanges of some sort are 
a necessity to prevent car-wheels from running off from the rails 
on which they may be moving. But the flanges, although a 
necessity, are also a source of compUcation in that they require 
some special mechanism which will, when desired, guide the 
wheels out from the controlling influence of the main-line rails. 
This must either be done by raising the wheels high enough 
so that the flanges may pass over the rails, or by breaking the 
continuity of the rails in such a waj^ that channels or "flange 
spaces'' are formed through the rails. An ordinary stub-switch 
breaks the continuity of the main-liuQ rails in three places, two 
of them at the switch-block and one at the frog. The Wharton 
switch avoids two of these breaks by so placing inclined planes 
that the wheels, rolling on their flanges, will surmount these 
inclines until they are a little higher than the rails. Then the 
wheels on the side toward which the switch runs are guided 
over and across the main rail on that side This rise being ac- 
complished in a short distance, it becomes impracticable to 
operate these switches except at slow speeds, as any sudden 
change in the path of the center of gravity of a car causes very 
destructive jars both to the switch and to the rolling stock. The 
other general method makes a break in one main rail (or both) 
at the switch-block. In both methods the wheels are led to one 
side by means of the '^ lead rails," and finally one line of wheels 
passes through the main rail on that side by means of a "frog." 
There are some designs by which even this break in the main 
rail is avoided, the wheels being led over the main rail by means 
of a short movable rail which is on occasion placed across the 
main rail, but such designs have not come into general use. 

297. Frogs. Frogs are provided with two channel-ways or 
"flange spaces'' through which the flanges of the wheels move, 

335 



336 



RAILROAD CONSTRUCTION. 



§297. 



Each channel cuts out a parallelogram from the tread area. 
Since the wheel-tread is always wider than the rail, the wing 
rails will support the wheel not only across the space cut out by 
the channel, but also until the tread has passed the point of the 
frog and can obtain a broad area of contact on the tongue of the 
frog. This is the theoretical idea, l)ut it is very imperfectly 




Fig. 134. — Diagrammatic Design of Frog. 



realized. The wing rails are sometimes subjected to excessive 
wear owing to ^'hollow treads'^ on the wheels — owing also to 
the frog being so flexible that the point ^' ducks'' when the wheel 
approaches it. On the other hand the sharp point of the frog 
will sometimes cause destructive wear on the tread of the wheel. 
Therefore the tongue of the frog is not carried out to the sharp 
theoretical point, but is purposely somewhat blunted. But 
the break which these channels make in the continuity of the 
tread area becomes extremely objectionable at high speeds, 
being mutually destructive to the rolling stock and to the frog. 
The jarring has been materially reduced by the device of '' spring 
frogs'' — to be described later. Frogs were originally made of 
cast iron — then of cast iron with wearing parts of cast steel, 
which were fitted into suitable notches in the cast iron. This 
form proved extremely heavy and devoid of that elasticity of 
track which is necessary for the safety of rolling stock and 
track at high speeds. The present standard practice is to build 
the frog up of pieces of rails which are cut or bent as required. 
There are always four pieces for single-pointed frogs. For 
heavy work they are assembled by bolting them together, the 
flangeways being provided by the use of fillers made of cast 
iron, cast steel or rolled steel. For still heavier work the above 
combination is riveted to a base plate. For hght or street rail- 
way work, the rails are riveted to a base plate without using 



'i 



fit 



ml 



*t 



n 



h 



ODJ 



,') 1 





(To face page 336.) 

Plate VIII. — Some Types of Fbogs. 
(As made by Ramapo Iron Works.) 



X( 



§ 298. SWITCHES AND CROSSINGS. 337 

fillers. For details, study Plate VIII. The operation of a 
spring-rail frog is evident from the figure. Since a siding is 
usually operated at slow speed, while the main track may be 
operated at fast speed, a spring-rail frog will be so set that the 
tread is continuous for the main track and broken for the 
siding. This also means that the spring-rail wiU only be moved 
by trains moving at a (presumably) slow speed on to the 
siding. For the fast trains on the main Hne such a frog is 
substantially a '^ fixed " frog and has a tread which is practically 
continuous. 

298. To find the frog number. The frog number (n) equals 
the ratio of the distance of any point on the tongue of the frog 
from the theoretical point of the frog divided by the width of 
the tongue at that point, i.e. ==hc-^ab (Fig. 134). This value 
may be directly measured by applying any convenient unit of 
measure (even a knife, a short pencil, etc.) to some point of the 
tongue where the width just equals the unit of measure, and then 
noting how many times the unit of measure is contained in the 
distance from that place to the theoretical point. But since c, 
the theoretical point, is not so readily determinable with exacti- 
tude, it being the imaginary intersection of the gauge lines, it 
may be more accurate to measure de, ah, and hs; then n, the frog 
nimiber, = /is -7- (ab + de) . If the frog angle be called F, then 

n=hc~ah=hs-^(ah-\-de) =i cot ^F; 
i.e., cot iF = 2n. 

299. Stub switches. The use of these, although once nearly* 
universal, has been practically abandoned as turnouts from 
main track except for the poorest and cheapest roads. In some 
States their use on main track is prohibited by law. They have 
the sole merit of cheapness with adaptability to the circum- 
stances of very light traffic operated at slow speed when a con- 

I siderable element of danger may be tolerated for the sake of 
j economy. The rails from Ato B (see Fig. 135*) are not fastened 



* The student should at once appreciate that in Fig. 135, as well as in 
nearly all the reinaining figures in this chapter, it becomes necessary to 
use excessively large frog angles, short radii, and a very wide gauge in 
order to illustrate the desired principles with figures which are sufficiently 
small for the page. In fact, the proportions used in the figures are such 
that serious mechanical difficulties would be encountered if they were 
,' used. These difficulties are here ignored because they can be neglected 
in the proportions used iu practice. 



338 



RAILROAD CONSTRUCTION. 



§299. 



to the ties; they are fastened to each other by tie-rods which 
keep them at the proper gauge; at and back of B they are 
securely spiked to the ties, and at A they are kept in place by 




Fig. 135.— Stub Switch. 

the connecting bar (C) fastened to the switch-stand. One great 
objection to the switch is that, in its usual form, when operated 
as a trailing switch, a derailment is inevitable if the switch is 
misplaced. The very least damage resulting from such a derail- 
ment must include the bending or breaking of the tie-rods of the 
switch-rail. Several devices have been invented to obviate this 
objection, some of which succeed very well mechanically, al- 
though their added cost precludes any economy in the total cost 
of the switch. Another objection to the switch is the looseness 
of construction which makes the switches objectionable at high 
speeds. The gap of the rails at the head-block is always con- 
siderable, and is sometimes as much as two inches. A driving- 




FiG. 136. — Point Switch. 



wheel with a load of 20000 to 30000 pounds, jumping this gap 
with any considerable velocity, will do immense damage to the 



§300. 



SWITCHES AND CROSSINGS. 



339 



farther rail end, besides producing such a stress in the construc- 
tion that a breakage is rendered quite Hkely, and such a breakage 
might have very serious consequences. 

300. Point switches. The essential principle of a point switch 
is illustrated in Fig. 136. As is shown, one main rail and also 
one of the switch-rails is unbroken and immovable. The other 
main rail (from A to F) and the corresponding portion of the 
other lead rail are substantially the same as in a stub switch. 
A portion of the main rail (AB) and an equal length of the oppo- 
site lead rail (usually 16.5 to 22 feet long) are fastened together 
by tie-rods. The end at A is jointed as usual and the other end 
is pointed, both sides being trimmed down so that the feather 
edge at B includes the web of the rail. In order to retain in it 

as much strength as possible, the point- 
rail is raised so that it rests on the base 
of the stock-rail, one side of the base of 
the point-rail being nearly cut away. 
As may be seen in Fig. 137, although 
the influence of the point of the rail in 
moving the wheelrflange away from the 
stock-rail is really zero at that point, 
yet the rail has all the strength of the 
web, more than one-half that of the 
base, and is also reinforced. The planing 
runs back in straight lines, until at about 
six or seven feet back from the point 
the full width of the head is obtained. The full width of 
the base will only be obtained at about 13 feet from the 
point. The A. R. E. A. standard switch rail is always cut on 
the basis that the distance between gauge lines at the heel of 
[ the switch (the distance MN in Fig. 143) is 6i inches and that 
I the " point " is i inch wide. Then, using four standard 
1 lengths, 11, 16|, 22 and 33 feet, the angles vary from 2° 36' 19" 
1 to 0° 52' 05", as shown in Table III. 




Fig. 137. 



301. Switch-stands. The simplest and cheapest form is the 
'^ ground lever,'' which has no target. The radius of the circle 
described by the connecting-rod pin is precisely one-half the 
throw. From the nature of the motion the device is practically 



340 



RAILROAD CONSTRUCTION, 



§301, 



self-locking in either position, padlocks being only u^ed to pre- 
vent malicious tampering. 



-cfi^ 








^ ft 


nia 


m 


Hi llllll 


U| 




N ]-< )m 








Fig. 138. — Gkound Lever for Throwing a Switch, 



STEEL SHAFT i;3^,°'^^- 



.i..i 




Fig. 139. — Ramapo Patent Switch Stand. Non-automatic. 



§ 302. 



SWITCHES AND CROSSINGS. 



341 



In Fig. 139 is shown a design in which the arc of the throwing 
lever is parallel to the track, an important feature in quick 
switching work. 

302. Tie-rods. These are fastened to the webs of the rails by 
means of lugs which are bolted on, there being usually a hinge- 




2^S^ 



-^ 



Fig. 140. — Forms of Tie-rods. 

it between the rod and the lug. Two such tie-rods (three for 

0-foot switch) are generally necessary. The first rod is some- 

les made without hinges, which gives additional stiffness to 

3 comparatively weak rail-points. The old-fashioned tie-rod, 

ving jaws fitting the base of the rail, was almost universally 

ed in the days of stub switches. One great inconvenience in 

eir use hes in the fact that they must be slipped on, one by one, 

^er the free ends of the switch-rails. 




j i '^guard rail 

Fig. 141. — Standard Guard-rail. 



303. Guard-rails. As shown in Figs. 135 and 136, guard-rails 
are used on both the main and switch tracks opposite the frog- 
point. Their function is not only to prevent the possibility of 
the wheel-flanges passing on the wrong side of the frog-point, 
but also to save the side of the frog-tongue from excessive wear. 
The flange-way space between the heads of the guard-rail and 
wheel-rail should equal If inches. Since this is less than the 
space between the heads of ordinary (say 80-pound) rails when 



342 



RAILROAD CONSTRUCTION. 



§304. 



placed base to base, to say nothing of the |" required for spikes, 
it becomes necessary to cut the flange of the guard-rail. The 
length of the rail should be 16 feet 6 inches, the middle portion 
being straight for a length of 3 feet 6 inches, and the ends, each 
being 6 feet 6 inches long, curved out so that the side of the 
rail head at each end is 4 inches from the main rail head, when 
the flange-way at the center is If inches. See Fig. 141. 



MATHEMATICAL DESIGN OF SWITCHES. 

In all of the following demonstrations regarding switches, 
turnouts, and crossovers, the lines are assumed to represent the 
gauge-lines — i.e., the lines of the inside of the head of the rails. 

304. Design with circular lead-rails. The simplest method 
is to consider that the lead-rails curve out from the main track- 




Fig. 142. 

rails by arcs of circles which are tangent to the main rails and 
which extend to the frog-point F. The simple curve from D to F 
is of such radius that (r+^g) vers F = gj in which F=the frog 
angle ^ = gauge, L = the " lead " (BF)j and r=the radius of the 
center of the switch-rails. 



r+ig = 



g 



vers F 



(69) 



Also, 



BF-^BD = cot^F; BD = g; BF=L. 
/. L=g cot F 



(70) 



§ 304. SWITCHES AND CROSSINGS. 343 

Also, L = {r-\-ig)smF; (71) 

QT = 2rsmiF (72) 

These formulae involve the angle F. As shown in Table III, 
the angles (F) are always odd quantities, and their trigonometric 
functions are somewhat troublesome to obtain closely with 
ordinary tables. The formulae may be simplified by substitut- 
ing the frog-number n, from the relation that n = i cot iF, 
Since 

r—ig = L cot F and r+ig=L cosec F, , 

then r^iLicotF+cosecF) 

= 1^ cot |F(cot F+cosec F) ^''^)#r- 
= igf cot^ |F, since (cot a+cosec a) =cot Ja 
-=2gnK (73) 

Also, L=2gn, (74) 

from which r=nXL,. (75) 

These extremely simple relations may obviate altogether the 
necessity for tables, since they involve only the frog-number and 
the gauge. On account of the great simplicity of these rules, 
they are frequently used as they are, regardless of the fact that 
the curve is never a uniform simple curve from ST\atch- block to 
frog. In the first place there is a considerable length of the 
gauge-line within the frog, which is straight unless it is pur- 
posely curved to the proper curve while being manufactured, 
which is seldom if ever done — except for the very large-angled 
frogs used for street-railway work, etc. It is'also doubtful whether 
the smtch-rails (BA^ Fig. 135) are bent to the computed curve 
when the rails are set for the switch. The switch-rails of point 
switches are straight^ thus introducing a stretch of straight track 
which is about one-fifth of the total length of the lead-rails. The 
effect of these modifications on the length and radius of the lead- 
rails will be developed and discussed in the following sections. 

The throw (t) of a stub switch depends on the weight of the 
rail, or rather on the width of its base. The throw must be at 



344 RAILROAD CONSTRUCTION. § 305. 

least f" more than that width. The head-block should there- 
fore be placed at such a distance from the heel of the switch (B) 
that the versed sine of the arc equals the throw. These points 
must be opposite on the two rails, but the points on the two rails 
where these relations are exactly true will not be opposite. 
Therefore, instead of considering either of the two radii (r + ig) 
and (r— iy), the mean radius r is used. Then (see Fig. 142) 

rers KOQ=f-^r, 

and the length of the switch-rails is 

QK=rsmKOQ (76) 

Stub-switches are generally used with large frog angles. For 
small frog angles (large frog-numbers) the values of QK are so 
great that the length of rail left unspiked is too great for a safe 
track. If this were obviated by spiking down a portion of the 
lead the theoretical accuracy of the switch would be lost. 

The use of stub switches may now be considered obsolete. 
But the above demonstration has been retained in this edition 
for its educational value as an introduction to the more com- 
plicated method which is now the standard. 

305. Standard design, using straight frog-rails and straight 
point-rails. It becomes necessary in this case to find a curve 
which shall be tangent to both the point-rail and the frog-rail. 
The curve therefore begins at ilf , its tangent making an angle of 
a ( varying from 0° 52' to 2° 36') with the main rail, and runs to 
H. FJ = W= the length of the " wing-rail " from the theoret- 
ical point of the frog (F) to the toe, / or J'. FK = K = the 
length from the theoretical point to the heel of the frog. MN 
=H = the " heel distance,'' or the distance of the gauge line of 
,the switch-rail at the heel from the gauge hne of the main track 
rail. "^ ! 

The central angle of the curve equals {F—a). The angle of 
the chord HM with the main rails is therefore 



JM = 



l{F-a)+a = i{F+a)] 

g-W^inF-H 

sin HF+a) ' 



§ 305. SWITCHES AND CROSSINGS. 345 

JM 



r+hg 



2sin|(F-a) 

g-W ^mF-H 
2 sin i(F+a) sin h{F-a) 

g-W sin F-H 



(77) 



cos a — COS F ' 

DAT = s cos a, — . . . (78) 

in which S = length of switch-rail. 

BF = L = JM cos3(^+«) +TF cos>+aS cos a 

= (g-W sin F-H) cos i{F-{-a)-{-W cos F+ *Sf cos a. . (79) 

It may be more simple, if (r-\-ig) has already been computed, 
to write 

L = 2(r+ig)smi{F-a)cosi{F+a)+WcosF+Scosa 
= (r-^ig)(smF-sma)+W cosy+S cosa. . .- . u (80) 

The above equations for L give the distance from^the actual 
(blunt) point of the switch-rail to the theoretical point of the frog. 
The lead (U) given in Table III is the distance from the actual 
point of the switch-rail to the actual (blunt) point of the frog. 
The difference (U — L) is the '' frog bluntness," which in each 
case equals the width of the frog point (J inch = .04166 foot) mul- 
tiplied by the frog number. The values of the frog bluntness 
for the various frogs is given in the second column of Part B, 
Table III. 

The value of MN = HhsiS been standardized by the A. R. E. A. 
as 6i inches for all lengths of switch-rail and for all values of a. 
The point of the switch-rail (at D) is invariably J-inch thick. 
When it is necessary to calculate MN for other standards of 
construction, it may be computed (caUing aS= length of switch- 
rail) to be 

MN=S sin a + (thickness of point of switch rail). 



346 



RAILROAD CONSTRUCTION. 



§305. 



The length of the wing rail of the frog {W = FJ) is given for 
each frog in the third column of Table III, Part B. The several 
values of F and a are also given in Table III. g is the gauge 
= 4 feet 8i inches = 4.7083 feet. 

The solution of Eq. 77-80 for various frog angles will give a 
series of ^' theoretical leads," as given in Table III, Part B. 
The table also gives the " closure values,'^ or the lengths of the 
arc MJ and of the straight rail M V. But these closure lengths 
are invariably such odd quantities that rails must be cut and 




Fig. 143. 



more or less rail must be wasted. By shortening the radius of 
the connecting curve very slightly and inserting a very short 
length of tangent either between the curve and switch-rail at 
Mj or between the curve and wing-rail at /, all of which will 
change very slightly the length of lead, the closure lengths can 
be made such that the rail cutting and wastage is minimized, 
and yet the combinations of curves and tangents are mathe- 
matically perfect. The detailed method of computing these 
combinations is tedious and will not be elaborated here, but a 
series of results developed by the A. R. E. A. is given under the 
heading of [' practical leads " in Table III, Part C. 



§ 306.. SWITCHES AND CROSSINGS. 347 

The above computations and tabular values assume that the 
two switch points (at B and D) are directly opposite. This 
would always mean that the straight rail {BF) is somewhat 
shorter than the curved rail from D to F, In the maximum case 
the difference is less than 5 inches. Therefore, assuming that 
rails are obtainable at even-foot lengths down to 27 feet, or 
24 feet for a No. 4 frog switch, the system of practical leads never 
requires more than one rail cutting. But even this is some- 
times avoided by using for the straight-rail closure the same 
number and lengths of uncut rails as are specified for the closure 
of the curved part. The chief effect of this is that the point of 
the switch-rail will be located a few inches below its normal posi- 
tion at B and that the gauge at the switch-point will be slightly 
widened when the switch is open. This effect is possibly an 
advantage rather than a disadvantage. 

306. Design for a turnout from the OUTER side of a curved 
track. Fig, 144 is a diagram of what the construction would be 




Fig. 144. 

if the switch-rails were circular throughout. Before the inven- 
tion of point switches and when stub switches were in universal 
use, the lead-rails were considered to be circular, both for straight 
and for curved main track. If Eqs. 70 and 75 and the corre- 
sponding Eqs. 77 to 80 are solved for any given frog, it is found 
that the lead, when using straight switch-rails and straight frog- 
rails, is considerably less than when using circular lead-rails 
throughout; also the curvature is considerably sharper. But 
stub-rail switches are obsolete and the mathematical solutions 
used for them cannot be utilized, even approximately, for point 
switches. If such a diagram as Fig. 144 is worked out in detail, 
as has been done in previous editions, it is found that 



348 RAILROAD CONSTRUCTION. § 307. 

(a) the lead (BF) is almost identical with that computed from 
Eq. 70 or 74, when the main line is straight. 

(6) the degree of curve (d) of the circular switch-rails would be 
very nearly equal to the degree of curve (d') of the circular 
switch-rails for a straight track minus the degree of curve (D) 
of the main track; or, d = d' — D. 

These statements are more exactly true when the degree of 
curvature of the main track is small. Even for a 10° curve on 
the main track the errors are not large. It has been found to be 
a needless refinement to compute the precise mathematical 
properties of the switch-rails from a curved main track, any- 
more than as given by the two principles stated above : There- 
fore 

(a) the length of the lead is assumed to be the same as that for 
a straight track, using the same frog, and 

(b) the degree of curve of the switch-rails is found as stated 
above — in principle (h). As the curvature of the main track 
sharpens, the curvature of the switch-rails becomes less until 
they become straight. For still sharper main track, the center 
of curvature is on the same side. This is illustrated in Fig. 145, 
if we consider the sharper curved track to be the main track 
and the easier curve the switch. The above rule is still appli- 
cable, the algebraic sign of the result showing the location of the 
center. 

307. Design for a turnout from the INNER side of a curved 
track. As in the previous section. Fig. 145 illustrates the dia- 




FiG. 145. 



gram for circular lead rails. It may be shown that the degree 
of the turnout (d) is nearly the sum of the degree of the main 



§308. 



SWITCHES AND CROSSINGS, 



349 



track (D) and the degree (d') of a turnout from a straight track 
when the frog angle is the same. The discrepancy in this case is 
somewhat greater than in the other, especially when the cm-va- 
ture of the main track is sharp. If the frog angle is also large, 
the curvature of the turnout is excessively sharp. If the frog 
angle is very small, the liabihty to derailment is great. Turn- 
outs to the inside of a curved track should therefore be avoided, 
unless the curvature of the main track is small. 

308. Connecting curve from a straight track. The *^ con- 
necting curve " is the track lying between the frog and the side 




Fig. 146, 



track where it becomes parallel to the main track (FS in Fig. 
146 or 147). Call d the distance between track centers. The 
angle KOiS = F (see Fig. 146). Call r' the radius of the con- 
necting curve. Then 

d—g —K sin/ 



fr'-k) = 



vers F 
FQ = (r'-ig) sin F+K cos/ 



(81) 
(82) 



In these equations (and in several that follow) K is the distance 
from the theoretical point of the frog to the heel. The length, 
for each standard frog, is found in Table III, Part B. 



350 RAILROAD CONSTRUCTION. § 309. 

r 309. Connecting curve ^from a curved track to the OUTSIDE. 
When the main track is curved, the required quantities are the 
radius of the connecting curve from K to S, Fig. 147, and its 
length or central angle. 

The accm-acy of all these computations on switches and frogs 
in cm-ved main track is vitiated by the fact that the frog-rails 
are straight. The design might be mathematically more perfect 
if the main track curve were transformed into two curves on 
either side of the frog which had centers separated as far as the 




Fig. 147. 

length of the frog, but this would introduce a very great and 
needless complication and is never done. The more simple solu- 
tion is to consider that the frog-rail is a chord of the original 
curve, which (a) narrows the track gauge by an amount equal to 
the middle ordinate of that chord and which (6) is not tangent 
to the curve at either end. For all- ordinary curvature neither 
of these theoretical defects is vitally objectionable or even appre- 
ciable. In Fig. 147 KC is practically perpendicular to one frog- 
rail and KOi is exactly perpendicular to the other frog-rail. 
Therefore, the\ngle CKOi equals the frog angle F. While the 
following calculations are amply precise for practical purposes, 
the discrepancy from strict mathematical accuracy should be 
noted and properly valued. 
In the triangle CSK 

CS+CK:CS-CK::tsiJi i(CKS+CSK):tsin i{CKS-CSK); 

but i{CKS+CSK) =90-^4^; and, since the triangle OiSK is 
isosceles, ^lCKS-CSK) = iF; 



§310. 



SWITCHES AND CROSSINGS. 



351 



.-. 2R-{-d+K sin F : d-g-K sin F ::cot ixp : tan §F 

I'.cotiF : tan J^; 

2n(d-g-K sin F) 

.*. tan h^l/ = — 

'^ 2R+d-i-K sin F 

From the triangle COiK we may derive 

r-ig : R-\-ig+K sin F :: sin xp : sin {F+xp)] 

sin \l/ 



(83) 



Also 



T-\g^{R+hg+KsmF). 

sm (r +^; 

X^=2(r~§^)sinKi^+\^). . . 



(84) 



(85) 



310. Connecting curve from a curved track tp the inside. 

As above, it may readily be deduced from the triangle CKS (see 
Fig. 148) that 




Fig. 148. 

CK-^CS : CK-CS:: tan ^{CSK-^CKS) : tan i(CSK-CKS); 
(2R-d-KsinF) : {d-g-K sin F:: cot JiA : tan |F; 

2n(d-^-i^sin F) 



tan i0= .... 

From triangle COiK, 

OyK : Ci^::sin ^/^ : sin {F-xp)] 
(r-ig) : (i2-|sf-i^ sin F) ::sin ^ : sin (F-ip); 

sin t/' 



(86) 



(.r-ig) = iR-ig-KsmF) 



sin (F-«^) 



(87) 



352 
Also 



RAILROAD CONSTRUCTION. 



KS==2{r-ig)smi{F-^)' 



§310. 



(88) 




Fig. 149. 



Two other cases are possible, (a) r may increase until it 
becomes infinite (see Fig. 149), then F = \p. In such a case 
we may write, by substituting in Eq. 86, 



2R-d-K sin F = ^n''{d-g-K sin F). 



(89) 




Fig. 150. 

This equation shows the value of R which renders this case pos- 
sible. (6) xp may be greater than F. As before (see Fig. 150). 

{2R-d-K sin F) : (d-g-K sin F) ::cot ix/y : tan ^F; 

2n{d-g-K sin F) 



tan ixp 
the same as Eq. 86, but 



2R-d-KsmF 



ir+ig = {R-ig-Ksmf) 



sin \l/ 



sin i^p-F) 



(90) 



§311 



SWITCHES AND CROSSINGS. 



353 



Problem. To find the dimensions of a connecting curve run- 
ning to the INSIDE of a curved main track; number 9 frog, 4° 30' 



curve, d = lS\ g = 4:' 


8i". 




Solution. 
[Eq. 86] d =13.000 
5.816 


i^=10'0"i^sinF =1.108 
a =4.708 


log 2n =1.25527 


7.184 

i2 =1273.6 

272=2547.2 

(d+K sin F) =14.108 


5.816 

2i2 -d-X sin F =2533.1 
log =3.40365 
co-log =6.59635 


log 7.184 =0.85636 

co-log =6.59635 
log tan i|/'=8.70799 




« 


1,^=2° 55' 20" 
^=5° 50' 40" 
F =6° 21' 35" 



Since F>4/, we must use Eq. 87, rather than Eq. 90. 



i^ =2.354 
K sin F= 1.108 



i? -la -i^ sin F = 1270.1 



(F-,/.) =1855"; log 



= 3.462 



= 3.26834 
4.68557 



F-^=0° 30' 55' 

log =3. 10384 
log sin ^=9.00787 



7.95391 
co-log =2.04608 



co-log =2.04608 



r-y=U381.2 
r =14383.5 
d=0°24' 



4.15779 



[Eq. 88]. 

UF-^) =927.5"; log=2. 96731 

4.68557 



sin^i^-^) =7.65289 



2 0.30103 

r-lg 4.15779 
7.65289 



iTS =129.33 



2.11171 



311. Crossover between two parallel straight tracks. (See 
Fig. 151.) The turnouts are as usual. The cross-over track 
may be straight, or it may be a reversed curve. The reversed 
curve shortens the total length of track required, but is somewhat 
objectionable. The first method requires that both frogs 
must be equal. The second method permits unequal frogs, 
although equal frogs are preferable. The length of straight 
crossover track is FiT, 



354 









I 

1 

1 
1 

J_ 


z 

Di 


D2 

/ 

1 
1 
1 

1 

1 _ 


T 




J 


1 


a 


1 



RAILROAD CONSTRUCTION. § 312 

i^iT sin Fi+gf COS Fi = d-gf; 
d-g 



FiT=- 



sin Fi 



^ cot Fin 



(91) 



The total distance along the track may 
be derived as follows: 

DZ=DiFi+D2F2+F2Y 

= DFi+D2F2+XY-XF2; 



XY = (d-g) cot Fi; 
XF2 = g-^smF2; 

DiZ = 2DiFi-{-{d-g)cotFi 



g 



sin F2 



.(92) 



Fig. 151. 



312. Crossover between two parallel curved tracks. Using 
a straight connecting curve. This solution has limitations. 

If one frog (Fi) is 



chosen, F2 must be 
determined, being a 
function of Fi. If 
Fi is less than some 
limit, depending on 
the width {d) between 
the parallel tracks, 
this solution becomes 
impossible. In Fig. 
152 assume Fi as 




Fig. 152. 



known. Then KiN = g sec Fi. In the triangle NOK2 we 
have 

sin NK2O : sin K2NO ::N0 : K2O; 

smK2NO = cosFi; iVX2O = 90°+F2; 
.-.sin iVZaO = cos F2. • 

NO =R+id-hg-Ki sin Fi-g sec Fi; K20 = R-id-\-i9 

+K2 sin F2; 

„ R+jd-^g-Ki si n Fi-g sec Fi 

.-. cos F2 = cos Fi — rm — r^ — - — ^ • • ^^^^ 

R-id-{-ig+K2smF2 



§312. SWITCHES AND CROSSINGS. 355 

The solution of this equation involves the frog angle F2, which is 
the angle sought, but there is little error in considering in this 
solution that K2 sin F2 is numerically equal to Ki sin i^i and 
solving accordingly. If the computed value of F2 is very different 
from Fi, it would be more precise to recompute Eq. 93 by sub- 
stituting for K2 sin F2 the more exact quantities obtainable 
from the first trial solution. The relative position of the frogs 
Fi and F2 may be determined as follows : 

iVO2K = 180°-(90°-Fi)-(90°+F2)=Fi-F2. 
Then GFi =2{R-^\d- Ig) sin § (Fi - F2) +Zi cos Fu . (94) 

There is a theoretical, but practically inappreciable, inac- 
curacy in Eq. 94, since the chord GFi is really the sum of two 
chords of which one is the chord from the point G to the point 
where ON produced intersects the gauge line. After locating 
(r, the point radially opposite, on the outer gauge line of the inner 
track, may be located, from which the frog-point F2 is located 
at a distance of K2 cos F2. Note that these frog-points referred 
to are the theoretical points. Due allowance must be made 
during location for the ^' frog bluntness.'^ 

In general, the value of F2 computed from Eq. 93 is not the 
angle of any standard number-frog, and a strict compliance with 
theory would require that the frog should be made to order. 
This is needlessly expensive and the nearest size frog may gen- 
erally be used without appreciable error. 

Example. A crossover between parallel tracks on a 6° curve, 
the track spacing d being 13 feet. Fi assumed a No. 9 frog. 

[Eq. 93] 

/i;=955.37 i^=2.35 1^1=10 ft. 

fd= 6.5 irismFi=l.ll sin Fi = .11077 

- 8.20 8.20 
953.67 log=2. 97940 



jR =955.37 
\g= 2.35 
Ki sin F2 = 1.11 (assumed = to Ki sin Fi) 

958.83 
-^d= -6.5 

952.33 log = 2.97879 

0.00061 
log cos Fi . . 9 . 99732 

Fi = 5° 35' 30" log cos 5° 35' 30" 9 . 99793 



356 RAILROAD CONSTRUCTION. §313. 

This angle is within 8 minutes of the angle of a No. 10 frog, 
which could be used without appreciable error. The point K2 
would be shifted laterally .023 foot, or about J inch, but there 
would be no visible irregularity in alinement. 

NOK2 =Fi -F2 =6° 21' 35" -5° 35' 30" =0° 46'. 
[Eq. 94] 22+^^=961.87 



^^=-2.35 


2 . . log =0.30103 


959.52 . 


log =2.98205 




sin ^N0K2 =sm 0° 23' =7.82545 




12.84 log =1.10853 




2ricosFi= 9.94 




GFi =22.78 



It is instructive to note that if the same crossover problem is 
worked out for a straight track, as in § 311, using No. 9 frogs 
on both tracks, the distance between frog points, measured 
parallel with the track, is nearly the same as in the above prob- 
lem, especially when the distance 12.84, measured on the outer 
track, is reduced by bringing it in to the center line. This is 
analogous to the statement, previously made, that the lead of a 
switch on a curved track is nearly the same as that for a straight 
track. 

It is theoretically possible to find two standard frog angles 
which may be so located that the connecting curve consists of 
straight lines and circular curves, which connect tangentially, 
making perfect alinement, but such methods are very com- 
plicated and the above method is sufiiciently exact for practical 
purposes. 

313. Practical rules for switch-laying. A consideration of 
the previous sections will show that the formulae are compara- 
tively simple when the lead-rails are assumed as circular; that 
they become complicated, even for turnouts from a straight 
main track, when the effect of straight frog and point rails is 
allowed for, and that they become hopelessly complicated when 
allowing for this effect on turnouts from a curved main track. 
It is also shown (§ 306) that the length of the lead is practically 
the same whether the main track is straight or is curved with 
such curves as are commonly used, and that the degree of curve 
of the lead-rails from a curved main track may be found with 
close approximation by mere addition or subtraction. From 
this it may be assumed that if the length of lead (L) and the 



§313. 



SWITCHES AND CROSSINGS. 



357 



radius of the lead-rails (r) are computed from Eq. 77 and 80 for 
various frog angles, the same leads may be used for curved main 
track; also, that the degree of curve of the lead-rails may be 
found by addition or subtraction, as indicated in § 306, and that 
the approximations involved will not be of practical detriment. 
In accordance with thig plan Table III has been computed from 
Eq. 77, 78 and 80. The leads there given may be used for all 
main tracks, straight or cin-ved. The table gives the degree of 
curve of the lead-rails for straight main track; for a turnout to the 
inside, add the degree of curve of the main track; for a turnout 
to the outside y subtract it. 

But there are complications resulting from practical and eco- 
nomical switch construction. A committee of the A. R. E. A., 
in 1910, adopted certain standards 
in details, which, when applied to 
Eqs. 77 to 80 give the values for 
switch dimensions as quoted in the 
second section of Table III. They 
adopted four lengths of switch-rails. 
In each case the " point '^ is always 
J" thick. The gauge line at the 
other end is always to be placed 
6i" from the gauge line of the main 
rail, and the planing is so done that 
when in this position the switch- 
rail lies against the main rail. 
Therefore the angle a is always an 
angle whose sine equals 6 inches (or 
0.5 foot) divided by the length of 
the switch-rail in feet. In Fig. 153, 
the point D is not on the gauge line of the main rail but at a 
point i" away from it, and the point M 6J" away from it. The 
straight rail BF consists of a point-rail at one end, the ^'closure 
rails," and one of the wing rails of the frog at the other end. 
The closure rails will in general consist of one rail cut to a com- 
puted length and one or more rails from 24 to 33 feet long, the 
lengths being in even feet. The curved rail DF will also con- 
sist of a point-rail, a frog wing-rail, and one or more lengths 
of closure rail, but the closure rails in this case are slightly longer 
than those for the straight rail. Since it is always practically 
easier to measure to the ^'actual point '^ of a frog (see Fig. 134), 



\ J'. 


K' 

\ i 
\\r 

\ \ > 

^ V 

M' 

\\ 

\\ 

\\ 




M 


B 


D 



Fig. 153. 



358 



RAILROAD CONSTRUCTION. 



§313. 



rather than to the theoretical point, Table III gives the distance 
L', which is the distance L, =jBF, plus the "frog bluntness," 
which is found by multiplying |" ( = 0.0417 foot) by the irog 
number. 

The curvature for a curved switch-rail (for a straight track) 
is most readily determined by measuring off a series of ordinates 
whose origin is at the switch-point Z>, Fig. 153, the points being 
the center and the quarter points of the actual curve. These 
ordinates, as computed on the basis of '^ practical leads," by the 
A. R. E. A. committee, are quoted below. It should be remem- 
bered that the system of practical leads usually involves a very 
short tangent adjacent to either M or J, and that the hne MJ 
for " practical leads " is not entirely an arc, 

TABLE XXV. — RECTANGULAR COORDINATES TO THE QUARTER 
AND CENTER POINTS ON THE GAUGE SIDE OF CURVED RAIL, 
REFERRED TO POINT OF SWITCH-RAIL AS ORIGIN. 



Frog 
No. 


Measured along main rail. 


Measured perpendicular to 
main rail. 


X 


Xi 


X2 


Y 


Yi 


Yi 


4 
5 
6 

7 
8 
9 

9^ 
10 
11 

' 12 
15 
16 

1 18 
20 
24 


17.74 

17.78 
19.07 

26.72 
28.37 
28.75 

30.31 

30.28 
40.74 . 

43.99 
55.49 
58.16 

58.73 
61.84 
67.82 


23.44 
24.54 
27.13 

36.93 
39.91 
40.98 

43.35 
44.05 
56.47 

60.65 
77.98 
81.76 

84.46 

90.21 

100.21 


29.75 
31.27 
35.15 

47.11 
51.45 
53.19 

56.37 
57.81 
72.19 

77.28 
100.45 
105.35 

110.10 
118.59 
132.59 


0.97 
0.95 
1.01 

0.97 
1.02 
1.02 

1.06 
1.06 
1.08 

1.15 
1.01 
1.04 

1.04 
1.08 
1.27 


1.67 
1.61 
1.74 

1.71 
1.78 
1.76 

1.82 
1.84 
1.84 

1.90 

1.78 
1.82 

1.82 

1.88 
1.97 


2.79 
2.62 
2.72 

2.74 
2.91 
2.75 

2.83 
2.85 
2.87 

2.91 

2.84 
2.87 

2.86 
2.93 
3.00 



If the position of the switch-block is definitely determined, 
then the rails must be cut accordingly; but when some freedom 
is allowable (which never need exceed 16.5 feet and may require 
but a few inches), one rail-cutting may be avoided. Mark on 
the rails at B, F, and D; measure off the length DN and locate 
the point M at the distance H from N, If the frog must be 
placed during the brief period between the running times of 



§ 314. SWITCHES AND CROSSINGS. 359 

trains, It will be easier to joint up to the heel of the frog (the 
point K'j Fig. 153), a piece of rail, the farther end of which will 
just reach the next joint and also joint up to the toe of the frog 
the straight closure rail and the point-rail. Then, when all is 
ready, the rails are loosened from the ties back to B, the joint 
beyond the frog is removed and the whole rail back to B is swung 
outward. The new combination is shoved into place and spiked, 
even the point-rail being temporarily spiked to hold it in place 
as a main track rail, until the other switch-rail and the tie rods 
can be placed. When the frog is thus in place, the point J 
becomes located. The curved closure rails, as called for in 
Table III, should prove to be just long enough, when properly 
curved, to fill in the gap between M and J, Using the proper 
pairs of values for X and Y as given above, the three values of X 
may be measured on the main track rail from the point D, and 
the corresponding offsets will give points on the ciu-ved switch- 
rail. The old main track rail which was bent outward from B 
may be utilized as the other switch-rail and set to gauge from the 
rail just located. 

Example. — Given a main track on a 4° curve — a turnout to 
the outside, using a No. 9 frog; gauge 4' 8J"; TF = 6'.00; ^ = 6J"; 
S = W 6" and a = l° 44' 11" Then for a straight track r 
would equal 605.18 [d = 9° 28' 42"]. For this curved track d 
will be nearly 9° 29' -4° = 5° 29', or r wiU be 1045.3. U for a 
straight track would be 72.28, and is here considered to be the 
same. The closure rails have a total arc length of 49.59, and 
will here be taken the same. Note that the curved and straight 
closure rails each have odd lengths which are made by one cut 
of a 33-foot rail. This avoids all rail waste and also one rail- 
cutting and the boring of holes. 

314. Slips. Track movements in crowded yards are facili- 
tated by using " slips '^ (see Fig. 154), which may be '' single " 
or *' double." The crossing of two rails is done either by oper- 
ating two movable rails or by using fixed " frogs," but a com- 
parison of the continuity of the running rails, using ordinary 
frogs (see Fig. 134) and these frogs, will show their radical 
difference. These slips can be used for frog angles from No. 6 
to No. 15. The levers are so connected that the several opera- 
tions necessary to set the rails for any desired train movement 
are accomplished by one motion, 



360 



RAILROAD CONSTRUCTION. 



§314. 




Fig. 154. — Single and Double Slips. 



§ 316, 



SWITCHES AND CROSSINGS. 



361 




CROSSINGS. 

315. Two straight tracks. When two 
straight tracks cross each other, four frogs 
are necessary, the angles of two of them 
being supplementary to the angles of the 
other. Since such crossings are sometimes 
operated at high speeds, they should be 



very strongly constructed, and the angles should preferably be 
90° or as near that as possible. The frogs will not in general 
be " stock " frogs of an even number, especially if the angles are 
large, but must be made to order with the required angles as 
measured. In Fig. 155 are shown the details of such a crossing. 
Note the fillers, bolts, and guard-rails. 

316. One straight and one curved track. Structurally the 
crossing" is about the same as above, but the frog angles are all 
unequal. In Fig. 156, R is known, and the angle M, made by 



362 



RAILROAD CONSTRUCTION. 



§317. 



the center lines of the tracks at their point of intersection, is 
also known. m = NCM, NC = R cos M. 

R cos M+ig 



{R-hg) cos Fi = NC-\-ig] :. cos Fi = 
Similarly cos r 2 = 7r7~x ^ cos 1 3 



^g 



R+hg 



cos F4 = 



R 

R cos M — ^g 

R+ig ' 
R cos M — \g 



R-hg 

F^F,= (R-\-igysinF^-(R-ig) sin F,; 1 
HF4 = (R-ig){sm F,-sm Fi). J 




(95) 



(96) 



Fig. 156. 

317. Two curved tracks. The four frogs are unequal, and 
the angle of each must be computed. The radii Ri and R2 
are known; also the angle M. ri, r2, r^ and r^ are therefore 
known by adding or subtracting Jgr, but the lines are so in- 
dicated for brevity. Call the angle MCiC2 = Ci, the angle 
MC2Ci = C2y and the line CiC2^c. Then 
i(Ci+C2)=90°-W 

^""^ tani(Ci-C2)=cot|M|p^'. . . . (97) 



R^-\-R] 



Ci and C2 then become known and 



C = C1C2 = R2'~. ~ 

Sin Ci 



(98) 



§317. 



SWITCHES AND CROSSINGS. 



363 



In the triangle FiCid, call Kc+ri+r^) =si; Si = \{c+ri+rd; 




Fig. 157. 



S3 = Kc+ri+r3); and S4 = Kc+r2+r3). Then, by formula 29, 
Table XIV, 

2(si-ri)(si-r,) 



Similarly 



vers F^ 



r.r^ 



1'4 



vers F^ = -^-^ — j^ , 



r->r 



2' 4 



vers i^Q 



vers 



2G^3-n)(g3-n) 



^ ^ 2(g4-r2)(g4-r3) 



sinCiC2^,=sini^,^%- 



(99) 



sin C1C2F2 =sin i^g— ; 
c 

&va.Ffi^C^='Wa.Fp^•, 



(100) 



sin i^jCiCj =sinF2^, 
c 



/. F,C,F,=F,Cfi,-F,Cfi,) ..... (101) 
i from which the chords F^F^ and F^F^ are readily computed. 



^ 



364 



RAILROAD CONSTRUCTION. 



§317, 



F1F2 and i^2^4 ^^^ nearly equal. When the tracks are straight 
and the gauges equal, the quadrilateral is equilateral. 

Problem. Required the frog angles and dimensions for a cross- 
ing of two curves (Z)i=4°; 2)2 = 8°) when the angle of their tan- 
gents at the point of intersection =62° 28' (the angle M in 
Fig. 157). 



Solution 



Eq. 97. 



i?i = 1432.7; i^2 = 1910.1; 
Ti =J?2 + i9' = 1910. 1+2. 35 = 1912. 45; 
^2 =-R,-k = 1910. 1-2. 35 = 1907.75; 
^3 =i^i + isr = 1432. 7 + 2. 35 = 1435. 05; 
U =2^1 -49^ = 1432. 7 -2. 35 = 1430.35. 

log cot JM =0.21723 
R,-R,=477A; log =2.67888 

7^2 + ^^1=3342.8; log = 3. 52411; co-log = 6 . 47589 

J(Ci-C2) =13° 15' 07"; tan 13° 15' 07" =9.37200 
i(C, + C2) =58° 46' [i{C, + C2) =90° -JM] 





Ci=72°01'07" 




C2 = 45°30'53" 


Eq. 98. 


logi?2=3.28105 




log sin M= 9. 94779 


log sin Cj 


=9.97825; co-log =0.02 175 


c = C,C, = 1780.7; 


log(7i(72=3.25059 


Eq. 99. 




c=1780.7 ' 


c = 1780.7 


c=1780.7 


c=1780.7 


ri = 1912.45 


r2= 1907.75 


ri = 1912.45 


r2= 1907.75 


r4= 14.30. 35 


r4= 14.30. 35 


r3= 1435.05 


r3= 1435.05 


2|5123.50 


2|5118.80 


2|5128.20 


2|5123.50 


51 = 2561.75 


S2 = 2559.40 


S3 = 2564. 10 


S4 = 2561.75 


«i-ri= 649.30 


»2-r2= 651.65 


S3 — ri= 651.65 


S4-r2= 654.00 


«i-r4=1131.40 


-r4= 1129.05 


53-^3=1129.05 


S4-r3=n26.70 






log 2 = 0.30103 




(si-ri); log 649.30 = 2.81244 




(si-r4); log 1131.40 = 3.05361 


ri = 1912.45; log=3.2S159; 


co-log = 6. 71841 


U = 1430 . 35 ; log = 3 . 15544 ; 


co-log = 6. 84456 


Fi=62° 25' 31"; 




log vers 62° 


25' 31" = 9. 73006 




log 2 = 0.30103 




(«2-r2); log 651.65 = 2.81401 




(s2-r4); log 1129.05 = 3.05271 


r2=1907.75; log = 3. 28052^ 


• co-log = 6. 71948 


r4=1430.35; log = 3. 15544s 


co-log = 6. 84456 


F2 = 62° 33' 55": 




log vers 62° 


33' 55" = 9. 73180 



§317. 



SWITCHES AND CROSSINGS. 



365 



ri = 1912.45; log = 3. 28159; 
r3= 1435.05; log = 3. 15686; 
F3 = 62° 21' 57"; 



r2=1907.75; log = 3. 28052; 
rs = 1435 . 05 ; log = 3 . 15686 ; 
F4 = 62° 30' 14''; 



log 2 = 0.30103 

(s3-ri); log 651.65 = 2.81401 

(53-^3); log 1129.05 = 3.05271 

co-log = 6. 71841 

co-log = 6. 843 13 

log vers 62° 21 ' 57'" = 9 .72930 

log 2 = 0.30103 

(s4-r2); log 654.00 = 2.81558 

(^4-^3); log 1126.70 = 3.05181 

co-log = 6. 71948 

co-log = 6.84313 

log vers 62" 30' 14" = 9. 731 03 



As a check, the mean of the frog angles = 62° 27' 54 ', which is within 6" of 
the value of M, 



Eq. 100. 



(7iC2F4 = 45° 37' 51"; 



log c = 3. 25059; 



Ci 02^2 = 45° 28' 17"- 

^2^2^4 = 45° 37' 51" -45* 28' 17" = 0° 09' 34". 



log sin 1^4 = 9.94794 

log r3 = 3. 15686 

co-log c = 6. 74940 

sin CiCzFi = 9 . 85421 

log sin 1^2 = 9.94818 

log r4 = 3. 15544 

co-log c = 6. 74940 

sinC7iC2F2 = 9.85303 



log 2 = 0.30103 

log r2 = 3. 28052 

i(0° 09' 34" ) = 0° 04' 47" ; log sin = (| • 6|557 

log ^^1^4 = 0.72500 

^ sin Fi = 9. 94763 

log ri =3.28159 

co-log c = 6. 74940 

sin FiCi Co = 9 . 97863 

sin F2 = 9. 94818 

log r2 = 3. 28052 

co-log c = 6. 74940 

sinF2CiC2 = 9.97811 

log-2 = 0. 30103 
log r4 = 3. 15544 

i(0° 12' 44" ) = 0° 06' 22" ; log sin = /'^ • ^^^57 

V 2. 58206 
FiF,=5.298; logFiF,- 0.72411 

As a check, 1^2^* and ¥^¥2 are Yery nearly equal, as they should 
be. 



^■2^4 = 5.309 ; 
Eq. 101. 

FiCiC2 = 72° 10" 22"5 



Fi,CiC2=71°57'38"l 

F,CiF2 = 72° 10' 22" -71° 57' 38" = 0° 12' 44" . 



366 KAILROAD CONSTRUCTION. § 317. 

The foregoing problems on switches, connecting curves and 
crossings cover only a few of the most common of the problems 
encountered by the engineer. For the solution of a far wider 
range of problems, the engineer is referred to " Track Formulae 
and Tables," by S. S. Roberts. [Wiley & Sons.] 



ite 



CHAPTER XII. 
MISCELLANEOUS STRUCTURES AND BUILDINGS. 
WATER-STATIONS AND WATER-SUPPLY. 

318. Location. The water-tank on the tender of a locomo- 
tive has a capacity of from 3000 to 10000 gallons — sometimes less, 
rarely very much more. The consumption of water is very vari- 
able, and will correspond very closely with the work done by 
the engine. On a long down grade it is very small; on a ruling 
grade, going up, using full stroke, an engine with 28-in. cylinders, 
30-in. stroke, 180 lbs. boiler pressure, will use 4.59 lbs. of steam, 
or water, per stroke or 18.36 pounds per revolution. With 
63-in. drivers, the circumference is 16.5 feet and there will be 
320 revolutions per mile. The engine wiU use 5875 lbs. or 700 
gallons of water per mile. This engine has a tank capacity of 
9000 gallons, which would permit rimning about 12 miles at full 
stroke. But it is very rare that a locomotive must work for 
such long distances at full stroke. After starting and attaining 
fuU normal speed, the valves may be set to cut off at one-fourth 
stroke, or even at one-fifth or one-sixth for high speed running. 
With ordinary grades, such an engine might average 200 gallons 
per mile, in both directions. A quoted numerical case is that of 
a 106-ton engine using 7,500,000 gallons during an annual mileage 
of 45000 miles. This means an average of 167 gallons per mile. 
Observations were taken in 1910, on the N. Y. Central R.R., 
where the grades are moderate, showing that the heavy pas- 
senger trains of eight to twelve cars consumed 80 to 100 gallons 
of water per mile and that freight trains of about fifty loaded 
cars consumed from 110 to 130 gallons per mile. These figures 
are far less than those given above, but the grades on the N. Y. 
Central are very light. 

Freight engines, running at lower speeds and longer cut-off, 
require more frequent water-tanks than passenger engines. 
Even before a road is built, the water-tank requirements and the 
minimum spacing may be computed on the basis of the steam 
consumption (see § 454), of the locomotives with which it is 
expected to handle the estimated trafiic of the road. Usually 
tanks will be located at intervals of 10 to 20 miles. 

367 



383 RAILROAD CONSTRUCTION. §319. 

In the early history of some of the Pacific railroads it was nec- 
essary to attach one or more tank-cars to each train in order to 
maintain the supply for the engine over stretches of 100 miles and 
over where there was no water. Since then water-stations have 
been obtained at great expense by boring artesian wells. The 
individual locations depend largely on the facility with which a 
sufficient supply of suitable water may be obtained. Streams 
intersecting the railroad are sometimes utihzed, but if such a 
stream passes through a hmestone region the water is apt to be 
too hard for use in the boilers. More frequently wells are dug or 
bored. When the local supply at some determined point is 
unsuitable, and yet it is necessary to locate a water-station there, 
it may be found justifiable to pipe the water several miles. The 
construction of municipal water-works at suitable places along 
the line has led to the frequent utiUzation of such suppUes. In 
such cases the railroad is frequently the largest single consumer 
and obtains the most favorable rates. When possible, water- 
stations are located at regular stopping points and at division 
termini. 

319. Required qualities of water. Chemically pure water is 
unknown except as a laboratory product. The water supplied 
by wells, springs, etc., is always more or less charged with cal- 
cium and magnesium carbonates and sulphates, as well as other 
impurities. The evaporation of water in a boiler precipitates 
these impurities to the lower surface of the boiler, where they 
sometimes become incrusted and are difficult to remove. The 
protection of the iron or steel of a boiler from the fierce heat of 
the fire depends on the presence of water on the other side of the 
surface, which will absorb the heat and prevent the metal from 
assuming an excessively high temperature. If the water side 
of the metal becomes covered or incrusted with a deposit of 
chemicals, the conduction of heat to the water is much less 
free, the metal will become more heated and its deterioration or 
destruction will be much more rapid. An especially common 
effect is the production of leaks around the joints between tubes 
and tube-sheets and the joints in the boiler-plates. Such injury 
can only be prevented by the application of one (or more) 
of three general methods— (a) the mechanical cleaning of the 
boilers, (h) the chemical purification of the water before its mtro- 
duction into the boiler, and (c) the use of some '' boiler com- 
pound " which is introduced directly into the boiler and which 



§ 320. MISCELLANEOUS STRUCTURES AND BUILDINGS. 369 ^ 

causes precipitation of the harmful ingredients as non-incrusting 
solids which can be readily blown out. 

320. Mechanical cleaning, as a sole dependence is impracticable 
except in the comparatively rare localities where the water is so 
" soft " that no incrusting deposits will be made and such pre- 
cipitation as does take plUce is of such a character that it is 
removable by blowing out the boiler. There are many rail- 
roads, especially the smaller ones, which do not give any chem- 
ical treatment to any of their engine water-supply, and yet 
which are not fortunate enough to obtain even approximately 
soft water. The only method by which such roads can prevent 
a great waste of heat and the rapid deterioration of boiler tubes 
and sheets is by frequent mechanical cleaning. 

321. Chemical purification before the water enters the boiler 
has the advantage of removing the troublesome ingredients, 
leaving nothing further to be done except the occasional removal, 
by blowing out, of the suspended matter or harmless matter 
precipitated by boiling. Sodium carbonate is the most common 
reagent. It is commercially sold as '^ soda crystals, sal soda, 
washing soda, Scotch soda, concentrated crystal soda, sesqui- 
carbonate of soda, crystal carbonate of soda, black ash, soda ash 
and pure alkali." Although often chemically impure, it can 
now readily be obtained with a purity of 97 to 99%. The chem- 
icals which are most common as incrustants are calcium and 
magnesium carbonates and sulphates. The effect of sodium 
carbonate on calcium sulphate is to produce soluble sodium sul- 
phate — which is non-incrustant — and calcium carbonate, which 
precipitates into a sludge at the bottom of the water softener 
tank. The action on magnesium sulphate is similar. When 
this is done in a purifying tank, the purified water is drawn off 
from the top of the tank and supplied pure to the engines. The 
precipitants are drawn off from the settling-basin at the bottom 
of the tank. This purification, which makes no pretense of 
being chemically perfect, may be accomplished for a few cents 
per 1000 gallons. There are manufacturers which make a spe- 
cialty of machinery, working more or less automatically, which 
introduces into the raw water a measured amount of chemical 
which, by analysis, has been calculated to be necessary with 
that particular quality of water. In spite of the automatic 
features, such machinery needs constant attention, and the 
water, both raw and treated, needs frequent analysis to 



370 



RAILROAD CONSTRUCTION. 



§321. 



insure efficiency, since 
change. 

Sodium hydrate 



the character of the raw water may 



or 



soda, 



has the same general 
quickly and 



caustic 
chemTcXeffect as sodium carbonate, and acts mo^ 

required per unit ot scaling ui ^ Manual of the Amer. 



by dividing the grains per g 



l>y<^-r%TlKo by Twelve In order to obtain the fuU 

T.B.E XXVI. QUANTITY O. P.K. B..™S KEQ.XB.D TO 
REMOVE ONE POUND OF INCRUSTING 
FROM THE WATER. 



REAGENTS 

OR CORROSIVE MATTER 



Incrusting or corrosive 

substance held m 

solution. 



Amount of reagent (pure). 



0.57-lb. lime plus 1.08 lbs. soda ash . . 

1.27 lbs. lime 

0.56-lb. Hme 

0.78-lb. soda ash 

0.96-lb. soda ash 

0.65-lb. soda ash 

0ll-lb'umeVus0.881bVsodaash.^^ 
059-lb lime plus l.U lbs. soda ash 
oil-lb. lime plus 0.72-lb. soda ash.. 



Sulphuric acid. • • • • ■ 
Free carbonic acid. . . • 
Calcium carbonate. . . . 

Calcium sulphate 

Calcium chloride 

Calcium nitrate • • • • • 

Magnesium carbonate 

Magnesium sulpha^te . 

Magnesium chloride . 

Magnesium nitrate. . • • l^-^g'J^s^bariSmbydrate 

Calcium carbonate..^. . .^3.15 ids. d^^.^^ hydrate 



Foaming] 

matter 
increased 



2' 62 lbs. barium hydrate. 
2.32 lbs. barium sulphate. 



1.45 lbs. 

None 

None 

1.04 lbs. 

1.05 " 
1.04 '• 
None 
1.18 lbs. 
1.22 " 
1.15 " 
None 
None 
None 
None 



Magnesium carbonate. 
Magnesium sulphate . . 
Calcium sulphate * . . . 

of soda ash, or lor reacting on either -^f ^ "^"^^"jtme plus 0.34 lb. of 
of barium hydrate performs the --^ o ai8 Ib^^f ^^P^^^^ 
soda ash. and the lime treatment can be correspona. g y 



§ 322. MISCELLANEOUS STRUCTURES AND BUILDINGS. 371 

order to obtain efficient treatment of water and reduce scaling 
matter to the minimum. 

322. Foaming and priming. This phenomenon is the foaming 
or frothing of the water for a considerable height above its normal 
level in the boiler. The rapid flow of steam into the steam pipe 
in the dome mechanically carries some of this froth into the steam 
pipe and causes water to accumulate in the steam pipe and also 
in the cylinders, with considerable resulting loss in efficiency. 
Foaming in treated water is largely due to the presence of sodium 
salts as a result of treatment for incrusting sulphates, and this 
constitutes one of the objections to the use of soda in treating 
water. The presence of suspended matter in the water ag- 
gravates and even causes foaming. The constant withdrawal of 
the water from the boiler leaves these suspended solids in the 
boiler and they keep accumulating until the concentrations reach 
a critical point, which is about 100 grains per gallon. Beyond this 
point foaming will be experienced unless the water is changed, 
which is done by a systematic blowing-off and an occasional com- 
plete blowing-down and washing. But blowing-off involves the 
wastage of water which has been heated to boiler temperature 
and which has, perhaps, been chemically treated. Even the raw 
water costs something, perhaps several cents per 1000 gallons. 
The blowing-off required to keep the concentration below the 
proper limit may be so excessive that some anti-foaming agent 
may be necessary. The required effect is physical rather than 
chemical, the object being to reduce the surface tension, which 
is done chiefly by the use of oils, petroleum and castor oil being 
used. Tannic acids are also used for such a purpose. , 

323. Boiler compounds. Chemical treatment at special plants 
along the road is unquestionably the most efficient method, but it 
is costly. The use of boiler compounds, often patented, obviates 
the erection of any plant, but, since the water at each water- 
supply station has its own characteristics and it is impracticable 
to vary the chemicals used at each supply-station according to 
the character of the water, the treatment is very imperfect. 
Minute instructions to enginemen to introduce definite amounts 
of chemical at each water-station have proved unsatisfactory 
and impractical. Sometimes the chemical is mixed with enough 
water to partially suspend it and then it is thrown into the ten- 
der tank, this method having the advantage that a considerable 
part of the precipitation takes place promptly and the sludge 



372 



RAILROAD CONSTRUCTION. 



§324. 



never enters the boiler. Sometimes a siphon attached to the 
feed-pipe outside of the injector, or, perhaps, a special injector, 
leads from a reservoir in which the chemical, suspended in water, 
has been placed. Sometimes a stick or ^* brick " of the chemical 
is placed directly in the boiler, through a hand-hole, during one 
of its periodical cleanings. In spite of the inefficiency of the 
method, 70% of replies to a circular inquiry reported the use of 
some kind of boiler compound. The chemicals used, some of 
which are patented compounds, are in general the same as those 
used in the outside chemical plants. Sodium carbonate is the 
most common constituent. 

324. Tanks. Whatever the source, the water must be led 
or pumped into tanks which are supported on frames so that the 

bottoms of the tanks are 
about 12 feet above the 
rails. Wooden tanks hav- 
ing a diameter of 24 feet, 
16 feet high, and with a 
capacity of over 50000 
gallons, are frequently 
employed. Iron or steel 
tanks are also used. 

In Table XXVII is 
shown the capacity of 
cylindrical water-tanks in 
United States standard 
gallons of 231 cubic inches. 
From this table the di- 
mensions of a tank of 
any desired capacity may 
tanks are sometimes used 
The smaller sizes 




Fig. 158. — Water-tank. 



readily be found. Two or more 
rather than construct one of excessive size, 
shown in the table are of course too small for ordinary use, 
but that part of the table was filled out for its possible con- 
venience otherwise. On single-track roads where all engines 
use one track the tank may be placed 8' 5^' from the track 
center; this gives sufficient clearance and yet permits the use 
of a single swinging pipe which will reach from the bottom 
of the tank to the tender manhole. In Fig, 158 is illustrated 
one form of wooden tank. They are preferably manufactured 
by those who make a special business of it and who by the use 



§ 325. MISCELLANEOUS STKUCTUKES AND BUILDINGS. 373 



TABLE XXVII — CAPACITY OF CYLINDRICAL WATER-TANKS IN 
UNITED STATES STANDARD GALLONS OF 231 CUBIC INCHES. 



Height 






Diameter of tank in feet. 






in 
feet. 


10 


12 


14 


16 


18 


20 


22 


24 


6 
7 
8 
9 
10 


3525 
4113 
4700 
5288 
5875 


5076 
5922 
6768 
7614 
8460 


6909 

8061 

9212 

10364 

11515 


9024 
10528 
12032 
13536 
15041 


11421 
13325 
15229 
17132 
19036 


14101 
16451 
18801 
21151 
23501 


17062 
19905 
22749 
25592 
28436 


20305 
23689 
27073 
30457 
33841 


11 
12 
13 
14 
15 


6463 
7050 
7638 
8225 
8813 


9306 
10152 
10998 
11844 
12690 


12667 
13819 
14970 
16122 
17273 


16545 
18049 
19553 
21057 
22561 


20939 
22843 
24746 
26650 
28554 


25851 
28201 
30551 
32901 
35251 


31280 
34123 
36967 
39810 
42654 


37225 
40609 
43994 
47378 
50762 


16 
17 
18 
19 
20 


9400 

9988 

10575 

11163 

11750 


13536 
14383 
15229 
16075 
16921 


18425 
19576 

20728 
21879 
23031 


24065 
25569 
27073 

28577 
30081 


30457 
32361 
34264 
36168 
38071 


37601 
39951 
42301 
44652 
47002 


45498 
48341 
51185 
54028 
56872 


54146 
57530 
60914 
64298 
67682 


21 
22 
23 
24 
25 


12338 
12925 
13513 
14101 
14688 


17767 
18613 
19459 
20305 
21151 


24182 
25334 
26485 
27637 
28789 


31585 
33089 
34593 
36097 
37601 


39975 
41879 
43782 
45686 
47589 


49352 
51702 
54052 
56402 
58752 


59716 
62559 
65403 
68246 
71090 


71067 
74451 
77835 
81219 
84603 



of special machinery can insure tight joints. When it is incon- 
venient to place the tank near the track, or when there is a 
double track, a ^' stand-pipe ^^ becomes necessary. See § 327. 
One of the most difficult and troublesome problems is to prevent 
freezing, particularly in the valves and pipes Not only are the 
pipes carefully covered but fires must be maintained during cold 
weather. When the pumping is accompHshed by means of a 
steam-pump, supplied from a steam-boiler in the pump -house 
under the tank, coils of steam-pipe may be employed to heat the 
water or to heat the pipes Partial protection may be obtained 
by means of a double roof and double bottom, the spaces being 
filled with sawdust or some other non-conductor of heat. 

325. Pumping, (a) Steam-pumps. When coal is very cheap 
or '' when 100 lbs. of coal in the pumphouse is cheaper than one 
gallon of fuel oil in the storage tank,'' and especially when steam 
can be procured from the railroad repair-shop plant, direct-acting 
steam pumps may be preferable and more economical, but they 
always require skilled attendance. (6) Gasoline-engines. These 
have been so highly developed in recent years that they are very 
efficient and are nearly ^' fool-proof," so that they may be oper- 



374 



RAILKOAD CONSTBUCTION. 



§325. 





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al. 

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6 " 
al. 




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fe 




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asolin< 
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uel oil 
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ated by unskilled labor, 
although skilled attention 
is periodically necessary. 
But the rising cost of gas- 
oline has directed atten- 
tion to other fuels, (c) Oil- 
engines. Crude petroleum, 
when refined, will give off 
approximately the follow- 
ing: Ether, 2%; gasoline, 
6%; naphtha and benzine, 
8%; kerosene, 44%; 39° 
poWer distillate, 10%; gas 
oil, 10%; lubricating oils 
and petrolatum, 15%, and 
^' slops" 5%. The ^' fuel 
oil," as supplied for oil 
engines, is a mixture of the 
slops with enough of some 
other constituent, usually 
the " power distillate," 
which is at the time the 
cheapest, to make the 
gravity of the mixture 
about 29°. The fuel oil 
costs approximately 40% 
as much as gasoline. Gas- 
oline engines have been 
converted into fuel oil 
engines by attaching a 
mixing chamber in which 
the oil is heated by the 
exhaust of the engine. 
(d) Gas-engines, using 
natural gas. Where nat- 
ural gas is available at 
25 cents per 1000 cu.ft. or 
less, it is an economical 
fuel, (e) Electric power. 
Where this is obtainable 
at a low rate, it may be 



§ 326. MISCELLANEOUS STRUCTURES AND BUILDINGS. 375 

a cheaper source of power than steam, gasolene or fuel oil. The 
electric motor either operates a centrifugal pump, or a slow-speed 
motor is direct- connected to a triplex reciprocating pump. 

A Committee of the Amer. Rwy. Eng. Assoc, reported in 1915 
the preceding (see p. 374) tabular costs of pumping 240,000 
gallons per day of 10 hours. By comparing the data with 
that of any given locaUty a fair idea of relative costs and of 
the proper choice for that particular station may be made. 

326. Track tanks. These are chiefly required as one of the 
means of avoiding delays during fast-train service. A trough, 
made of steel plate, is placed between the rails on a stretch of 
perfectly level track. A scoop on the end of a pipe is lowered 
from under the tender into the tank while the train is in motion. 
The rapid motion scoops up the water, which then flows into the 
tender tank. They should preferably be located on tangents, 
although the Penn. R. R. has track tanks at Atglen on a 2° 
curve where the track has 4 inches superelevation. Since the 
inside width of the tank (19'0 is almost exactly \ of the gauge, 
the water is about \\ inches deeper on the side toward the inner 
rail, but this much lack of symmetry does not seem to have 
interfered with successful operation. The length of the tanks 
varies from 1200 to 2500 feet; the net inside width is usually 

19 inches. The scoops are usually 12 to 13 inches wdde, which 
gives allowance for swaying. The tanks are made of sheet steel 
Yq" to \" thick. The usual cross-section is that of a wide and 
shallow U, 19'' wide, 6'' to 1\" deep, reinforced on the sides with 
angles. The ties are usually dapped, especially for the deeper 
tanks, so that the upper edges will not be higher than the rail. 
At each end there is a double incHned plane on which the 
scoops may shde without catching if the scoop should be lowered 
too soon or if it is not raised before the far end of the tank is 
reached. Experiments have shown that, at a speed as low as 

20 m.p.h., more water is wasted by slopping over the sides than 
the amount collected by the scoop. At a speed of 45 to 50 m.p.h. 
the amount wasted becomes minimum and the amount scooped 
up becomes maximum. At higher speeds the amount scooped 
up decreases and the wastage increases. The best results show 
a wastage of at least one-eighth of the total. These same tests 
showed that at 45 to 50 m.p.h. the 13" scoop in a 19" tank will 
scoop up about 625 gallons per inch of immersion per 1000 feet 
of tank, or say 2500 gallons per 1000 feet for a 4-inch immersion. 



376 



RAILROAD CONSTRUCTION. 



§327, 



The amount scooped up is practically proportional to the depth 
of immersion when that depth is over 2| inches. Heating. 
The water must be heated in winter to prevent freezing. There 
are two general methods r (a) Live steam is forced into the 
tank through nozzles about 40 feet apart; (6) a '^ circulatory- 
system " by which steam is forced into a water main which feeds 
the tank in such a way that the water is in constant circulation 
through the main, into the tank and then back again into the 
main to be reheated. For the climatic conditions of the N. Y. 
Central R. R. a steam capacity of 100 H. P. is considered essen- 
tail to heat 7000 sq. ft. of tank surface, which means about 4400 
lineal feet of 19-inch tank, or two good-length tanks on a double 
track. On account of the great amount of water splashed over 
the track and its scouring action on any ordinary ballast, a 

large item in the cost of an in- 
stallation is the reconstruction 
of the track. The certainty of 
quick freezing in winter, at least 
in high latitudes, demands that 
a drainage system, to carry away 
the spilled water, shall be effec- 
tive and thorough. Scouring is 
prevented by a pavement of 
cobbles, 6-inch quarry spalls, or 
large flat stones, laid over the 
ballast. A layer of large stones 
under the ballast faciUtates 
drainage to numerous cross 
drains and to longitudinal drains 
laid between the tracks. For 
further details the student is re- 
ferred to a monograph by Geo. 
W. Vaughan, Eng. Main, of 
Way, N. Y. Central R. R., in Vol. 
XIV,Proc.Am.Rwy.Eng. Assoc. 
327. Stand-pipes. These are usually manufactured by those 
who make a specialty of such track accessories, and who can 
ordinarily be trusted to furnish a correctly designed article. In 
Fig. 159 is shown a form manufactured by the Sheffield Car Co. 
Attention is called to the position of the valve and to the device 
for holding the arm parallel to the track when not in use so that 




Fig. 159. — Stand-pipe. 



§ 328. MISCELLANEOUS STRUCTURES AND BUILDINGS. 377 

it will not be struck by a passing train. When a stand-pipe is 
located between parallel tracks, the strict requirements of clear- 
ance demand that the tracks shall be bowed outward slightly. 
If the tracks were originally straight, they may be shoved over by 
the trackmen, the shifting gradually running out at about 100 
feet each side of the stand-pipe. If the tracks were originally 
curved, a slight change in radius will suffice to give the necessary 
extra distance between the tracks. 

BUILDINGS. 

328. Station platforms. These are most commonly made of 
planks at minor stations. Concrete is used in better-class work, 
also paving brick. An estimate of the cost of a platform of paving 
brick laid at Topeka, Kan., was $4.89 per 100 square feet when 
laid flat and $7.24 per 100 square feet when laid on edge. The 
curbing cost 36 cents per linear foot. Cinders, curbed by timbers 
or stone, bound by iron rods, make a cheap and fairly durable 
platform, but in wet weather the cinders will be tracked into 
the stations and cars. Three inches of crushed stone on a 
cinder foundation is considered to be still better, after it is once 
thoroughly packed, than a cinder surface. 

Elevation. — The elevation of the platform with respect to 
the rail has long been a fruitful source of discussion. Some roads 
make the platforms on a level with the top of the rail, others 
3 inches above, others still higher. As a matter of convenience to 
the passengers, the majority find it easier to enter the car from 
a high platform, but experience proves that accidents are more 
numerous with the higher platforms, unless steps are discarded 
altogether and the cars are entered from level platforms, as is 
done on elevated roads. As a railroad must generally pay dam- 
ages to the stumbling passenger, they prefer to build the lower 
platform. Convenience requires that the rise from the platform 
to the lowest step should not be greater than the rise of the car 
steps. This rise is variable, but with the figures usually employed 
the application of the rule will make the platform 5 ins. to 15 ins. 
above the rail. 

Position with respect to tracks. — Low platforms are gen- 
erally built to the ends of the ties, or, if at the level of the top 
of the rail, are built to the rail head. Car steps usually extend 
^ ft. 6 ins. from the track center and are 14 ins. to 24 ins. above 



378 



RAILROAD CONSTRUCTION, 



§329. 



the rail. The platform must have plenty of clearance, and when 
the platform is high its edge is generally required to be 5 ft. 6 ins. 
from the track center. 

329. Minor stations. The Amer. Rwy. Eng. Assoc, recom- 
mend one general waiting room (without reference to separate 
waiting room for colored people), for a passenger station of 
medium size for the following reasons : (See 1915 Manual, p. 187) . 

(1) It permits the general waiting room to be properly pro- 
portioned. 

(2) It permits proper development of a retiring room for 
women, with private entrance to the lavatory. 



1 TOILET 


GEN'L WAITING ROOM 
41^ 


1 BAGGAGE & U 
1 EXPRESS 1 

1 


1 TOILET 

n WOMENS 
1 ROOM 


1 TICKET fl 
1 OFFICE 1 



Fig. 160. — Division of Floor Area Recommended for Passenger 
Stations with One General, Waiting Room. 



(3) It readily admits of the other rooms being properly pro- 
portioned. 

(4) It permits ease of access from the agent's office to the 
trains, to the baggage room and to the waiting room. 

(5) It permits the ticket office to be of proper size and location 
for general office purposes. 

(6) It admits of the station being contracted in size without 
detriment to facihties. 

(7) It offers economy in heating. 

In the Southern States a separate waiting room for colored 
people is provided and is sometimes even required by law. The 
older design, combining a residence for the agent with the station, 
is now obsolete for new construction, although many such still 
exist. *' Combination stations " (for both passenger and freight 
business) were formerly quite popular for very small stations and 



I 



§ 330. MISCELLANEOUS STRUCTURES AND BUILDINGS. 379 



J 



are still considered desirable when all responsible freight and 
passenger business must be handled by one man. But it is 
desirable to separate them whenever the volume of business will 
justify the employment of two responsible men. 

In Gillette's Handbook of Cost Data (1910 ed.), is given in 
detail the cost of several station buildings. Such figures can be 
utilized when unit prices are given or can be derived. For exam- 
ple, in one case the building was 24X60 ft., exclusive of plat- 
forms; there was no masonry foundation nor plastering. The 
summary was as follows: 



Materials. 


Total. 


Per cent. 


Per sq. ft. 
of floor. 


30,057 ft. B. M. at S13.23 (aver.) 

20 M shingles at SI. 10 


$296.97 
22.00 
55.75 
37.50 
16.10 
8.80 


33.2 
2.4 
6.1 
4.1 
1.8 
1.0 

48.6 

45.3 
0.6 
2.8 
1.8 
0.9 

51.4 
100.0 


21 ft. B. M. 


Mill work 


3 9 cents 


Hardware 

23 gal. paint at 70 cents 


2.6 " 
11'* 


1100 brick, at $8.00 per M 








Total materials 


$437.12 

$406.38 

5.00 

24.50 

16.00 

8.50 


30 4 " 


Labor: 

176.2 days' labor, building at $2.32 

2 days' labor, put up ladders, at $2.50. . 

14 days' labor, painting at $1.75 

4 days' labor, building chimney, at $4.00 
8 days' labor, filling cinders, at $1.20 . . . 


28 . 2 cents 
1 . 7 cents 


Total labor 


$460.38 

$897.50 
55.00 
38.50 


31.9 *• 


Total, materials and labor 




Freight, 65 tons, 200 miles |c. ton-m. . . . 
Tools (excessive in this case) 








Grand total. 


$990.00 


68 8 cents 







I 



The cost of lumber was very low and even the unit cost of 
labor (carpenters, $2.50; masons, $4.00; average of all, $2.32), 
were lower than must frequently be paid. But the figures can 
be utilized by noting the percentages of the various items to the 
total and applying local unit costs for material and labor. The, 
total cost per square foot ($0,688), is abnormally low, partly 
because of no masonry foundation nor cellar, which would add 
40 to 50 cents per square foot. Note also that no expenses were 
included for fighting, plumbing, or heating — except a chimney. 

FREIGHT HOUSES. 

330. Two types. The freight house, or freight room, at a sta- 
tion where the business is small, is merely a small ordinary build- 
ing or a room attached to the station building. As the business 



380 ■ EAILROAD CONSTRUCTION. § 331. 

becomes larger, efficient operation requires that two types of 
buildings must be designed — the inbound and the outbound 
freight house. These types agree in requiring certain details in 
common, but there are also differences. 

331. Fire-risk. A small freight house in the country usuaHy 
has a minimum of actual fire-risk and of valuable freight stored 
at any one time. This may justify an inexpensive type of frame 
building which is in no sense fireproof. On the other hand, a 
building in the heart of a city, closely surrounded by other build- 
ings and stored with a large amount of valuable freight, justifies 
an expensive type of fireproof construction. The term '^ fire- 
proof " is only relative. Certain devices and added expendi- 
tures will reduce more and more the probability of destructive 
fires. Certain principles of construction which reduce fire-risk 
are as follows: (a) Use of noncombustible materials for floor, 
side walls and roof; (h) avoidance of space under wooden main 
floor, between foundations, where combustible rubbish may 
accumulate; (c) fire-walls dividing large houses so that there is 
not more than 5000 square feet of floor between fire-walls; fire- 
walls to be never more than 200 feet apart; {d) minimum num- 
ber of doors through a fire-wall; no door larger than 80 square 
feet; all doors fireproof and automatically self-closing; (e) 
fireproofing protection of walls and roof for at least five feet 
each side of a fire-wall; (/) provision for fire stand-pipes and 
hose racks not more than 150 feet apart; the stand-pipe should 
run up about 8 feet above floor where there should be 50 feet of 
2-inch linen hose in a hose rack;* the valve should be in a pit 
{always accessible), and so far below floor level that there is little 
or no danger of freezing, since freight houses are ordinarily not 
heated. 

332. Dimensions. A freight house usually has a track on one 
side and a vehicle driveway on the other, the floor being utilized 
for the more or less temporary storage of freight, which in this 
case is always in " less than carload " (L. C. L.) lots, carload 
shipments being transferred directly between cars and vehicles. 
Since small shipments can usually be loaded into cars (outbound 
shipments) with less delay than the delivery of freight to vehicles 
(inbound shipments), the required space for outbound shipments 
can be less than that for inbound. Experience has shown that 
for outbound freight only, a width of 30 feet is desirable; for 
both outbound and inbound, the width may be 30 to 40 feet; 



§ 333. MISCELLANEOUS STRUCTURES AND BUILDINGS. 381 

for inbound only it should be 40 to 60 feet. Too great a width 
needlessly increases the amount of hand-trucking. The length 
is indefinite and should correspond to the amount of business to 
be handled. Freight houses are usually single-storied, except 
where galleries or partial second stories are built to accommodate 
offices, file and stationery rooms, toilet and locker rooms, the 
room for '^ over, short and damaged '' freight and the cooperage 
room for repairing broken packages. 

333. Platforms. The platform on the track side should 
preferably be 8 to 10 feet wide, which will avoid the necessity 
of spotting cars with their doors directly in front of freight-house 
doors. The platform should be not more than 4 feet above the 
top of the rail. Even this would be too high to permit opening 
the doors of refrigerator cars, which swing outward. An occa- 
sional refrigerator car could be handled, even with a high plat- 
form, by opening the doors before placing the car. The M. C. B. 
standard, for regular use of refrigerator cars, is '' not more than 
3 ft. 8 ins. The P. R. R. standard is 3 ft. 5 ins. The minimum 
distance from track center to edge of platform is 5 ft. 9 ins. 
The P. R. R. standard is 6 ft. If ins. If there is a platform on 
the driveway side, it should be 3 to 4 feet above the driveway 
level. At an outbound house, where the freight is delivered 
from the vehicle into the freight house, the height should be not 
more than 3 feet. Platforms should slope away from the house 
with a grade of about 1 in. to 8 ft. for drainage. 

334. Floors. The designed floor loading should be 250 lbs. 
per square foot. In § 347 are described several types of floors 
suitable for engine houses, many of which are also suitable for 
freight houses. In selecting a type, it should be remembered 
that hand-trucking is apt to be concentrated along certain 
rather narrow paths and that this wears out the floor surface, 
requiring premature renewals along these paths, unless these 
paths are overlaid with iron or steel plates. When a solid type 
of floor is used (supported on sub-soil), the flooring should be 
independent of the side walls, which avoids trouble due to floor 
settlement. For inbound freight houses the floor should slope 
about 1 inch in 8 feet from the track side toward the driveway 
side, the slope continuing to the outer edge of the driveway plat- 
form, since this is in the direction of traffic and aids it, but the 
track platform must slope the other way for drainage. For out- 
bound freight houses, the slope is exactly reversed. 



J 



382 RAILROAD CONSTRUCTION. § 335. 

335. Doors. Ordinary swinging doors are unsuitable. Lift- 
ing doors, counterbalanced, which sometimes fold as they lift, 
are used. Rolling metal shutters are, perhaps, most satisfactory, 
but are expensive. Sliding doors require that a guarded space 
be made so that stored freight does not interfere with the 
sliding. They also limit the possible total door width to less 
than half the side of the house. All lifting types permit opening 
up the whole side of the house (if desired), except the space oc- 
cupied by the posts. Continuous doors are particularly neces- 
sary when there is no platform between the house and the 
track. Doors should be at least 8 feet high. On the track side 
this is sufficient, since the car door cannot be higher. On the 
driveway side a greater height might be desirable. 

336. Roofs projecting over platforms. These are desirable as 
a protection when loading or unloading during storms. That 
over the driveway platform should be at least 10 feet above the 
platform or 14 feet above the driveway. When not forbidden 
by State laws, the roof may be extended beyond the edge of 
the track platform, but it should be, at least, 17 feet above the 
rail and 18 inches from the track center, thus leaving a walking 
space on top of the car. 

337. Lighting. Daylight lighting should be obtained by win- 
dows through the side-walls above the doors, or by vertical sashes 
in a monitor roof, which will also provide for ventilation. Sky- 
lights, especially when nearly flat, are expensive both for con- 
struction and for maintenance. Artificial lighting should be 
obtained from electricity, with wires run according to the strictest 
specifications of the National Board of Underwriters. Platforms 
should be illuminated. A series of push plugs should be placed 
along the platform wall face, from which extension cords with 
bulbs may be run to hght car interiors. 

338. Scales. Outbound houses need scales, with capacity of 
8000 lbs., to weigh outgoing freight. '' From 50 to 80 feet apart 
is good practice. '^ 

339. Ramps. These are slopes from the driveway level to 
the car level which facilitate the loading or unloading of agricul- 
tural implements and all heavy vehicles running on their own 
wheels. They are usually built at the end of an extension of 
the platform, with as low a grade as the circumstances will 
permit. 

" Buildings and Structures of American Railroads," by Walter 



§ 340. MISCELLANEOUS STRUCTURES AND BUILDINGS. 383 

G. Berg, although now (1916) somewhat old, contains many 
plans, showing considerable detail, of station and other buildings. 
^' Railroad Structures and Estimates " by J. W. Orrock, also 
shows some plans. 

340. Section houses. These are houses built along the right- 
of-way by the railroad company as residences for the trackmen. 
The hability of a wreck or washout at any time and at any part^ 
of the road, as well as the convenience of these houses for ordinary 
track labor, makes it all but essential that the trackmen should 
live on the right-of-way of the road, so that they may be easily 
called on for emergency service at any time of day or night. 
This is especially true when the road passes through a thinly 
settled section, where it would be difficult if not impossible to 
obtain suitable boarding places. It is in no sense an extrava- 
gance for a railroad to build such houses. Even from the direct 
financial standpoint the expense is compensated by the corre- 
sponding reduction in wages, which are thus paid partly in free 
house rent. And the value of having men on hand for emergen- 
cies will often repay the cost in a single night. Where the coun- 
try is thickly settled the need for such houses is not so great, and 
railroads will utilize or perhaps build any sort of suitable house, 
but on Southern or Western roads, where the need for such 
houses is greater, standard plans have been studied with great 
care, so as to obtain a maximum of durability, usefulness, com- 
fort, and economy of construction. (See Berg's Buildings, etc., 
noted above.) On Northwestern roads, protection against cold 
and rain or snow is the chief characteristic; on Southern roads 
good ventilation and durability must be chiefly considered. 
Such houses may be divided into two general classes — (a) those 
which are intended for trackmen only and which may be built 
with great simplicity, the only essential requirements being a 
living room and a dormitory, and (5) those which are intended 
for families, the houses being then distinguished as ^' dwelling- 
houses for employees." 

ENGINE HOUSES.* 

341. Form. When not more than three or four engines are 
to be housed at once and when no turntable is to be provided, 

* Condensed and abbreviated from Committee Report, Am. Ry. Eng. 
Assoc, 1915. 



. 



384 RAILROAD CONSTRUCTION. § 342. 

the rectangular form is preferable. All large engine houses are 
^' circular," with a turntable at the center of the circle, except 
some very large houses, which are really repair shops, where it 
seems advisable to install a transfer table. 

342. Doors. The clear opening should be not less than 13 
feet wide by 16 feet high. The doors should fold outward and 
should have such a design that a pilot door may be inserted. 

343. Length. The length of stall along the center line of the 
track should be 15 feet greater than the overall length of the 
longest locomotive, which will provide a walkway behind the 
tender, a trucking space in front of the pilot and a sufficient dis- 
tance in which to stop the engine so that the side rods will be in 
any desired position. 

344. Materials of construction. Wood was formerly very 
commonly used, but it is too inflammable. The walls should be 
made of brick, stone, or plain concrete — not reinforced, at least 
*' for that portion of the wall directly in line of track where 
engine is liable to run into it." The roof is the difficult problem, 
since wood is inflammable and iron or steel, even for framing, is 
very rapidly corroded by coal gas from the engines. Rein- 
forced concrete is the only thoroughly satisfactory material but 
" when the roof is of reinforced concrete, the columns and roof 
beams should be of the same material," i. e., it is useless to sup- 
port a reinforced concrete slab on steel beams. 

345. Engine pits. These ^' should be not less than 60 feet 
in length, with convex floor, with drainage toward the turntable. 
The walls and floors may be of concrete. Proper provision 
should be made for the support of the jacking timbers." The 
engine should stand with its tender toward the turntable. 

346. Smokejacks. Locomotives leave an engine house under 
their own steam, which requires starting their fires considerably 
beforehand, and the smoke must be removed. The precise 
position of the locomotive on the track is variable, since it must 
be adjusted to the place where the side rods are. in a proper 
position for repairs. A smoke jack is essentially a funnel whose 
base is at the minimum height above the track which will give 
the smokestack a proper clearance. The base should be 42 
inches wide and long enough for the adjustment as stated above, 
which means at least 10 feet. The sides should slope upward 
gradually to a flue whose area should be not less than 7 square 
feet. There should be a drip trough around the base of the jack. 



§ 347. MISCELLANEOUS STRUCTURES AND BUILDINGS. 385 

The material should be " non-combustible/^ but the choice is 
troublesome. Sheet iron, even when heavily painted, corrodes 
rapidly. Wood, covered with '^ fireproof paint," has been tried. 
Cast iron has been tried but is exceedingly heavy as well as ex- 
pensive. Asbestos is being used on several important roads. 
Patented designs, of which there are several, are used on the 
majority of roads. 

347. Floors, (a) Stone screenings. Subsoil should be good; 
all soft spots cleaned out and filled with good material; subsoil 
rolled. Foundation of cinders or gravel, 6 ins. thick. Top coat, 
2 inches of stone screenings, perhaps mixed with a little clay or 
crude oil, the surface being thoroughly rolled. Special founda- 
tions for machinery necessary. Surface is not good for heavy 
wheeling, (h) Planks. Subsoil same as above; 6 ins. cinders 
or gravel, with 4"X6" creosoted sleepers, spaced about 3 feet, 
embedded in upper surface of cinders; then 3-inch plank. 
Again, special foundations for machinery and at jacking-up 
places are necessary, (c) Creosoted wood-block. The wood 
blocks, 4 ins. deep, fiber vertical, should be laid on a 1-inch 
cushion coat of sand which is supported by a 6-inch layer of 
concrete. A 6-inch layer of cinders, as specified above, is also 
recommended as a bed for the concrete, but this may depend on 
the character of the subsoil. The joints should be filled with 
asphaltic mastic, and an expansion joint 1 inch wide should be 
provided every 50 feet, (d) Wood floor on concrete. Sleepers, 
spaced about 3 feet, trapezoidal, 4-inch top, 6-inch bottom, 4 
inches deep, embedded in a 6-inch layer of concrete, so that the 
sleepers project J inch above concrete. Then layer of 2-inch 
plank, covered with IJ-inch maple flooring, (e) Brick. Same 
as (c) except that bricks are used in place of wood block. (/) 
Concrete. Same foundation as above; 6-inch course of concrete 
overlaid with 1-inch surface coat (1:2) laid on before base has 
taken initial set. (g) Asphalt. Same as (/) except that surface 
coat is li inches of rock mastic. Expert workmen are needed 
for satisfactorily mixing and laying the asphalt, but the floor is 

I ideal. 

I 348. Drop pits are necessary, where pairs of truck, driving and 

I trailer wheels may be dropped from their journals and removed 

; from the engine for repairs or renewals. 

I 349. Heating. The primary object of heating is to thaw out 
the engines so that they may be returned to service as quickly 



386 KAILROAD CONSTRUCTION. § 350. 

as possible, rather than to heat the building, whose general tem- 
perature should be kept at 50° to 60°. Therefore heat should 
be concentrated at the pits. Hot air should be forced through 
permanent ducts, preferably laid under the floor. The outlets 
should have dampers, which may be closed when men are working 
in the pits. Fresh air should be drawn from outdoors and no 
recirculation permitted. The air should be heated by passing 
over coils containing exhaust steam, supplemented by live steam, 
if necessary. The air passes out of the building through annular 
openings around the smokejacks, and also through openings 
between the wall plates and the roof rafters. These openings 
should extend entirely around the building. 

350. Window lighting. Skylights are undesirable because of 
preponderant disadvantages. The windows in the outer walls 
should be as large, wide and as high as safe construction will 
permit, the sill not more than 4 feet from the floor. Windows 
should be placed over the locomotive doors. Windows set into 
locomotive doors cause heavy maintenance charges on the doors. 

351. Electric lighting. Numerous lights should be provided 
to avoid shadows. Plugged outlets for incandescent lights in 
alternate spaces between pits should be provided. 

352. Piping. Pipes for air, steam and water supply should be 
provided, and where desired, piping for a washout and refilling 
system should be installed. Where this system is installed, the 
blow-off lines should be led to a central reservoir; where it is not 
used, the blow-off lines should be led outside the house. The 
steam outlet should be located near the front end of the boiler. 
The blow-off pipe, the air, the washout and refilling water and the 
cold water connections should be near the front end of the fire- 
box. Connections need only be provided in alternate spaces 
between stalls. 

353. Tools. There should ordinarily be facilities provided for 
hand tools and for the location of a few machine tools, prefer- 
ably electrically driven. 

354. Hoists. Hoists with differential blocks are generally 
used for handling heavy repair parts, and suitable provision 
should be made for supporting them. 

355. Turntables. The turntable should be long enough to 
balance the engine when the tender is empty. The deck form is 
preferable to the through form. Power should be provided at 
turntables having great service. Electric power is best and least 



§ 356. MISCELLANEOUS STRUCTURES AND BUILDINGS. 387 

expensive when it is available. Compressed air, supplied either 
by a pumping plant or by the locomotive itself, is sometimes 
used. The turntable pit should be thoroughly drained and prefer- 
ably paved. The circle wall should be of concrete or brick, with 
proper supports and fastenings for rails on the coping. The cir- 
cle rail should preferably bear directly on concrete base. The 
use of wood ties and tie-plates supported by masonry is desirable 
for the circle rail under some conditions. Easy access to the 
parts of a turntable for the oiling of bearings, painting and inspec- 
tion should be provided in the design of the turntable pit, unless 
ample provision is made in the turntable itself. 



LOCOMOTIVE COALING STATIONS. 

356. Hand shoveling. For roads of the smallest traffic, par- 
ticularly at terminals where locomotives lie overnight, hand 
shoveling direct from coal cars or from platforms provided with 
a jib crane and one-ton buckets, is the most economical. 

357. Locomotive crane. A locomotive crane, equipped with 
buckets, provides an efficient method of transferring coal from 
the coal car to a tender, particularly when the crane can be 
profitably employed at other times. 

358. Coaling trestle. This method requires a trestle with an 
approach not exceeding 5%, so that coal may fall from bottom- 
dumping cars into a pocket and then be discharged through 
chutes into the tender on a track on either side of the trestle. 
This method is satisfactory when two coaling tracks are sufficient 
and when there is available space for the approach track. 

359. Coal conveyors. When more than two coaling tracks 
are essential, a conveyor system may be preferable. The coal is 
brought to the plant in bottom-dumping gondola cars, which 
.dump the coal on to a conveyor which conveys it up and drops 
it into the bin, from which it may fall either into the tender or 
into an elevated conveyor car which runs it across a system of 
parallel tracks and dumps it into a tender, spotted there for the 
purpose. Incidentally, such a plant usually has also an ash 
conveyor onto which ashes are dumped from the engine. This 
conveyor carries the ashes to a place where the conveyor buckets 
dump them into a waiting gondola car, which when full is hauled 
away. 



388 



RAILROAD CONSTRUCTION. 



§360. 



360. Oil houses * should be fireproof and should be separated 
from other buildings. Above ground there should be a masonry 
building, 20'X40', or perhaps less, with one fireproof door and 
one or more windows, having wire glass. This room contains a 
row of pumps, one for each kind of oil; also a series of inlet pipes 
in the floor leading to tanks in the basement. The floor should 
be 4 feet above the track rail outside and there should be a 



COMPOSITION ROOJ' 

REINFORCED CONCRETE 



y,, 9" I-BEAM ABOUT 6% FT. CENTERS 08 % 
^ REINFORCED CONCRETE BEAMS. "^ 



* -WIRE GLASS WINDOW 



I 



-200- 



I 



REINFORCED CONCRETEv 



r^-r^l 



4J^" 



y 



OPENING FOR VENTILATION 
WITH WIRE NETTING. 



1^ 
Mi 



13 8.RICK WALL 



.SLIDING DOOR 
TIN CLAD 



SLOPE-'Vz 



FOR PIPING OIL 
FROM TANK CARS. 

ONE PIPE FOR EACH 

KIND OF OIL. 



CONCRETE 



F^:>:;:| -----V- 



■f \ I 




Fig. 161. — Cross-section of Typical Oil-house. 

platform between the house and the track. The storage space 
for oil is entirely in the basement and includes the area under the 
floor and also the area under the platform. The height depends 
on the required storage space for tanks. A series of pipes, one 
for each kind of oil, pass through the outer vertical face of the 
platform, for the convenient emptying of tank cars into the 
storage tanks. The inlet pipes through the floor are only for 
small quantities of oil drawn from barrels. 

The delivery system from the storage tanks to the faucets 
should be such that the oil can be delivered quickly and measured 
automatically. The delivery should also be such that there will 



* Condensed from the Manual of the Am. Rwy. Eng. Assoc, 1915 Ed. 



§ 361. MISCELLANEOUS STRUCTURES AND BUILDINGS. 389 

be a minimum of dripping at the faucet and that the dripping 
may drain back to the storage tanks. Openings for ventilation 
should be provided above the level of the top of the tanks. 
Lighting, when required, should be by electricity and heating by 
steam . For fire protection purposes a live-steam line should be run 
to the oil storage space, controlled by a valve outside the house. 

361. Section tool houses. For small-traffic roads these 
should be 10'X14', the short dimension parallel with the track, 
with double swinging doors, swinging out on the end nearest the 
track. For roads of larger traffic the dimension parallel with the 
track should be 18 to 20 feet and the other dimension 12 to 14 
feet. There should be a sliding door, 8 feet in clear, at extreme 
end, on track side, to permit the storing of hand car. A sliding 
wooden shutter (instead of glass) may serve as a window for 
fair weather. It should not be made so convenient and com- 
fortable that it will become a lounging place for trackmen in 
stormy or wintry weather. The building should be of wooden 
frame construction, resting on wooden posts, or on masonry piers 
if the location can be considered permanent. Drop siding on 
the sides and some kind of prepared roofing will usually be most 
economical. 

362. Sand houses. Sand is a necessity in the operation of 
locomotives. Ordinarily it is obtained in a more or less moist 
and caked condition. It must be made thoroughly dry, so that 
it will flow readily through a pipe having sufficient slope. The 
plant consists essentially of a " wet storage bin/' about 12'X16', 
which adjoins a '^ drying room '' of about the same size. This 
room contains a screen, which is usually necessary to screen out 
the coarser particles; also a furnace to dry the sand, and a coal 
bin. For small traffic roads it may be sufficient to store the dry 
sand in a bin or even in buckets which are lifted by hand to the 
engine. For heavier traffic it may be justifiable to raise the 
sand to a bin or hopper whose lowest point is at least 22 feet 
above the rail, from which the sand may flow through a jointed 
pipe, somewhat similar to a water-supply pipe, directly into the 
sand box on the engine. Of course the bottom of the hopper 
must have sufficient slope so that the sand will always flow over 
it. The sand is hoisted to the hopper, either by some mechan- 
ical conveyor system, or is forced through a pipe by compressed 
air. The building should be located about 8 feet from the nearest 
track center. 



390 RAILROAD CONSTRUCTION. § 363. 

363. Ash pits. A locomotive must dump the ashes from its 
ash pan at frequent intervals. The operation is usually timed 
to be done at terminal or divisional points, just before taking 
on water, coal, etc. These several plants are, therefore, grouped 
together in the yard. When there are no facilities for removing 
ashes by a conveyor at the same time that coal is being loaded on 
to the tender (see §§ 356-359), the ashes are dumped into a pit. 
The poorest roads dump them on the track under the engine, but 
this burns the ties, is dangerous, and is uneconomical, since they 
must be immediately removed. The simplest form of ash pit 
is made by dropping the ties about a foot, and then laying the 
rails on a pair of stringers about 12''X12". The stringers and 
ties must be covered with sheet iron to protect them from hot 
ashes. The capacity of such a pit is so small that the ashes must 
be removed quite frequently, which must 'usually be done by 
hand shoveling over the side of a gondola car on an adjacent 
track. The next development is a deeper pit, with concrete 
walls. Even then, the rails must be fastened to longitudinal 
wooden stringers, protected with sheet iron, or to cast-iron chairs 
which, are embedded in the concrete. The ashes may be shov- 
eled out by hand after the locomotive has passed, or they may be 
dropped from the ash pan into buckets or small cars, which run 
on a narrow track at the bottom of the pit, and which may be 
lifted out by a jib crane. Another development is to widen the 
pit, running one rail on one wall and the other rail on a series 
of cast-iron columns. The pit has much greater capacity 
and the ashes may be hoisted out at any time, even if the loco- 
motive is still on the ash track. Great economy in the disposal 
of ashes is obtained when it is practicable to construct a de- 
pressed track, with its track center about 14 feet away from the • 
ash track and 9 feet or more lower. The ashes may then be 
dropped onto a platform about 3 feet below the ash track, the 
platform extending to the top of a vertical retaining wall whose 
face is 5 ft. 6 ins. from the center of the depressed track, and from 
there the ashes are easily shoveled over the side of a gondola car 
placed on the lower track. No lifting of the ashes by hand is 
necessary. As in the previous plan, one rail of the ash track is 
supported by a wall, while the rail toward the depressed track is 
supported on cast-iron columns. The platform space is thus 
10 to 11 feet wide. 

Ashes should be quenched promptly after being deposited, 



§ 364. MISCELLANEOUS STRUCTURES AND BUILDINGS. 391 

SO as to reduce their heating effect even on metal and masonry. 
This requires a hose and a water supply. The pits should be 
graded so as to drain to a sump, which should have an overflow 
sufficiently above the bottom so that periodical cleaning out will 
suffice to keep the drain pipe from getting clogged with detritus 
from the ashes. 

SNOW STRUCTURES. 

364. Snow-fences. Snow structures are of two distinct 
kinds — fences and sheds. A snow-fence implies drifting snow — 
snow carried by wind — and aims to cause all drifting snow to be 
deposited away from the track. Some designs actually succeed 
in making the wind an agent for clearing snow from the track 
where it has naturally fallen. A snow-fence is placed at right 
angles to the prevailing direction of the wind and 50 to 100 feet 
away from the tracks. When the road line is at right angles to 
the prevailing wind, the right-of-w^ay fence may be built as a 
snow-fence — high and with tight boarding. Hedges have some- 
times been planted to serve this purpose. When the prevailing 
wind is oblique, the snow fences must be built in sections where 
they will serve the best purpose. The fences act as wind break- 
ers, suddenly lowering the velocity of the wind and causing the 
snow carried by the wind to be deposited along the fence. 
Portable fences are frequently used, which are placed (by per- 
mission of the adjoining property owners) outside of the right- 
of-way. If a drift forms to the height of the portable fence the 
fence may be replaced on the top of the drift, where it may act 
as before, forming a still higher drift. WTien the prevailing 
w^nd runs along the track line, snow-fences built in short sec- 
tions on the sides will cause snow to deposit around them 
while it scours its way along the track line, actually clearing 
it. Such a method is in successful operation at some. places on 
the White Mountain and Concord divisions of the Boston & 
Maine Railroad. Snow-fences, in connection with a moderate 
amount of shoveling and plowing, suffice to keep the tracks 
clear on railroads not troubled with avalanches. In such cases 
snow-sheds are the only alternative. 

365. Snow-sheds. These are structures which will actually 
keep the tracks clear from snow regardless of its depth outside. 
Fortunately they are only necessary in the comparatively rare 
situations where the snowfall is excessive and where the snow 



392 



RAILROAD CONSTRUCTION. 



§365, 



is liable to slide down steep mountain slopes in avalanches. 
These avalanches frequently bring down with them rocks, trees, 
and earth, which would otherwise choke up the road-bed and 
render it in a moment utterly impassable for weeks to come. 
The sheds are usually built of 12" X 12" timber framed in about 
the same manner as trestle timbering; the ^' bents" are some- 
times placed as close as 5 feet, and even this has proved insuffi- 
cient to withstand the force of avalanches. The sheds are there- 




. Level-fall shed 

Fig. 1G2. — Snow-sheds — Oanadian Pacific Railroad. 

fore so designed that the avalanche will be defected over them 
instead of spending its force against them. Although these 
sheds are only used in especially exposed places, yet their length 
is frequently very great and they are liable to destruction by fire. 
To confine such a fire to a limited section, ^^fire-breaks" are 
made — i.e., the shed is discontinued for a length of perhaps 100 
feet. Then, to protect that section of track, a V-shaped de- 
flector will be placed on the uphill side which will deflect all 
descending material so that it passes over the sheds. Sohd crib 



§ 366. MISCELLANEOUS STRUCTURES AND BUILDINGS. 393 

work is largely used for these structures. Fortunately suitable 
timber for such construction is usually plentiful and cheap 
where these structures are necessary. Sufficient ventilation 
is obtained by longitudinal openings along one side immediately 
under the roof. '^ Summer '^ tracks are usually built outside 
the sheds to avoid the discomfort of passing through these semi- 
tunnels in pleasant weather. The fundamental elements in 
the design of such structures is shown in Fig, 162, which illus- 
trates some of the sheds used on the Canadian Pacific Railroad. 

FENCES. 

366. Wire fences. The following is condensed from the con- 
clusions adopted by the Amer. Rwy. Eng. Assoc, and incor- 
porated in their 1915 Manual. The recommended standard 
right-of-way fence is a wire fence, supported on wood or concrete 
posts. The wiring is to consist of five to nine longitudinal 
strands, with vertical stay wires spaced 12 to 24 inches apart. 
The longitudinal and vertical wires are to be locked or fastened 
with a mechanical lock which will prevent slipping either longi- 
tudinally or vertically, or the wires shall be electrically welded. 
The wire shall be galvanized so as to stand the following test: 
'' The galvanizing shall consist of an even coating of zinc, which 
shall withstand one-minute immersion tests in a solution of 
commercial sulphate of copper crystals and water, the specific 
gravity of which shall be 1.185 and whose temperature shall be 
from 60° to 70° F. Immediately after each immersion the 
sample shall be washed in water and wiped dry. If the zinc is 
removed, or a copper-colored deposit formed at the end of the 
fourth immersion, the lot of material from which the sample is 
taken shall be rejected. The fence shaU be so fabricated as not 
to remove the galvanizing or impair the tensile strength of the 
wire.'' Electrically welded fencing should be galvanized after 
it has been fabricated. 

367. Types. Class A fence has 9 horizontal smooth wires 
whose spacing, starting at the ground, is 5, 4, 4J, 5, 5J, 6, 7, 8 
and 9 inches. To make it *' hog-tight " the bottom space (5") 
is reduced to 3 inches and a barbed wire is inserted midway in 
the 3-inch space. The top and bottom smooth wires are No. 7 
gauge wire and the 7 intermediate wires are No. 9. The ver^ 
tical stay wires, spaced 12 inches, shall be No. 9 gauge. 



394 RAILROAD CONSTRUCTION. § 368. 

Class B fence has 7 horizontal wires, and 2 vertical wires 
spaced 18 inches — all wires No. 9 gauge. The spacing, starting 
at the ground, is 7, 6J, 7, 7J, 8, 8i and 9 inches. 

Class C fence has 5 horizontal wires, and 2 vertical wires 
spaced 24 inches — all wires No. 9 gauge. The spacing, starting 
at the ground, is 9, 7J, 8, 8i and 9 inches. 

Class D fence has 5 horizontal wires and no vertical stay 
wires, the wires being No. 9 gauge. The spacing, starting at 
the ground, is 10, 10, 10, 12 and 12 inches. 

368. Posts. End, corner, anchor and gate posts shall be at least 
8 feet long and set 3 feet 4 inches in the ground, even if blasting 
must be resorted to. Intermediate posts shaU be at least 7 feet 
long and set 2 feet 4 inches in the ground. Where rock is en- 
countered at intermediate post holes, the intermediate posts, if 
of wood and not more than two in succession, may be set on sills, 
6"X6''X4'0'', braced on both sides by braces 2''X6"X3' 0''. 
End, corner, anchor and gate posts, when of wood, shall be 8 
inches in diameter at the small end; when of concrete, shall be 
6 inches square at the top, 8 inches square at the base and shall 
be remforced with four |-inch square twisted rods. Intermediate 
wood posts shall be at least 4 inches in diameter at the small end; 
intermediate concrete posts shall be 4 inches thick at the top, 
5^ inches at the bottom and reinforced with three (or four, 
dependmg on design) J-inch square twisted rods. 

369. Braces. End, corner, anchor and gate posts shall be 
braced by 4" X4'' sawed lumber, or round posts at least 4 inches 
in diameter, or by concrete struts, 4"X4'', reinforced with four 
J-inch twisted rods. The strut braces shall extend from a point 
about 12'' below the top of the braced post to a point about 12" 
from the ground line at the adjacent intermediate post. In 
addition, a tie, made of a double strand of No. 9 galyanized soft 
wire, looped around the end, corner, anchor or gate post near the 
ground line, and around the next intermediate or hne post about 
12 inches from the top, shall be put on and twisted until the top 
of the next intermediate or line post is drawn back about 2 
inches. 

370. Concrete posts. These are recommended. They may 
be made of one part of cement to four parts of pit gravel; or 
one part cement, two parts sand and four parts of stone of low 
absorption or screened gravel, the aggregate in any case being 
not less than J" nor more than J". The molds should be oiled 



§ 371. MISCELLANEOUS STRUCTURES AND BUILDINGS. 395 

or soaped and should be vibrated while concrete is poured to 
make the concrete more compact. The concrete should have a 
^' quaking " consistency. The pouring should not be done out 
of doors in freezing weather. The concrete should not be ex- 
posed to sun, should be sprinkled every day for 8 or 10 days 
and should have 90 days for curing. They should be packed 
in sawdust or straw for shipment. Posts are usually made taper- 
ing and the cross-section is variously a square, a rectangle, or 
an isosceles triangle, the corners being chamfered. The rein- 
forcement should be placed not more than J'' from the surface 
and should be wired by bands spaced about 12''. The fencing 
is sometimes fastened to the posts merely by wires tied tightly 
about the post or may be fastened to metal lugs which are 
embedded in the soft concrete during molding. 

371. Construction details. Wood posts shall be anchored 
by gaining and spiking two cleats, 2"X6"X2' 0", on the side of 
the post below the ground line. Staples shall be 1 inch long for 
hard wood, and IJ inch for soft wood, made of No. 9 galvanized 
steel wire. They shall be driven diagonally with the grain of 
the wood, the top wires double-stapled. Staples, No. 9 wire, 
1 inch long, weigh 108 to the pound; IJ inch long, 72 to the 
pound. 

Wire. No. 7 wire is 0.177 inch in diameter, weighs 439 pounds 
to the mile, or 12.05 feet to the pound. No. 9 wire is 0.148 inch 
in diameter, weighs 306 pounds to the mile or 17.24 feet to the 
pound. Smooth wire is preferable to barbed. A heavy smooth 
wire or a plank should be used at the top of a barbed-wire fence. 
Wires shall be placed on the side of the post away from the 
track. Splicing shall be done as follows: " The ends of the 
wires shall be carried 3 inches past the splicing tools and 
wrapped around both wires backward from the tool for at least 
five turns, and after the tool is removed, the space occupied by 
it shall be closed by pulling the ends together." After erection, 
wood posts should be sawed off, on a one-fourth pitch, the high 
side being next to the wire and 2 inches above it. 

Gates should be hinged to swing away from the tr&.ck; should 
be at least 12 feet w4de and 4 feet 6 inches above the ground; 
should swing shut by gravity, and the free end should overlap 
the post so that it cannot be swung open toward the track. 
All-metal construction is preferable. 



396 RAILROAD CONSTRUCTION. § 372. 

SIGNS. 

372. Highway signs. The crossing sign recommended by 
the Amer. Rwy. Eng. Assoc, is essentially as follows: Two 
wooden blades, 12 inches wide, 8 feet long, with mitered ends, 
are placed diagonally, with an angle of 50° between the blades, 
on an 8''X8''X16' 0'' wooden post sunk 4 feet in the ground. 
The lower 9 feet is painted black, the upper 7 feet white. The 
blades are painted white with black letters and a |-inch black 
border around the blades. The border and lettering is on both 
sides. The lettering is Egyptian style 9 inches high with the 
exception of the connecting terms, as ^^ for the" in the recom- 
mended sign, which should be 4 inches high. The recommended 
wording is '' RAILROAD CROSSING " on one blade and 
" LOOK OUT FOR THE LOCOMOTIVE " on the other blade. 
The width of band of the letters is IJ inches. If two railroads 
parallel each other within 400 feet, another blade marked 
" TWO CROSSINGS " should be added. The laws in some 
states prescribe what the lettering shall "be. 

373. Trespass signs. The specifications for these signs are 
applicable to many other pubHc -warnings which must be dis- 
played. A cast-iron plate, i inch thick, stiffened on the back by 
f-inch diagonal cast ribs and having the letters and border cast 
on the front by raising the surface about I inch, is set on an iron 
post 10 feet long, which is embedded 2 feet in a block of con- 
crete, which serves as foundation. The letters should be about 
2 inches high. A socket is cast on the rear side of the plate of 
such dimensions that it will set on the pipe and be fastened with 
a J-inch set screw. The posts may be made of 2J-inch wrought 
iron pipe or of good second-hand boiler tubes, which should 
be filled with cement grout. The face of the letters and the 
borders should be painted black while the background is painted 
white. The tablet will usually be about 30 inches wide by 18 
inches high with rounded corners, although the dimensions will 
vary in accordance with the lettering to be placed on it. The 
following tj^espass signs frequently need to be displayed: 



RAILROAD PROPERTY 

TRESPASSING 

FORBIDDEN UNDER 

PENALTY OF LAW 



DANGER 
DO NOT 
TRESPASS ON THE 
RAILROAD 



§ 374. MISCELLANEOUS STRUCTURES AND BUILDINGS. 397 



DANGER 

DO NOT 

TRESPASS ON THIS 

BRIDGE 



374. Marker posts. Mile posts are most economically made, 
considering their durability, of skeletonized cast iron. The post 
is made up of two slabs of cast iron J inch thick, 8 feet long, the 
width tapering from 10 inches to 12 inches, the two slabs being 
formed in one piece and connected at intervals by |-inch webs 
and a top and bottom plate. They should be set 3 feet 6 inches 
in the ground and have a 4-inch slab of concrete or a heavy, flat 
stone as a base. The mile post numbers should be cast in raised 
letters on the face, the letters being 4| inches high. The two 
faces should be at right angles with each other and shoiild each 
stand at an angle of 45° with the track. They should be set at 
least 8 feet from the gauge line of the nearest rail and 11 feet 
away, where it is practicable. The numbers should be so set 
that, on approach, the distance to the terminus or division point 
beyond will be indicated. 

The separating line between divisions is indicated to track 
men by an iron sign, called a division post, which is structurally 
the same as that of the mile posts. The two divisions are 
indicated by raised lettering on the faces of the posts. Of 
course there must be a variation in the lettering or numbering 
and a special post must be cast for each location of division post 
or mile post. 

Whistle signs are made similarly except that there is but one 
slab, suitably reinforced with ribs, and which faces in the desired 
direction. The letter W 7f ins. high is cast in raised letters 
near the top. The ring sign is made similarly by using the letter 
R. The separating line between sections is indicated to the 
trackmen by a cast-iron sign, called a section post, which is made 
similarly to the Trespass Signs, except that the tablet is much 
smaller. Such a sign will have two consecutive numbers, for 
example, 24-25, to indicate that the sign is at the separating 
line between section 24 and section 25. 

375. Bridge warning. When possible the headroom beneath 
overhead bridges is made at least 22 ft., which will make it 
safe for a trainman to stand on the top of a freight car which is 



398 



RAILROAD CONSTRUCTION. 



§375. 




i 



§ 375. MISCELLANEOUS STRTTCTUEES AND BUILDINGS. 399 

passing under the bridge but if i« r.^f t 

that an^ount of he:S;om uZ* l^okMT': *^ '^^^ 

warning for trainmen is necessary Thp,. °"^«"°\«ta°ces, a 

to be spanned, two poles wiU be user^ Z7 T ^ ^^^ 

iron rods which swing on ring-bolts which are nin ^hZ \ 
wooden arm or hanger. The distance frn,; T °"^'' ^ 

bridge or tunnel should be ab utToOt" SS TT ^^ 
somewhat on the grade, since that affects the tiLe of 'the T"^' 
freight train in passing the interval. *^^ ^"^''^^e 



CHAPTER XIII. 
YARDS AND TERMINALS. 



,76 Value Of proper design. A large part of the total cost of 
^rL andlhe »r. grouped into . t,m without regard to order 

the c^r is hauled out into a yard occupying valuable ground is 
denied overThe yard tracks for a considerable aggregate nuleage 
Sre starting for its destination, where the same process is re- 
betore sta-rung terminal expenses are 



§ 377. YARDS AND TERMINALS. 401 

the necessary changes, and the difficulty of making the changes 
while the yard is being used, only prolong the bad state of 
affairs and an inefficient makeshift is frequently adopted. As- 
sume that an improvement in the design of the yard will permit 
a saving of the use of one switching engine, or for example, that 
the work may be accomplished with three switching engines in- 
stead of four. Assuming a daily cost of $25, we have in 313 
working days an annual saving of $7825, which, capitalized at 
5%, gives $156,500, enough to reconstruct any ordinary yard.* 

377. Divisions of the subject. The subject naturally divides 
itself into three heads — (a) Yards for receiving, classifying, and 
distributing freight cars, called more briefly freight yards; (h) 
yards and conveniences for the care of engines, such as ash tracks, 
turn-tables, coal-chutes, sand-houses, water-tanks, or water 
stand-pipes, etc., and (c) passenger terminals. 

FREIGHT YARDS. 

378. General principles. It should be recognized at the start 
that at many places an ideally perfect yard is impossible, or at 
least impracticable, generally because ground of the required 
shape or area is practically unobtainable. But there are some 
general principles which may and should be followed in every yard 
and other ideals which should be approached as nearly as possi- 
ble. Nevertheless every yard is an independent problem. Be- 
fore taking up the design of freight yards, it is first necessary to 
consider the general object of such yards and the general princi- 
ples by which the object is accomplished. These may be briefly 
stated as follows: 

1. A yard is a device, a machine, by which incoming cars are 
sorted and classified — some sent to warehouses for unloading, 
some sent to connecting railroads, some made up for local dis- 
tribution along the road, some sent for repairs, and, in short a 
device by which all cars are sent through and out of the yard as 
quickly as possible. 

2. Except w^hen a road's business is decreasing, or when its 
equipment is greater than its needs and its cars must be stored, 
efficiency of management is indicated by the rapidity with which 
the passage of cars through the yard is accomplished. 

3. When a yard is the terminal of a "division," the freight 

* Estimate of Mr." H. G. Hetzler, C, B. <fc Q. Ry., now Pres. Chi. & 
West. Ind. Rwy. 



402 RAILROAD CONSTRUCTION. § 378. 

trains will be pulled into a "receiving track'' and the engine and 
caboose detached. The caboose will be run on to a "caboose 
track/' which should be conveniently near, and the engine is run 
off to the engine yard. If the train is a " through" train and no 
change is to be made in its make-up, it will only need to wait for 
another engine and perhaps another caboose. If the cars are to 
be distributed, they will be drawn off by a switching engine to 
the "classification yard." 

4. The design of a yard is best studied by first picking out the 
ladder tracks and the through tracks which lead from one divi- 
sion of the yard to another. These are tracks which must always 
be kept open for the passage of trains, in contradistinction to 
the tracks on which cars may be left standing, even though it is 
only for a few moments, while drilling is being done. Such a set 
of tracks, which may be called the skeleton of the yard, is shown 
by heavy lines in Fig. 164. Each line indicates a pair of rails. 
The tracks of the storage yards are shown by the lighter lines. 

5. There is a distinct advantage in having all storage tracks 
double-ended — except "team tracks." Team tracks are those 
which have spaces for the accommodation of teams, so that load- 
ing or unloading may be done directly between the cars and teams. 
To avoid the necessity of teams passing over the tracks, these are 
best placed on the outskirts of the yard and consist of short stub- 
sidings arranged in pairs. But storage tracks should have an 
outlet at each end so as to reduce the amount of drilling neces 
sary to reach a car which may be at the extreme end of a long 
string of cars. This is done usually by means of two "ladder" 
tracks, parallel to each other, which thus make the storage 
tracks between them of equal length. 

6. The equality of length of these storage tracks is a point in- 
sisted on by many, but on the other hand, trains are not always of 
uniform length even on any one division. Loaded trains and 
trains of empties will vary greatly in length, and the various 
styles and weights of freight engines eitiployed necessitate other 
variations in the weights and lengths of trains hauled. With 
storage tracks of somewhat variable length a larger percentage 
of track length may be utilized, there will be less hauling over a 
useless length of track, and (assuming that the plot of ground 
available for yard purposes has equally favorable conditions for 
yard design) more business may be handled in a yard of given 
area. 



§378. 



YARDS AND TERMINALS. 



403 




404 RAILROAD CONSTRUCTION. § 379- 

7. Although not absolutely necessary, there is an advantage 
in having all frog numbers and switch dimensions uniform. 
No. 8 frogs are recommended. Sharper-angled frogs make 
easier riding, less resistance and less chance of derailment, but 
on the other hand require longer leads and more space. No. 
7 and even No. 6 frogs are sometimes used on account of economy 
of space, but they have the disadvantages of greater tractive 
resistance, greater wear and tear on track and rolling stock, and 
greater danger of derailment. 

8. The spacing of ^' body tracks '' (the parallel storage tracks 
which are headed by ladder tracks), should be 13 to 14 ft. and 
when they are parallel to a main track, or important running 
track, the first body track should be at least 15 ft. from the main 
track. 

9. When practicable, caboose tracks should be so located 
that cabooses can be placed on and removed from them in the 
order of their arrival, and should be so graded that cabooses 
can be dropped by gravity on to the rear of trains made up for 
departure. 

10. " Bad-order '' tracks are those onto which damaged cars 
may be conveniently placed and from which they may be easily 
run to double-ended ^^ repair tracks," which should have a 
capacity of about 15 cars each, and laid out in pairs which are 
spaced 16 and 24 ft. alternately. 

11. Car capacity should be rated at 42 ft. of track per car. 
379. Hixmp yards. A great economy in the movement of 

cars in a classification yard is obtained by the use of humps. 
A hump is a grade summit in a receiving track which has such an 
elevation that cars will run by gravity from it to any desired 
point in the classification yard. If a yard is practically level, 
an engine must push or " kick '' every separate '' cut '^ of the 
train on to its particular track, which involves not only a great 
waste of time but also a very large switch-engine mileage. By 
pushing the cars over the hump and successively cutting off, or 
uncoupling, one or more cars which are to be run down a ladder 
track to any one body track, the cars quickly acquire a desired 
velocity on a short stretch of perhaps 4% grade. This grade 
reduces to about 1% along the ladder track and through the 
switches, which allows for the added resistance through them, and 
then the grade is dropped to about 0.5% or less. The 4% grade, 
for about 50 ft., followed by a vertical curve about 150 ft. long 



; 



§ 380. YARDS AND TERMINALS. 405 

at the end of which the grade is reduced to 1%, develops the 
required velocity in the car, the 1% grade maintains it and the 
momentum thus acquired is sufficient to move the cars to the 
farthest point of the body tracks. A brakeman, or " rider," 
accompanies each car, or group of cars. To avoid the great 
waste of time required for these riders to walk back to the hump, 
it has been found economical in some large yards to have a track 
for the exclusive use of a car, especially fitted for easy jumping on 
or off, operated, perhaps, by a switching engine, or possibly by 
gasoline, which picks up the riders and carries them back to the 
hump. The aggregate time saved justifies the expenditure. 
Since empty cars have a greater tractive resistance per ton than 
loaded cars, they require a steeper grade to maintain the same 
velocity, and, therefore, when tracks are set aside for the use of 
empty cars, the grade leading to such empty tracks should be 
increased if possible. To operate such a hump efficiently, the 
yard clerk makes up a triple (or quadruple) list for each freight 
train arriving at the yard for distribution. One of these lists 
is given to the man cutting off the cars at the top of the hump, 
and one to the towerman, if the switches are operated from the 
tower, or one to each switch tender if the switches are hand- 
operated. Each list contains in the first column the consecutive 
number of the cut, in the second column the number of the track 
on which that cut of cars is to be placed, and in the third column 
the number of cars cut. Cut No. 1 is the first car (or cars) to 
go over the hump. The grade from the receiving track to the 
hump should be such that one engine can push the maximum 
train over the hump. Since track resistance is greater in winter 
than in summer, the summit of the hump may be raised in winter 
sufficiently to develop the required added gravity force, and 
lowered again when the added height is not needed. The length 
of track required to be raised is not very great, while the saving 
in not being obliged to lift every train the required extra height, 
during the many months each year when the extra height is not 
needed, usually justifies the two changes each year. 

380. Relation of yard to main tracks. Safety requires that 
there should be no connection between the yard tracks and the 
main tracks except at each end of the yard, where the switches 
should be amply protected by signals. Sometimes the main 
tracks run through the yard, making practically two yards — one 
for the traffic in either direction — but this either requires a double 



406 RAILROAD CONSTRUCTION. § 381. 

layout of tracks and houses (such as ash tracks, coal-chutes, sand- 
houses, etc.), or a very objectionable amount of crossing of the 
main- line tracks. The preferable method is to have the main hne 
tracks entirely on the outside of the yard. A method which is in 
one respect still better is to spread the main tracks so that they 
run on each side of the yard. In this case there is never any 
necessity to cross one main track to pass from the j^ard to the 
other main track; a train may pass from the yard to either 
main track and still leave the other main track free and open. 
The ideal arrangement is that by which some of the tracks cross 
over or under all opposing tracks. By this means ail connections 
between the 3^ard and the main tracks maybe by ^'trailing" 
switches; that is, trains will run on to the main track in the 
direction of motion on that main track. Of course all this 
applies only to double main track. 

An important element of yard design is to have a few tracks im- 
mediately adjoining the main tracks and separate from the yard 
proper on which outgoing trains may await their orders to take 
the main track. When the orders come, they may start at once 
without any delay, without interfering with any yard operations, 
and they are not occupying tracks which may form part of the 
s\^stem needed for switching. 

381. Minor freight yards. The term here refers to the sub- 
stations, only found in the largest cities, to which cars will be sent 
to save in the amount of necessary team hauling and also to re- 
lieve a congestion of such loading and unloading at the main 
freight terminal. The cars are brought to these yards sometimes 
on floats (as is done so extensively at various points around New 
York Harbor), or they are run do^Ti on a long siding runningl 
perhaps through the city streets. But the essential feature off 
these yards is the maximum utilization of every square foot of 
yard space, which is always very valuable and which is frequently 
of such an inconvenient shape that a great ingenuity is required 
to obtain good results. There is generally a temptation to use 
excessively sharp curves. When the radii are greater than 175 
feet no especial trouble is encountered. Curves wdth radius as 
short as 50 feet have been used in some yards. On such curves 
the long cars now generally used make a sharper angle with each 
other than that for which the couplers were designed and spe- 
cial coupler-bars become necessary. The two general methods 
of construction are (a) a series of parallel team tracks (as pre- 



§381 



YARDS AND TERMINALS. 



407 




408 RAILROAD CONSTRUCTION. § 382. 

viously described and as illustrated further in Fig. 165), and (h) 
the "loop system/' as is illustrated in Fig. 166. 

382. Transfer cranes. These are almost an essential feature 
for yards doing a large business. The transportation of built- 
up girders, castings for excessively heavy machinery, etc., which 
weigh five to thirty tons and even more, creates a necessity for 
machinery which will easily transfer the loads from the car to 
the truck and vice versa. An ordinary "gin-pole" will serve the 
purpose for loads which do not much exceed five tons. A fixed 
framework, covering a span long> enough for a car track and a 
team space, with a trolley traveling along the upper chord, is the 
next design in the order of cost and convenience. Increasing 
the span so that it covers two car tracks and two team spaces 
will very materially increase the capacity. Making the frame 
movable so that it travels on tracks which are parallel to the 
car tracks, giving the frame a longitudinal motion equal to two 
or three car lengths, and finally operating the raising and travel- 
ing mechanism by power, the facility for rapidly disposing of 
heavy articles of freight is greatly increased. Of course only a 
very small proportion of freight requires such handhng, and the 
business of a yard must be large or perhaps of a special character 
to justify and pay for the installation of such a mechanism. 
Figs. 165 and 166 each indicate a transfer crane, evidently of the 
fixed type. 

383. Track scales. The location of these should be on one of 
the receiving tracks near the entrance to the yard, but not on the 
main track nor on any track where drilling must be done. It is 
usually best to have a " dead track " over the scales — i.e., a 
track which has one rail on the solid side wall of the scale pit 
and the other supported at short intervals by posts which come 
up through the scale platform and yet do not touch it. These 
rails and the regular scale rails switch into one track by means of 
point rails a few feet beyond each end of the scales. The switches 
should be normally set so that all trains will use the dead track, 
unless the scales are to be operated. It has been found possible 

' in a gravity yard to weigh a train with very little loss of time by 
running each car slowly and separately by gravity over the 
scales and weighing them as they pass over. 



§383. 



YARDS AND TEBMINALS. 



409 




r; ^ Entrance 



East 135th St. 

Fig. 166.— Minor Freight Yard on a Harbor Front. 



410 



BAILROAD CONSTRUCTION. 



§384. 




FiQ. 167. — Engine Yard and Shops. Urbana. Iul. 



^ 







iTo face p. 411.) (Published through courtesy of Union Switch and Signal Co.) 



§ 384. YARDS AND TERMINALS. 411 

ENGINE YARDS. 

384. General principles. Engine yards must contain all the 
tracks, buildings, structures, and facilities which are necessary 
for the maintenance, care, and storage of locomotives and for pro- 
viding them with all needed supplies. The supplies are fuel, 
water, sand, oil, waste, tallow, etc. Ash-pits are generally neces- 
sary for the prompt and economical disposition of ashes; engine- 
houses are necessary for the storage of engines and as a place 
where minor repairs can be quickly made. A turntable is 
another all but essential requirement. The arrangement of all 
these facilities in an engine yard should properly depend on the 
form of the yard. In general they should be grouped together 
and should be as near as possible to the place where through 
engines drop the trains just brought in and where they couple on 
to assembled outgoing trains, so that all unnecessary running 
light may be avoided. Switching engines should be able to 
dump ashes, take their supplies and pass around waiting road 
engines. In Figs. 164 and 167 are shown two designs which 
should be studied with reference to the relative arrangement of 
the yard facilities. 

PASSENGER TERMINALS. 

(Passenger terminals are one of the logical subdivisions of 
this chapter, but their construction does not concern one engineer 
in a thousand. The local conditions attending their construction 
are so vaiied that each case is a special problem in itself — a prob- 
lem which demands in many respects the services of the archi- 
tect rather than the engineer. The student who wishes to pursue 
this subject is referred to an admirable chapter m " Buildings and 
Structures of American Railroads," by Walter G. Berg, Chief 
Engineer of the Lehigh Valley Railroad.) 



CHAPTER XIV. 

BLOCK SIGNALING. 
GENERAL PRINCIPLES. 

385. Two fundamental systems. The growth of systems of 
block signaling has been enormous within the last few years — • 
both in the amount of it and in the development of greater per- 
fection of detail. The development has been along two general 
lines: (a) the manual, in which every change of signal is the 
result of some definite action on the part of some signalman, but 
in which every action is so controlled or limited or subject to 
the inspection of others that a mistake is nearly, if not quite, 
impossible; (6) the automatic, in which the signals are oper- 
ated by mechanism, which cannot set a wrong signal as long as the 
mechanism is maintained in proper order. The fundamental 
principles of the two systems will be briefly outlined, after which 
the chief details of the most common systems will be pointed out. 

386. Manual systems. Small traffic roads are usually operated 
on the basis of the '^ train-order system.'^ A ^Hrain dispatcher'^ 
controls the movement of every train on his division and telegraphs 
orders to men (who are frequently station agents) at various points 
along the line, who transmit these orders to the trainmen as the 
trains reach these points. A train-order signal station, whether at 
a regular trafiic station or in a special cabin, has " train-order 
signals '' which, when in the stop position, inform the engineman 
and conductor that they are to receive orders at the telegraph 
office; the clear position informs them that there are no orders 
for them. When more than one train is allowed on a single 
track between two consecutive train-order stations, the engine- 
man and conductor of each train has strict orders with reference 
to the other train, for example, that the trains are to pass at 
some siding where there is no telegraphic station. A very strict 
code of rules has been developed which, when literally followed, 
ensures safety of operation, but these rules cannot eliminate 
the human element, or the liability of personal negligence 
or error. When such a system is applied to a double-track 
road, or even to a single-track road, with train-order signal 

412 



§ 387. BLOCK SIGNALING. 413 

stations located so frequently that only one train will be al- 
lowed between two consecutive offices at once, it virtually 
becomes a block system even though it is not called such. 
When such a system is adhered to rigidly, it is called an absolute 
Mock system But when operating on this system, a delay of 
one train will necessarily delay every other train that follows 
closely after. A portion, if not all, of the delay to subsequent 
trains may be avoided, although at some loss of safety, by a 
system of permissive blocking. By this system an operator 
may give to a succeeding train a "clearance card" which per- 
mits it to pass into the next block, but at a reduced speed and 
with the train under such control that it may be stopped on 
very short notice, especially near curves. One element of the 
danger of this system is the discretionary power with which it 
invests the signalmen, a discretion which may be wrongfully 
exercised. A modification (which is a fruitful source of colli- 
sions on single-track roads) is to order two trains to enter a 
block approaching each other, and with instructions to pass 
each other at a passing siding at which there is no telegraph- 
station. When the instructions are properly made out and 
literally obeyed, there is no trouble, but every thousandth or 
ten thousandth time there is a mistake in the orders, or a mis- 
understanding or disobedience, and a collision is the result. The 
telegraph line, a code of rules, a corps of operators, and sig- 
nals under the immediate control of the operators, are all that 
is absolutely needed for the simple manual system. 

387. Development of the manual system. One great diffi- 
culty with the simple system just described is that each operator 
is practically independent of others except as he may receive 
general or specific orders from a train-dispatcher at the division 
headquarters. Such difficulties are somewhat overcome by a 
very rigid system of rules requiring the signalmen at each station 
to keep the adjacent signalmen or the train-dispatcher in- 
formed of the movements of all trains past their own stations. 
When these rules (which are too extensive for quotation here) 
are strictly observed, there is but little danger of accident, and 
a neglect by any one to observe any rule will generally be appar- 
ent to at least one other man. Nevertheless the safety of trains 
depends on each signalman doing his duty, and a Httle careless- 
ness or forgetfulness on the part of any one man may cause an 
accident. The signaling between stations may be done bv 



414 RAILROAD CONSTRUCTION. § 387. 

ordinary telegraphic messages or by telephone, but is frequently 
done by electric bells, according to a code of signals, since these 
may be readily learned by men who would have more difficulty 
in learning the Morse code. 

In order to have the signalmen mutually control each other, 
the "controlled manual" system has been devised. The first 
successful system of this kind which was brought into exten- 
sive use is the "Sykes" system, of which a brief description 
is as follows: Each signal is worked by a lever; the lever is 
locked by a latch, operated by an electro-magnet, which, with 
other necessary apparatus, is inclosed in a box. When a signal 
is set at danger, the latch falls and locks the lever, which cannot 
be again set free until the electrp-magnet raises the latch. The 
magnet is energized only by a current, the circuit of which is 
closed by a "plunger" at the next station ahead; just above 
the plunger is an "indicator," also operated by the current, 
which displays the words clear or blocked. (There are varia- 
tions on this detail.) When a train arrives at a block station 
(A), the signalman should have pre^dously signaled to the station 
ahead (B) for permission to free the signal. The man ahead (B) 
pushes in the "plunger" on his instrument (assuming that the 
previous train has already passed him), which electrically opens 
the lock on the lever at the previous station (A), The signal 
at A can then be set at "safety." As soon as the train has 
passed A the signal at A must be set at " danger." A further 
development is a device by which the mere passage of the train 
over the track for a few feet beyond the signal will automati- 
cally throw the signal to "danger." After the signal once goes 
to danger, it is automatically locked and cannot be released 
except by the man in advance (B), who will not do so until the 
train has passed him. The "indicator" on B^s instrument 
shows "blocked" when A's signal goes to danger after the train 
has passed A, and B's plunger is then locked, so that he can- 
not release A's signal while a train is in the block. As soon as 
the train has passed A , B should prepare to get his signals ready 
by signaling ahead to C, so that if the block between B and C 
is not obstructed, B may have his signals at "safety" so that 
the train may pass B without pausing. The student should 
note the great advance in safety made by the Sykes system; 
a signal cannot be set free except by the combined action of 
two men, one the man who actually operates the signal and 



§ 388. BLOCK SIGNALING. 415 

the other the man at the station ahead, who frees the signal 
electrically and who by his action certifies that the block im- 
mediately ahead of the train is clear. 

A still further development makes the system still more " auto- 
matic '' (as described later), and causes the signal to fall to dan- 
ger or to be kept locked at danger, if even a single pair of wheels 
comes on the rails of a block, or if a switch leading from a main 
track is opened. 

388. Permissive blocking. " Absolute '' blocking renders ac- 
cidents due to coUisions almost impossible unless an engineer 
runs by an adverse signal. The signal mechanism is usually 
so designed that, if it gets out of order, it will inevitably fall to 
"danger,'^ i.e., as described later, the signal-board is counter- 
balanced by a weight which is much heavier. If the wire breaks, 
the counterweight will fall and the board will assume the hori- 
zontal position, which always indicates '' danger.'' * But it some- 
times happens that when a train arrives at a signal-station, the 
signalman is unable to set the signal at safety. This may be 
because the previous train has broken down somewhere in the 
next block, or because a switch has been left open, or a rail has 
become broken, or there is a defect of some kind in the electrical 
connections. In such cases, in order to avoid an indefinite 
blocking of the whole traffic of the road, the signalman may 
give the engineer a "caution-card" or a "clearance card," 
which authorizes him to proceed slowly and with liis train under 

1 complete control into the block and through it if possible. If 
he arrives at the next station without meeting any obstruction 
it merely indicates a defective condition of the mechanism, 

j which will, of course, be promptly remedied. Usually the next 

I section will be found clear, and the train may proceed as usual. 
On roads where the "controlled manual" system has received 

Jl its highest development, the rules for permissive blocking are 

I I so rigid that there is but little danger in the practice, unless 
' there is an absolute disobedience of orders. 

389. Automatic systems. By the very nature of the case, 
such systems can only be used to indicate to the engineers of 
trains something wit h reference to the passage of previous 

* This was written on the basis of the older system, in which the sema- 
phore swings through the lower right-hand quadrant. The most recent 
practice swings the semaphore through the upper right-hand quadrant. 
^ break in the wire holding the semaphore vertical will cause it to fall 
to horizontal position without the aid of a counterweight. 



416 RAILROAD CONSTRUCTION. § 389. 

trains. The complicated shifting of switches and signals which 
is required in the operation of yards and terminals can only be 
accomplished by '' manual " methods, and the only automatic 
features of these methods consist in the mechanical checks 
(electric and otherwise), which wdll prevent wrong combina- 
tions of signals. But for long stretches of the road, where it 
is only required to separate trains by at least one block length, 
an automatic system is generally considered to be more relia- 
ble. As expressed forcibly by a railroad manager, ''an auto- 
matic system does not go to sleep, get drunk, become insane,: 
or tell lies when there is any trouble." The same cannot always 
be said of the employes of the manual system. 

The basic idea of all such systems is that when a train passes 
a signal-station (A), the signal automatically assumes the ''dan-* 
ger'' position. This may be accomplished electrically, pneu- 
matically, or even by a direct mechanism. When the train 
reaches the end of the block at B and passes into the next one, 
the signal at B will be set at danger and the signal at A will be 
set at safet5^ The lengths of the blocks are usually so great' 
that the only practicable method of controlling from B a 
mechanism at A is by electricity, although the actual motive 
power at A may be pneumatic or mechanical. At one time 
the current from A to B was run only through wires. This^ 
method has the very positive advantage of reliability, definite 
resistance to the current, and small probability of short-circuit- 
ing or other derangement. But now all such systems use the 
rails for a track circuit and this makes it possible to detect the 
presence of a single pair of wheels on the track anywhere in the 
block, or an open switch, or a broken rail. Any such circum- 
stances, as well as a defect in the mechanism, will break or 
short-circuit the current and will cause the signal to be set at 
danger. To prevent an indefinite blocking of traffic owing to 
a signal persistently indicating danger, most roads employing 
such a system have a rule substantially as follows : When a train 
finds a signal at danger, after waiting one minute (or more, 
depending on the rules), it may proceed slowly, expecting to 
find an obstruction of some sort; if it reaches the next block 
without finding any obstruction and finds the next signal clear, 
it may proceed as usual, but must promptly report the case to 
the superintendent. Further details regarding these methods 
will be given later. See § 394. 



§ 390. BLOCK SIGNALING. 417 

390. " Distant " signals. The close running of trains that 
is required on heavy-traffic roads, especially where several 
branches combine to enter a common terminal, necessitates the 
use of very short blocks. A heavy train running at high speed 
can hardly make a '^ service" stop in less than 2000 feet, while 
the curves of a road (or other obstructions) frequently make 
it difficult to locate a signal so that it can be seen more than a 
few hundred feet away. It would therefore be impracticable 
to maintain the speed now used with heavy trains if the eugi- 
neer had no foreknowledge of the condition in which he will 
find a signal until he arrives within a short distance of it. To 
overcome this difficulty the ^'distant'' signal was devised. This 
is placed about 1800 or 2000 feet from the ''home" signal, and 
is interlocked with it so that it gives the same signal. The dis- 
tant signal is frequently placed on the same pole as the home 
signal of the previous block. When the engineer finds the 
distant signal ''clear/' it indicates that the succeeding home 
signal is also clear, and that he may proceed at full speed and 
not expect to be stopped at the next signal; for the distant 
signal cannot be cleared until the succeeding home signal is 
cleared, which cannot be done until the block succeeding that 
is clear. A clear distant signal therefore indicates a clear track 
for two succeeding blocks. When the engineer finds the distant 
signal blocked, he need not stop (providing the home signal is 
clear). It simply indicates that he must be prepared to stop 
at the next home signal and must reduce speed if necessary. 
It may happen that by the time he reaches the succeeding home 
signal it has already been cleared, and he may proceed without 
stopping. This device facilitates the rapid running of trains, 
with no loss of safet}^, and yet with but a moderate addition to 
the signaling plant. 

391. "Advance " signals. It sometimes becomes necessary 
to locate a signal a few hundred feet short of a regular passen- 
ger-station. A train might be halted at such a signal because 
it was not cleared from the signal-station ahead — perhaps a 
mile or two ahead. For convenience, an "advance" signal 
may be erected immediately beyond the passenger-station. 
The train will then be permitted to enter the block as far as 
the advance signal and may deliver its passengers at the station. 
The advanc^ signal is interlocked with the home signal back 
of it, and cannot be cleared until the home signal is cleared and 



418 RAILROAD CONSTRUCTION. § 392. 

the entire block ahead is clear. In one sense it adds another 
block, but the signal is entirely controlled from the signal station' 
back of it. 

MECHANICAL DETAILS. 

308. Signals. The primitive signal is a mere cloth flag. A 
better signal is obtained when the flag is suspended in a suit- 
able place from a fixed horizontal support, the flag weighted 
at the bottom, and so arranged that it may be drawn up and 
out of sight by a cord which is rim back to the operator's office. 
The next step is the substitution of painted wood or sheet metal 
for the cloth flag, and from this it is but a step to the standard 
semaphore on a pole, as is illustrated in Fig. 168. The simple 
flag, operated for convenience with a cord, is the signal em- 
ployed on thousands of miles of road, where they perhaps make 
no claim to a block-signal system, and where the trains are rua 
on the " train-order system." 

Semaphore boards. These are about 5 feet long, 8 inches 
wide at one end, and tapered to about 6 inches wide at the hinge 
end. The boards are fastened to a casting which has a ring to 
hold a red glass which may be swung over the face of a lantern, 
so as to indicate a red signal. "Distant'' signal-boards usually 
have their ends notched or pointed; the "home" signal-boards 
are square ended. The boards are always to the right of the 
hinge when a train is approaching them. The "home" signals 
are generally painted red and the "distant" signals green, 
although these colors are not invariable. The backs of the 
boards are painted white. Therefore any signal-board which 
appears on the left side of its hinge will also appear white, and 
is a signal for traffic in the opposite direction, and is therefore 
of no concern to an engineman. 

Poles and bridges. When the signals are set on poles, they 
are always placed on the right-hand side of the track. When 
there are several tracks, four or more, a bridge is frequently 
built and then each signal is over its own track. The signals 
for two tracks, operated in the same direction, may be placed 
on one pole by having a cross-piece which supports two " masts," 
see Fig. 168. In that figure the signals on the left-hand mast 
control the second track at the left of the signal; 'those on the 
right-hand mast control the track just to the left of the signal. 



(To face page 418.) 




Courtei>y of the Union Switch and Signal Co. 

Fig. 168. — Semaphores, 



{To face page 418.) "> 









i 



I 



Courtesy of the Union Switch and Signal Cto, 

Fig. 170.—" Banjo " Signals. 



§ 393. BLOCK SIGNALING. 419 

A train movement, from the switch track at the right of the sig- 
nal on to the main track, is controlled by the ^' dwarf " signal 
at the right of the switch track. The signals controlling the 
two tracks at the extreme left are not shown. The building at 
the left of the track in the extreme background is apparently 
the signal tower controlling this signal. 

In Fig. 169 is shown a " bridge " and the two signals (home 
and distant), for each track. The two pairs of signals on the 
two right-hand poles are extended to the right and show^ that 
the movement of trains on those tracks is away from the observer. 
The darkness of the blades in the picture shows that they are 
painted dark, probably orange or red. The other blades show 
light (because painted white), and extend to the left but would 
appear to the right to an engineman on either left-hand track 
coming toward the observer. Incidentally the picture shows, 
over the two right-hand tracks, the ropes of a " tickler " (see 
§ 375), to protect brakemen on the tops of cars w^hich will enter 
the tunnel shown in the background. 

" Banjo " signals. This name is given to a form of signal, 
illustrated in Fig. 170, in which the indication is taken from the 
color of a round disk inclosed with glass. The great argument 
in their favor is that they may be worked by an electric current 
of low voltage, which is therefore easily controlled; that the 
mechanism is entirely inside of a case, is therefore very light, 
and is not exposed to the weather. The argument urged against 
them is that it is a signal of color rather than form or position, 
and that in foggy weather the signal cannot be seen so easily; 
also that unsuspected color-blindness on the part of the engine- 
man may lead to an accident. Notwithstanding these objections, 
this form of signal is used on thousands of miles of line in this 
country. 

393 • Wii'es and pipes. Signals are usually operated by levers 
in a signal-cabin, the levers being very similar to the reversing- 
lever of a locomotive. The distance from the levers to the sig- 
nals is, of course, very variable, but it is sometimes 2000 feet. 
The connecting-link for the most distant signals is usually 
No. 9 wire; for nearer signals and for all switches operated 
from the cabin it may be 1-inch pipe. When not too long, one 
pipe will serve for both motions, forward and back. When 
wires are used, it is sometimes so designed (in the cheaper sys- 
tems) that one wire serves for one motion, gra\'ity being de- 



i 



420 KAILROAD CONSTRUCTION. § 393. 

pended on for the other, but now all good systems require two 
wires for each signal. 

Compensators. Variations of temperature of a material with 
as high a coefficient as iron will cause very appreciable differ- 
ence of length in a distance of several hundred feet, and a 
dangerous lack of adjustment is the result. To illustrate: A 
fall of 60° F. will change the length of 1000 feet of wire by 

1000 X 60 X. 0000065 =0.39 foot =4.68 inches. 

A much less change than this will necessitate a readjustment 
of length, unless automatic compensators are used. A com- 
pensator for pipes is very readil}^ made on the principle illus- 
trated in Fig. 171. The problem is to preserve the distance 
between a and d constant regardless of the temperature. Place 
the compensator half-way between a and d, or so that ah = cd, 
A fall of temperature contracts ah to ab\ Moving h to 6' will 
cause c to move to c', in which hV =cc\ But cd has also short- 
ened to c^d\ therefore d remains fixed in position. 

The regulations of the Am. Rwy. Eng. Assoc, require that 
"A compensator shall be provided for each pipe line over fifty 
(50) feet in length and under eight hundred (800) feet, with 
crank-arms eleven by thirteen (11X13) inch centers. From 
eight hundred (800) to twelve hundred (1200) feet in length, 
crank-arms shall be eleven by sixteen (11X16) inch centers. 
Pipe lines over twelve hundred (1200) feet in length shall be 
provided with an additional compensator. 

'^Compensators shall have one sixty (60) degree and one one 
hundred and twenty (120) degree angle-cranks and connecting 
link, mounted in cast iron base, having top of center pins sup- 
ported. The distance between center of pin-holes shall be 
twenty-two (22) inches." 

The compensator should be placed in the middle of the length 
when only one is used. When two are used they should be 
placed at the quarter points. Note that in operating through 
a compensator the direction of motion changes; i.e., if a moves 
to the right, d moves to the left, or if there is compression in ah 
there is tension in cd, and vice versa. Therefore this form of 
compensator can only be used with pipes which will withstand 
compression. It has seemed impracticable to design an equally 
satisfactory compensator for wires, although there are several 
designs on the market. 



§393, 



BLOCK SIGNALING. 



421 



The change of length of these bars is so great that allowance 
must be made for the temperature at the time of installation. 
On the basis of 50° as the mean temperature, the pipes are so 
adjusted that the distance between the points b and c of Fig. 171 
is made greater or less than 22 inches, according to the tem- 
perature of installation. For example, if the temperature were 
80° and the length of the piping were 900 feet, the length of the 
pipes should be adjusted so that he is less than 22 inches by an 
amount equal to 900 X (80°- 50°) X .0000065 = 0.1755 feet = 




Fi«. 171. — Standard Pipe Compensator, 



2.106 inches. The length should therefore be 19.9 inches in- 
stead of 22 inches. If the mean temperature was very different 
(say in Florida) some higher temperature should be taken as 
normal, so that the extreme range above and below the normal 
shall be approximately the same. 

Guides around curves and angles. When wires are required 
to pass around curves of large angle, pulleys are used, and a 
length of chain is substituted for the wire. For pipes, when 
the curve is easy the pipes are slightly bent and are guided 
through pulleys. When the angle is sharper, ''angles" are 
used. The operation of these details is self-evident from an 
inspection of Fig. 172. 



422 



RAILROAD CONSTRUCTION, 



§394. 



394. Track circuit for automatic signaling. The fundamental 
principle of the track circuit method of indicating a track obstruc- 
tion or breakage, using direct current, is as follows: A current 
of low potential is run from a battery at one end of a section 
through one line of rails to the other end of the section, then 
through a relay, and then back to the battery through the other 
line of rails. To avoid the excessive resistance which would 
occur at rail joints which may become badly rusted, a wire 




Fig. 172. — Deflecting-rods and Angle. 



suitably attached to the rails is run aroimd each joint. In 
order to insulate the rails of one section from the rails at either 
end and yet maintain the rails structurally continuous, the ends 
of the rails at these dividing points are separated by an insulator 
and the joint pieces are either made of wood or have some 
insulating material placed between the rails and the ordinary 
metal joint. The bolts must also be insulated. Whien the 
relay is energized by a current, it closes a local circmt at the 
signal-station, which will set the signal there at " safety.'^ The 
resistance of the relay is such that it requires nearly the whole 
current to work it. and to keep the local circuit closed. There- 
fore, when there is any considerable loss of current from one 
rail to the other, the relay will not be sufficiently energized, the 
local circuit will be broken, and the signal will automatically 
fall to danger. This diversion of current from one rail to the 
other before the current reaches the relay may be caused in 
several ways: the presence of a pair of wheels on the rails any- 
where in the section will do it; also the breakage of a rail; also 
the opening of a switch an5rwhere in the section; also the pres- 
ence of a pair of wheels on a siding between the " fouHng point " 
and the switch. (The " fouhng point '' of a siding is that point 
where the rails first commence to approach the main track.) 
In Fig. 173 is shown all of the above details as well as some others. 



§394. 



BLOCK SIGNALING. 



423 



(0 E 



ICQilll' 



At A, By and the ^^ fouling point '^ are shown the insulated joints. 
The batteries and signals are arranged 
for train motion to the right. When 
a train has passed the points near A, 
where the wires leave the rails for the 
relay, the current from the " track 
battery " at 5 will pass through the 
wheels and axles, and although no 
electrical connection is broken, so 
much ciu-rent will be shunted through 
the wheels and axles that the weak 
current still passing through the relay 
is not strong enough to energize it 
against its spring and the '^ signal- 
magnet " circuit is broken, and the 
signal A goes to ^' danger." At the 
turnout the rails between the fouling 
point and the switch are so connected 
(and insulated) that a pair of wheels 
on these rails will produce the same 
effect as a pair of the main track. This 
is to guard against the effect of a car 
standing too near the switch, even 
though it is not on the main track. 
When the train passes 5, if there is 
no other interruption of the current, 
the track battery at B again energizes 
the relay at A, the signal-magnet 
circuit at A is closed, and the signal 
is drawn to '' safety." 

About 1903 the application of alter- 
nating current to signahng circuits was 
invented. This not only permits the 
substitution of a. c. circuit for track 
batteries, but also makes it possible to 
utilize the track circuit method to in- 
dicate obstructions or rail breakages 
even when the track is the return cir- 
cuit for an electrified road. But an 
explanation of this development would 
be too long for this text-book. It is fig. 173. 





424 KAILKOAD CONSTRUCTION. § 394. 

given in a 548-page book called ^'Alternating Current Signaling," 
published by the Union Switch & Signal Co., Swissvale, Pa. 

This chapter also omits all references to " interlocking plants," 
which are essential features of the operation of large terminal 
yards. Even an elementary treatment of the present develop- 
ment of signaling and interlocking would require a large text- 
book, and, therefore, nothing more than the above brief outline 
will be here given. 



CHAPTER XV. 



ROLLING-STOCK. 



(It is perhaps needless to say that the following chapter is 
in no sense a course in the design of locomotives and cars. Its 
chief idea is to give the student the elements of the construc- 
tion of those vehicles which are to use the track which he may 
design — to point out the mutual actions and reactions of vehicle 
against track and to show the effect on track wear of varia- 
tions in the design of rolling-stock. The most of the matter 
given has a direct practical bearing on track-work, and it is con- 
sidered that all of it is so closely related to his work that the 
civil engineer may study it with profit.) 



WHEELS AND RAILS. 

395. Effect of rigidly attaching wheels to their axles. The 
wheels of railroad rolling-stock are invariably secured rigidly 
to the axles, which therefore revolve with the wheels. The 
chief reason for this is to avoid excessive wear 
between the axles and the wheels. 

Any axle must always be somewhat loose in 
its journals. A sidewise force P (see Fig. 174) 
acting against the circumference of the wheel 
will produce a much greater pressure on the 
axle at S and S\ and if the wheel moves on 
the axle, the wear at S and S^ will be exces- 
sive. But when the axle is fitted to the wheel 
with a ^'forced fit" and does not revolve, 
the mere pressure produced at S is harmless. 
When two wheels are fitted tight to an axle, 
as in Fig. 175, and the axle revolves in the jour- 
nals aa, a sidewise pressure of the rail against the wheel flange 
will only produce a slight and harmless increase of the journal 
pressure Q, although at Q there is sliding contact. Twdst- 

425 




426 



RAILROAD CONSTRUCTION. 



§396. 



ing action in the journals is thus practically avoided, since a 
small pressure at the journal-boxes at each end of the axle 
suffices to keep the axle truly in line. 



fe Q 



a 



Fig. 175. 




On the other hand, when the wheels are rigidly attached to 
their axles, both wheels must turn together, and when rounding 
curves, the inner rail being shorter than the outer rail, one 
wheel must slip by an amount equal to that difference of length. 
I'he amount of this slip is readily computable : 



Longitudinal slip 



2na 



360 



5(^2-^1) = 






(102) 



In which C is a constant for any one gauge, and .g= the track 
^auge = (r2— ri). For standard gauge (4.708) the shp is .08218 
foot per degree of central angle. This shows that the longitu- 
dinal slipping around any curve of any given central angle will 
be independent of the degree of the curve. The constant (.08218) 
here given is really somewhat too small, since the true gauge 
that should be considered is the . distance between the lines of 
tread on the rails. This distance is a somewhat indeterminate 
and variable quantity, and probably averages 4.90 fset, which 
would increase the constant to .086. The slipping may occur 
"by the inner wheel slipping ahead or the outer wheel slipping 
back, or by both wheels slipping. The total slipping will be 
constant in any case. The slipping not only consumes power, 
but wears both the wheels and the rail. But even these dis- 
advantages are not sufficient to offset the advantages resulting 
from rigid wheels and axles. 

396. Effect of parallel axles. Trucks are made with two or 
three parallel axles (except as noted later), in order that the 
axles shall mutually guide each other and be kept approximately 



§396. 



ROLLING-STOCK, 



427 



perpendicular to the rails. If the curvature is very sharp and 
the wheel-base comparatively long (as is notably the case on 
street railwavs at street corners), the front and rear wheels 




hH 



Fig. 177. 




'^'^'^^X^'' 




'1 



Fig. 179. 



will stand at the same angle (a) with the track, as showTi in 
Fig. 177. But it has been noticed that for ordinary degrees of 
curvature, the rear wheels stand radial to the curve (see Fig. 
178), and for steam railroad work this is the normal case. A\Tien 
the two parallel axles are on a curve (as shown), the wheels tend 
to run in a straight line. In order that they shall run on a curve 
they must slip laterally. The principle 

is illustrated in an exaggerated form in ^ '"' 

Fig. 179. The wheel tends to roll from a j- 
toward h. Therefore in passing along the 
track from a to c it must actually slip late- ''" 
rally an amount he which equals ac sin a. 
Let ^=length of the wheel-base (Figs. 177 and 178); r = radius 
of curve; then for the first case (Fig. 177), sina = ^-^2r; for 
the second and usual case (Fig. 178), sin a = t^r; for t = 5 feet 
and r=radius of a 1° curve, a = 0°G3' for the second case, a 
varies (practically) as the degree of curve. The lateral slipping 
per unit of distance traveled therefore equals sin a. As an 
illustration, given a 5-foot wheel-base on a 5° curve, a = 0° 15', 
sin a = .00436, and for each 100 feet traveled along the curve 
the lateral slip of the front wheels would be 0.436 foot. There 
would be no lateral slipping of the rear w^heels, assuming that 
the rear axle maintained itself radial. 

From the above it might be inferred that the flanges of the 
forward wheels wiU have much greater wear than those of the 
rear wheels. Since cars are drawn in both directions about 
equally, no difference in flange wear due to this cause wdll occur, 
but locomotives (except switching-engines) run forward almost 



428 



EAILROAD CONSTRUCTION. 



§397. 



exclusively, and the excess wear of the front wheels of the pilot - 
and tender-trucks is plainly observable. 

For a given curve the angle a (and the accompanying resist- 
ance) is evidently greater the greater the distance between 
the axles. On the other hand, if the two axles are very close 
together, there will be a tendency for the truck to twist and 
the w^heels to become jammed, especially if there is consider- 
able play in the gauge. The flange friction would be greater 
and would perhaps exceed the saving in lateral slipping. A 
general rule is that the axles should never be closer together 
than the gauge. 

Although the slipping per unit of length along the curve varies 
directly as the degree of curvature, the length of curve necessary 
to pass between two tangents is inversely as the degree of curve, 
and the total slipping between the two tangents is independent 
of the degree of curve. Therefore when a train passes between 

two tangents, the total slipping 
of the wheels on the rails, lon- 
gitudinal and lateral, is a quantity 
which depends only on the central 
angle and is independent of the 
radius or degree of curve. 

397. Effect of coning wheels. 
The wheels are always set on the 
axle so that there is some ^'play'' 
or chance for lateral motion be- 
tween the wheel-flanges and the 
rail. The treads of the wheel are 
also " coned.^' This coning and play 
of gauge are shown in an exagger- 
ated form in Fig. 180. When the 
wheels are on a tangent, although there will be occasional oscil- 
lations from side to side, the normal position will be the sym- 
metrical position in w^hich the circles of tread hh are equal. 
When centrifugal force throws the wheel-flange against the rail, 
the circle of tread a is larger than &, and much larger than c; 
therefore the wheels will tend to roll in a circle whose radius 
equals the slant height of a cone whose elements would pass 
through the unequal circles a and c. If this radius equaled the 
radius of the track, and if the axle were free to assume a radial 
position, the wheels would roll freely on the rails without any 




— 



Fig. 180. 



I 



§ 398. ROLLING-STOCK. ^ 429 

slipping or flange pressure. Under such ideal conditions, 
coning would be a valuable device, but it is impracticable to 
have all axles radial, and the radius of curvature of the track 
is an extremely variable quantity. It has been demonstrated 
that with parallel axles the influence of coning diminishes as 
the distance between the axle increases, and that the effect is 
practically inappreciable when the axles are spaced as they are 
on locomotives and car-trucks. The coning actually used is 
very slight (see Chapter XV, § 420) and has a different object. 
It is so slight that even if the axles were radial it would only 
prevent the slipping on a very light curve — say a 1° curve. 

398. Effect of flanging locomotive driving-wheels. If all the 
wheels of aU locomotives were flanged it would be practically 
impossible to run some of the longer types around sharp curves. 
The track-gauge is always widened on curves, and especially 
on sharp curves, but the widening would need to be excessive 
to permit a consolidation locomotive to pass around an 8° or 
10° curve if all the drivers were flanged. The action of the 
wheels on a curve is illustrated in Figs. 181, 182, and 184. All 
small truck-wheels are flanged. The rear drivers are always 
flanged and four-driver engines usually have all the drivers 
flanged. Consolidation engines have only the front and rear 
drivers flanged. Mogul and ten-wheel engines have one pair 
of drivers blank. On Mogul engines it is always the middle 
pair. On ten-wheel engines, when used on a road ha\dng sharp 
curves, it is preferable to flange the front and rear dri\'ing- 
wheels and use a ^' swing bolster" (see § 399); when the curva- 
ture is easy, the middle and rear drivers may be flanged and 
the truck made with a rigid center. The blank drivers have 
the same total width as the other drivers and of course a much 
wider tread, which enables these drivers to remain on the rail, 
even though the curvature is so sharp that the tread overhangs 
the rail considerably. 

399. Action of a locomotive pilot- truck. The purpose of 
the pilot-truck is to guide the front end of a locomotive around 
a curve and to relieve the otherwise excessive flange pressure 
that would be exerted against the driver-flanges. There are 
two classes of pilot-trucks — (a) those having fixed centers and 
(6) those ha^^ng shifting centers. This second class is again 
subdi^dded into two classes, which are radically different in 
their action — (b{) four-wheeled trucks having two parallel axles 



430 



RAILROAD CONSTRUCTION, 



§399. 



and (62) two-wheeled trucks which are guided by a "radius- 
bar.'' The action of the four-wheeled fixed-centered truck (a) 
is showTi in Fig. 181. Since the center of the truck is forced 




Fig. 181. — Fixed Center Pilot-truck. 
to be in the center of the track, the front drivers are drawn 
away from the outer rail. The rear outer driver tends to roll 
away from the outer rail rather than toward it, and so the effect 




Fig. 182. — Four-wheeled Truck — Shifting Center. 

of the truck is to relieve the driver-flanges of any excessive 
pressure due to curvature. The only exception to this is the 
case where the curvature is sharp. Then the front inner driver 
may be pressed against the inner rail, as indicated in Fig. 181. 

This limits the use of this type of 

wheel-base on the sharper curves. 

The next type — (Jb^ four-wheeled 

trucks with shifting centers — is 

ri_i p^ — I -^ j|[n|||i| h much more flexible on sharp 

"T -"i ILJJ^^' " ~ l ^ ^H hJ" curvature; it likewise draws the 

front drivers away from the outer 
rail. The relative position of the 
wheels is shown in Fig. 182, in 

Z, u^ which c' represents the position 
/^ of center-pin and c the displaced 
: / truck center. The structure and 

action of the truck is shown in 
Fig. 183. The "center-pin" (1) is 
supported on the "truck-bolster" (2), which is hung by the 
''links" (4) from the "cross-ties" (3). The links are therefore 




Fia. 183. — Action of Shifting 

Center. 



§399. 



ROLLING-STOCK. 



431 



in tension and when the wheels are forced to one side by the 
rails the links are inclined and the front of the engine is 
drawn inward by a force equal to the weight on the bolster 
times the tangent of the angle of inclination of the links. This 
assumes that all links are vertical when the truck is in the 
center. Frequently the opposite links are normally inclined to 
each other, which somewhat complicates the above simple relation 
of the forces, although the general principle remains identical. 

The two-wheeled pilot-truck with shifting center is illus- 
trated in Fig. 184. The figure shows the facihty with which 




Fig. 184. — Two-wheeled Truck — Shifting Center. 




Fig. 185. — Action of Two- 
wheeled Truck. 



an engine with long wheel-base may be made to pass around 
a comparatively sharp curve by omitting the flanges from the 
middle drivers and using this form of pilot-truck. As in the 
previous case, the eccentricity of 
the center of the truck relative 
to the center-pin induces a cen- 
tripetal force which draws the 
front of the engine inward. But 
the swing- truck is not the only 
source of such a force. If the 
"radius-bar pin'' were placed at 0' (see Fig. 185), the truck- 
axle would be radial. But the radius-bar is always made some- 
what shorter than this, and the pin is placed at 0, a considerable 
distance ahead of 0', thus creating a tendency for the truck 
to run toward the inner rail and draw the front of the loco- 
motive in that direction. This tendency will be objectionably 
great if the radius-bar is made too short, as has been practically 
demonstrated in cases when the radius-bar has been subse- 
quently lengthened with a resulting improvement in the running 
of the engine. This type of pilot truck is used on both Mogul 
and ConsoHdation locomotives and explains why these long 
engines can so easily operate on sharp curves. 



i^ 



432 



RAILROAD CONSTRUCTION. 



§400. 



400. Types of locomotive wheel-bases. The variations in 
locomotive service have developed all conceivable types as to 
total weight, ratio of total weight to weight on drivers, types of 
rmming gear, relation of steaming capacity to tractive power, 
etc. The method of classification on the basis of the running 
gear is very simple. The number of wheels on both rails of the 
pilot truck, if any, is placed as the first of three numbers. If 
there is no pilot truck, the character is used. This is followed 
by the number of drivers and then by the number of trailing 
wheels, if any. For example, a Pacific type engine has four 
wheels on the pilot truck, six driving wheels, and two trailing 
wheels under the rear of the boiler. The wheel-base is symbolized 
as 4-6-2. The most common types of locomotives, with their 
popular names and wheel base symbols, are 



American 4-4-0 

Columbia 2-4-2 

Atlantic 4-4-2 

Mogul 2-6-0 

Prairie 2-6-2 

Ten-wheel 4-6-0 

Pacific 4-6-2 

Six- wheel switcher. 0-6-0 



Consolidation 


•2-8-0 


Mikado 


2-8-2 


Mastodon 


4-8-0 


Santa Fe 


2-10-2 



Mallet A-B-B-A 

A =truckwheels, usually 2 or 
B = drivers, varying from 4 to 10 



The " Mallet " type of locomotive is one which combines 
suflScient flexibility to operate on ordinary railroad curves, wheel 
loads on the drivers which are not excessive, a very great increase 
in the total tractive power and yet operated by one engineman. 
In one respect it is like coupling two or three locomotives together, 
but the saving consists in reducing the number of enginemen 
and firemen which would be needed to run the two or three 
locomotives. Excluding freak variations, they are usually 
*^ four-cylinder compounds,'' one pair of cylinders discharging 
into the other pair and then exhausting. This type has from 
five to ten driving axles and has a length of engine wheel-base 
up to about 60 ft., but this wheel-base is flexible, so that it will 
bend on a curved track. Sometimes the boiler is made flexible 
by having a set of accordion-shaped steel rings forming a joint 
in the boiler shell. The boiler itself is on one side of this flexible 
joint and the feed-water heater, the reheater, and perhaps the 
superheater are on the other side of the joint. In this case each 
half of the flexible boiler is carried on a frame supported by one 
of the sets of driving wheels, the two frames being connected by a 
suitable joint. The boiler shell is made rigid; one end is rigidly 
attached to the frame carrying the high-pressure cylinders and 



§401. ROLLING-STOCK, 433 

the other end is supported on a bearing on the truck frame which 
carries the low-pressure cylinders and the drivers operated by 
them. The low-pressure truck frame swings around a pivot in 
the fixed frame. This flexibility has been made so great that 
these locomotives are operated successfully on 20° curves. The 
Baldwin Locomotive Works have developed this type still 
further by building a locomotive for the Erie R. R. which has 
three wheel frames, mutually flexible with each other, the third 
frame being under the tender. Each wheel frame has eight 
driving wheels. The total load carried by the twenty-four 
drivers is 761,600 lbs. or an average of 31,733 lbs. per driver. 
There are six cylinders of equal size. The two cylinders on the 
center frame use high-pressure steam and exhaust into the other 
four cylinders. The total weight of locomotive and tender is 
853,050 lbs. On a test trip it pulled a train with a total length 
of 8547 ft. or 1.6 miles, the total weight of the train being 18,338 
tons. The maximum draw-bar pull, registered by the dyna- 
mometer car, was 130,000 lbs. The adhesion between the 
drivers and the rails must have been considerably more. Such 
engines are chiefly used for hauHng long trains of slow-speed 
freight. Their boilers cannot produce steam fast enough to 
develop their enormous tractive power at high speeds and the 
power falls off rapidly with increase in speed. They are fre- 
quently equipped with automatic stokers for burning coal, or 
with oil-burning outfits, since the great amount of power devel- 
oped can only be produced by the consumption of a corresponding 
amount of fuel, and a fireman would be physically incapable of 
shoveling coal as rapidly as the production of such an amount of 
power would demand. 

LOCOMOTIVES. 
GENERAL STRUCTURE. 

401. Frame. .The frame or skeleton of a locomotive con- 
sists chiefly of a collection of forged wrought-iron bars, aa 
shown in Figs. 186 and 187. These bars are connected at the 




Fig. 186. — Engine -frame. 
front end by the "bumper" (c), which is usually made of wood. 



434 



RAILROAD CONSTRUCTION. 



§402. 



A little further back they are rigidly connected at hb by the 
cylinders and boiler-saddle. The boilers rest on the frames 
at aaaa by means of ^'pads/' which are bolted to the fire-box, 
but which permit a free expansion of the boiler along the frame. 
This expansion is sometimes as much as x\". On a '^ con- 
solidation" engine (frame shown in Fig. 187) it is frequently 




Fig. 187. — Engine -frame — Consolidation Type. 

necessary to use vertical swing-levers about 12'' long instead 
of "pads." The swinging of the levers permit all necessary 
expansion. At the back the frames are rigidly connected by 
the iron "foot-plate." The driving-axles pass through the 



"jaws" dddd, which hold the axle-boxes. The frame-bars 
have a width (in plan) of 3'' to 4". The depth (at a) is about 
the same. Fig. 186 shows a frame for an "American" type 
of locomotive; Fig. 187 shows a frame for a '^ Consolidation" 
type (see §400). 

402. Boiler. A boiler is a mechanism for transferring the 
latent heat of fuel to water, so that the water is transformed 
from cold water into high-pressure steam, which by its expan- 
sion will perform work. The efficiency of the boiler depends 
largely on its ability to do its work rapidly and to reduce to 
a minimum the waste of heat through radiation. The boiler 
contains a fire-box (see Fig. 188), in which the fuel is burned. 
The gases of consumption pass from the fire-box through the 
numerous boiler-tubes into the "smoke-box" S and out through 
the smoke-stack. The fire-box consists of an inner and outer 
shell separated by a layer of water 3" to 5" thick. The ex- 
posure of water-surface to the influence of the fire is thus very 
complete. The efficiency of this transferal of heat is somewhat 
indicated by the fact that, although the temperature of the 
gases in the fire-box is probably from 3000° to 4000° F., the 
temperature in the- smoke-box is generally reduced to 500° to 
600° F. If the steam pressure is 180 lbs., the temperature of 
the water is about 380° F., and, considering that heat will not 
pass from the gas to the water unless the gas is hotter than the 
water, the water evidently absorbs a large part of the theo- 
retical maximum. Nevertheless gases at a temperature of 



I 



§403. 



ROLLING-STOCK, 



435 



600° F. pass out of the smoke-stack and such heat is utterly 
wasted. 

The tubes vary from If" to 2", inside diameter, with a thick- 
ness of about O'MO to 0".12. The aggregate cross-sectional 




Fig. 188. — Locomotive -boiler. 



area of the tubes should be about one-eighth of the grate area. 
The number will vary from 140 to 375. The length varies from 
11' to 21', but the length is virtually determined by the type and 
length of engine. 

403. Fire-box. The fire-box is surrounded by water on the 
four sides and the top, but since the water is subjected to the 




Fig. 189. 



Fig. 190. 



boiler pressure, the plates, which are 1^" to f " thick, must be 
stayed to prevent the fire-box from collapsing. This is easily 
accomplished over the larger part of the fire-box surface by 



436 



RAILROAD CONSTRUCTION. 



§403, 



having the outside boiler-plates parallel to the fire-box plates 
and separated from them by a space of 3" to 5". The plates 




V V V V ^ V 



t::* ^-' v^ 



Am^^ 



are then mutually held by " stay-bolts/^ See Fig. 189. These 
are about |" in diameter and spaced 4'' to 4^". The A'' hole, 
drilled Uf' deep, indicated in the figure, will allow the escape 



§403. 



ROLLING-STOCK, 



437 



of steam if the bolt breaks just behind the plate, and thus calls 
attention to the break. The stay-bolts are turned down to a 
diameter equal to that at the root of the screw-threads. This 
method of supporting the fire-box sheets is used for the two 
sides, the entire rear, and for the front of the fire-box up to the 
boiler-barrel. The '' furnace tube-sheet^' — the upper part of 
the front of the fire-box — is stayed by the tubes. But the top 
of the fire-box is troublesome. It must always be covered 
with water so that it will not be " burned '^ by the intense heat. 
It must therefore be nearly, if not quite, flat. There are three 
general methods of accomplishing this. 




Fig. 192. — **Belpaire" Fire-box. 
Half -section through AB. Half -section through CD. 



(a) Radial stays. This construction is indicated in Fig. 190. 
Incidentally there is also shown the diagonal braces for resist- 
ing the pressure on the back end of the boiler above the fire- 
box. It may be seen that the stays are not perpendicular to 
either the crown-sheet or the boiler-plate. This is objection- 
able and is obviated by the other methods. 

(b) Crown-bars. These bars are in pairs, rest on the side 
furnace-plates, and are further supported by stays. See Fig. 
191. 

(c) Belpaire fire-box. The boiler above the fire-box is rect- 
angular, with rounded corners. . The stays therefore arc per- 
pendicular to the plates. See Fig. 192. 

Fire-brick arches. These are used, as shown in Fig. 193, to 
force all the gases to circulate through the upper part of the fire- 
box. Perfect combustion requires that all the carbon shall be turned 
into carbon dioxide, and this is facihtated by the forced circulation. 



438 



RAILROAD CONSTRUCTION. 



§404. 



Water-tables. The same object is attained by using a water- 
table instead of a brick arch — as shown in Fig. 191. But it has 
the further advantages of giving additional heating-surface and 
avoiding the continual expense of maintaining the bricks. One 
feature of the design is the use of a number of steam- jets which 
force air into the fire-box and assist the combustion. 



fDPPEB LriTE OfTUBES 





Fig. 193.— Fire-brick Arch. 



Fig. 194. — ^Wootten Fire-box. 



404. Area of grate. The older types of engines, as represented 

by the " American," ^' Mogul " or ^' Consolidation " type, 
always had the fire-box set between the drivers, which practically 
meant that the maximum effective inside width of the fire-box 
was limited to about 3 ft. 5 ins. for standard-gauge locomotives. 
The maximum distance over which a fireman can properly 
control a fire is perhaps 10 to 11 ft., but such extreme lengths 
are objectionable. The grate area was thus quite definitely 
limited. The Wootten fire-box, illustrated in Fig. 194, obtained 
a fire-box eight feet mde by raising it above the level of the 
drivers, as shown, but this required that the drivers should be 
objectionably small in diameter, except for low-speed engines, 
or that the fire-box would be set objectionably high. The last 
difficulty has been solved by engines of the " Columbia," " At- 
lantic," '' Pacific," " Mikado," and '' Santa Fe " types, all of 
which have a pair of trailing wheels, 36 to 45 ins. in diameter, 
set back of the driving wheels and under the fire-box, which may 
thus be widened to 7 or 8 ft., the entire fire-box being placed back 
of the driving wheels. 

405. Superheaters. Inside of a boiler the steam has a tem- 
perature corresponding to its pressure. For example, if the 
pressure is 180 lbs., the temperature is about 379° F. When the 
steam of a locomotive is superheated, the steam is conducted 
from the throttle to the cylinders through pipes which are pur- 



^ 



§ 406. ROLLING-STOCK. 439 

posely placed in the path of the flue gases on their way to the 
smokestack. A simple form of superheater is a series of tubes 
and drums located in the smokebox. Here the temperature is 
perhaps 600° F., which is sufficient to heat the steam from 30° 
to 50° above the boiler temperature and to produce substantial 
economies. In another more effective but more costly type a 
considerable number of the ordinary 2i-inch boiler tubes are 
replaced by 5J-inch tubes, inside of each of which is a pipe loop 
extending from the smokebox headers to within a short dis- 
tance of the fire-box, where the temperature approaches the 
fire-box temperature, which is perhaps 2000° F. The hve steam 
passes through these loops and is so heated that, even after it 
reaches the cylinder, it has a superheat of 150° to 200° over the 
boiler temperature, but since its pressure is substantially the 
boiler pressure, the quantity (or weight) of steam required to fill 
the cylinder at that temperature and pressure is much less 
than the quantity of steam at the same pressure but lower tem- 
perature. Superheating also has the advantage of making the 
steam more dry and of preventing condensation in the cyhnders 
until the steam has lost in temperature at least the amount of its 
superheat. Superheating is chiefly advantageous for use with 
passenger engines, when they must work at high power for long, 
continuous runs. An economy of 15 to 25% in coal consumption 
(and even 30% in some tests), can ordinarily be obtained by the 
use of superheaters, but the economy is somewhat offset by the 
additional cost for installation and for subsequent repairs and 
maintenance. 

406. Reheaters. A reheater is substantially the same as a 
superheater in its general principle of construction. When steam 
has been exhausted from a high-pressure cylinder, the tempera- 
ture and pressure are both considerably lower than their boiler 
values. If the steam is to be again used, an economy is obtained 
and the steam is dried by passing it through a reheater. They 
are generally used on Mallet engines to reheat the steam in its 
passage from the high-pressure to the low-pressure cylinders. 

407. Coal consumption. No form of steam-boiler (except 
a boiler for a steam fire-engine) requires as rapid production of 
steam, considering the size of the boiler and fire-box, as a 
locomotive. The combustion of coal per square foot of grate 
per hour for stationary boilers averages about 15 to 25 lbs. and 
seldom exceeds that amount. An ordinary maximum for a 



440 RAILROAD CONSTRUCTION. § 407. 

locomotive is 125 lbs. of coal per square foot of grate-arjea per 
hour, and in some recent practice 220 lbs,, have been used. Of 
course such excessive amounts are wasteful of coal, because 
a considerable percentage of the coal will be blown out of the 
smoke-stack unconsumed, the draft necessary for such rapid 
consumption being very great. The only justification of such 
rapid and wasteful coal consumption is the necessity for rapid 
production of steam. The best quality of coal is capable of 
evaporating about 14 lbs. of water per pound of coal, i.e., change 
it from water at 212° to steam at 212°; the heat required to 
change water at ordinary temperatures to steam at ordinary 
working pressure is (roughly) about 20% more. From 6 to 9 lbs. 
of water per pound of coal is the average performance of ordinary 
locomotives, the efficiency being less with the higher rates of 
combustion. Some careful tests of locomotive coal consump- 
tion gave the following figures: when the consumption of coal 
was 50 lbs. per square foot of grate-area per hour, the rate of 
evaporation was 8 lbs. of water per pound of coal. When the 
rate of coal consumption was raised to 180, the evaporation 
dropped to 5 lbs. of w^ater per pound of coal. It has been 
demonstrated that the efficiency of the boiler is largely increased 
by an increased length of boiler-tubes. The actual consump- 
tion of coal per mile is of course an exceedingly variable quan- 
tity, depending on the size and type of the engine and also on 
the work it is doing — whether climbing a heavy grade with its 
maximum train-load or running easily over a level or down 
grade. A test of a 50-ton engine, running without an}^ train at 
about 20 to 25 miles per hour, showed an average consumption 
of 21 lbs. of coal per mile. Statistics of the Pennsylvania Rail 
road show a large increase (as might be expected, considering 
the growth in size of engines and weight of trains) in the aver- 
age number of pounds of coal burned per ^ram-mile — some of 
the figures being 55 lbs. in 1863, 72 lbs. in 1872, and nearly 
84 lbs. in 1883. Figures are published showing an average 
consumption of about 10 lbs. of coal per passenger-car mile, 
and 4 to 5 lbs. per freight-car mile. But these figures are always 
oblained by dividing the total consumption per train-mile by 
the number of cars, the coal due to the weight of the engine 
being thrown in. Wellington developed a rule, based on the 
actual performance of a very large number of passenger- trains, 
that the number of pounds of coal per mile = 21.1 +6.74 times 



I 



§ 408. ROLLING-STOCK. ^44T 

the number of passenger-cars. The amount of coal assigned 
to the engine agrees remarkably with the test noted above 
For freight-trains the amount assigned to the engine should 
be much greater (since the engine is much heavier), and that 
assigned to the individual cars much less, although the great 
increase in freight-car weights in recent years has caused an 
increase in the coal required per <*rt. 

There is a physical limit to the amount of coal which can be 
shovelled into a firebox by a fireman. Tests have shown that 
the average fireman can handle about 4000 lbs. of coal per hour 
and keep up such work almost indefinitely. For a short time 
he can shovel coal at the rate of 80 or 90 lbs. per minute, and 
this may be necessary to keep up steam while the train is going 
over some hump, but it must be followed by some relief which 
will make the average about the same. Automatic stokers have 
been devised for locomotives which can feed as much as 6000 lbs. 
of coal per hour when the grate area is less than 70 square feet 
and up to 8000 lbs. per hour when the grate area is 70 square feet 
or over. These are necessary on some of the most powerful 
locomotives in order to produce steam fast enough to develop 
their maximum capacity. 

408. Oil-burning locomotives. In 1912 over one-sixth of all 
the locomotives west of the Mississippi River used oil as fuel. 
Some of the advantages in using oil are as follows: (1) the 
British thermal units in one pound of oil vary from about 19,000 
to 21,000; those in a pound of coal vary from perhaps 14,000 for 
the very best down to 5000 for the poorer grades of lignite found 
in the western parts of the United States, and this means a great 
reduction in the cost of carrying and storing fuel, measured in 
heat units; (2) the cost of handling fuel is reduced and that of 
disposing of ashes is eliminated; (3) engine repairs are reduced 
in many respects, although it is said that the increased cost of 
fire-box repairs, due to the intense heat of the oil flame, offsets 
any reduction in other items; (4) the fires can be more easily 
controlled and waste of heat reduced during stoppages or when 
drifting down grade; (5) wayside fires due to sparks are alto- 
gether eliminated; (6) there is a practical limitation (see § 407), 
to the amount of coal that one fireman can feed to a fire; but 
there is no such limitation when using oil; (7) there is an equality 
in cost of heat units when a 42-gallon barrel of oil, weighing 7.3 
lbs. per gallon, costs 60 cents and a ton (2000 lbs.) of coal, having 



442 RAILROAD CONSTRUCTION. §409. 

two-thirds as many heat units per pound, costs $2.61, or 4.35 
times as much. The other items of difference almost invariably 
favor the oil and might make it more desirable even when the 
ratio of cost seemed to favor the coal. The extensive use of oil 
west of the Mississippi River is due to the fact that in many 
localities a very suitable quality of crude oil is plentiful and 
cheap while coal is expensive and of low calorific power. 

409. Heating-surface. The rapid production of steam requires 
that the hot gases shall have a large heating-surface to which 
they can impart their heat. From 50 to 75 square feet of 
heating-surface is usually designed for each square foot of 
grate-area. A more recently used rule is that there should be 
from 60 to 70 square feet of tube heating-surface per square 
foot of grate-area for bituminous coal. 40 or 50 to 1 is more 
desirable for anthracite coal. Almost the whole surface of 
the fire-box has water behind it, and hence constitutes heating- 
surface. Although this surface forms but a small part of the 
total (nominally), it is really the most effective portion, since 
the difference of temperature of the gases of combustion and 
the water is here a maximum, and the flow of heat is therefore 
the most rapid. The heating-surface of the tubes varies from 
85 to 93% of the total, or about 7 to 15 times the heating-surface 
in the fire-box. By dividing the total weight of a well-designed 
engine (exclusive of tender) by the number of square feet of 
heating-surface (fire-box and tubes), we get a quotient which 
varies from 60 to 80 or over. For example, a light engine, weigh- 
ing only 96,450 lbs. had a total heating surface of 1449 square 
feet, or about 67 lbs. per square foot. On the other hand, a 
Mikado engine, weighing 297,500 lbs., had 4359 square feet of 
heating surface, or 68 lbs. per square foot. 

410. Loss of efficiency in steam pressure. The effective 
work done by the piston is never equal to the theoretical energy 
contained in the steam withdrawn from the boiler. This is due 
chiefly to the following causes: 

(a) The steam is " wire-drawn," i.e., the pressure in the 
cylinder is seldom more than 85 to 90% of the boiler pressure. 
This is due largely to the fact that the steam-ports are so small 
that the steam cannot get into the cylinder fast enough to exert 
its full pressure. Partially closing the throttle, so that the 
steam will be used less rapidly, also wire-draws the steam. 

(b) Entrained water. Steam is always drawn from a dome 



§411. ROLLING-STOCK. 443 

placed over the boiler so that the steam shall be as far above 
the water-surface as possible, and shall be as dry as possible. 
In spite of this the steam is not perfectly dry and carries with 
it water at a temperature of, say, 361°, and pressure of 140 lbs. 
per square inch. When the pressure falls during the expan- 
sion and exhaust, this hot water turns into steam and absorbs 
the necessary heat from the hot cjdinder-w^alls. This heat is 
then carried out by the exhaust and wasted. 

(c) The back pressure of the exhaust-steam, which depends 
on the form of the exhaust-passages, etc. This amounts to 
from 2 to 20% of the power developed. 

(d) Clearance-spaces. When cutting off at full stroke . this 
waste is considerable (7 to 9%), but when the steam is used 
expansively the steam in these clearance-spaces expands and 
so its power is not wholly lost. 

(c) Radiation. In spite of all possible care in jacketing the 
cylinders, some heat is lost by radiation. 

(/) Radiation into the exhaust-steam. This is somewhat 
analogous to (b). Steam enters the cylinder at a temperature 
of, say, 361°; the walls of the cylinder are much cooler, say 250°; 
some heat is used in raising the temperature of the cylinder- 
walls; some steam is vaporized in so doing; when the exhaust 
is opened the temperature and pressure fall; the heat tem- 
porarily absorbed by the c^^inder-walls is reabsorbed by the 
exhaust-steam, re-evaporating the vapor pre\dously formed, 
and thus a certain portion of heat-energy goes through the 
cylinder vrithout doing any useful work. With an early cut-off 
the loss due to this cause is very great. 

The sum of all these losses is exceedingly variable. They 
are usually less at lower speeds. The loss in initial pressure 
(the difference between boiler pressure and the cylinder pres- 
sure at the beginning of the stroke) is frequently over 20%, 
but this is not all a net loss With an early cut-off the average 
cylinder pressure for the whole stroke is but a small part of 
the boiler pressure, yet the horse -power developed may be as 
great as, or greater than, that developed at a lower speed, later 
cut-off, and higher average pressure. 

411. Tractive power The work done by the two cylinders 
during a complete revolution of the drivers evidently = area of 
pistons X average steam pressure X stroke X 2X2. The resist- 
ance overcome evidently = tractive force at circumference of 



444 RAILROAD CONSTRUCTION. § 412. 

drivers times distance traveled by drivers (which is the cir- 
cumference of the drivers) Therefore 



-1 



area pistons X average steam pressure 

^ ^. - ^ XstrokeX2x2. 

Tractive force == ] ; t >-j-^ . 

circumference of drivers 

Dividing numerator and denominator by n (3.1415), we have 

r (diam piston) ^ X average steam 

\ pressure X stroke . 

Tractive force = ) ^. ^-r-. — , . (103) 

( diameter of driver 

which is the usual rule Although the rule is generally stated 
in this form, there are several deductions In the first place 
the net effective area of the piston is less than the nominal on 
account of the area of the piston-rod. The ratio of the areas 
of the piston-rod and piston varies, but the effect of this reduc- 
tion is usually from 1.3 to 1.7%. No allowance has been made 
for friction — of the piston, piston-rod, cross-head, and the 
various bearings This would make a still further reduction 
of several per cent. Nevertheless the above simple rule is 
used, because, as will be shown, no great accuracy can be 
utiHzed. 

The maximum draw bar pull is limited by the adhesion between 
the driving wheels and the rails. This is usually about one- 
fourth of the weight. The use of sand may increase it to one- 
third. But this ratio is important only when starting or at very 
low speeds. The adhesion is always ample for the much lower 
cylinder power which can be developed at higher speeds. This 
is considered more fully in Chapter XVIII, 

RUNNING GEAR. 

412. Equalizing-levers. The ideal condition of track, from 
the standpoint of smooth running of the rolling stock, is that 
the rails should always lie in a plane surface. While this con- 
dition is theoretically possible on tangents, it is unobtainable 
on curves, and especially on the approaches to curves when the 
outer rail is being raised. Even on tangents it is impossible 
to maintain a perfect surface, no matter how perfectly the 
track may have been laid. In consequence of this, the points 



§412. 



ROLLING-STOCK, 



445 



of contact of the wheels of a locomotive, or even of a four- 
' wheeled truck, will not ordinarily lie in one plane. The rougher 
and more defective the track, the worse the condition in this 
respect. Since the frame of a locomotive is practically rigid, 
and the frame rests on the driver-axles through the medium of 
springs at each axle-bearing, the compression of the springs 
(and hence the pressure of the drivers on the rail) will be varia- 
ble if the bearing-points of the drivers are not in one plane 
This variable pressure affects the tractive power and severely 
strains the frame. Applying the principle that a tripod will 
stand on an uneven surface, a mechanism is emploj^ed which 




r>l~~^-:^ 



Fig. 195. — Action of Equalizing-levers. 

virtually supports the locomotive on three points, of which one 
is usually the center-bearing of the forward truck. On each 
side the pressure is so distributed among the drivers that even 
if a driver rises or falls with reference to the others, the load 
carried by each driver is unaltered, and that side of the engine 
rises or falls by one nth of the rise or fall of the single driver, 
where n represents the number of wheels. The principle in- 
volved is shown in an exaggerated form in Fig. 195. In the 
diagram, ikfAT^ represents the normal position of the frame when 
the wheels are on line. The frame is supported by the hangers 
at a, c, /, and h, ah, de, and gh are horizontal levers vibrating 
about the points H^ Kj and L, which are supported by the 
axles. While it is possible with such a system of levers to make 



446 RAILROAD CONSTRUCTION. § 412. 

MN assume a position not parallel with its natural position, 
yet, by an^extension of the principle that a beam balance loaded 
with equal weights will always be horizontal, the effect of rais- 
ing or lowering a wheel will be to move MN parallel to itself. 
It only remains to determine how much is the motion of MN 
relative to the rise or drop of the wheel. 

The dotted Unes represent the positions of the wheels and 
levers when one wheel drops into a depression. The wheel 
center dreps from p to g, a distance m. L drops to L', a 
distance m (see Fig. 195, 6); M drops to M', an unknown dis- 
tance X) therefore aa' =x] hV =x] cc' =x] dd' = 3x = ee''y i'f = x] 
.'. gg' — 5x] hh' =x) LU = ^{gg' -\'hh') = l(fix)=m] .\ x = \m) 
i.e., MN drops, parallel to itself, 1/n as much as the wheel 
drops, where n is the number of wheels. The resultant effect 
caused by the simultaneous motion of two wheels with refer- 
ence to the third is evidently the algebraic sum of the effects 
of each wheel taken separately. 

The practical benefits of this device are therefore as follows: 

(a) When any driver reaches a rough place in the track, a 
high place or a low place, the stress in all the various hangers 
and levers is unchanged. 

(6) The motion of the frame (represented by the bar MN 
in Fig. 195) is but 1/n of the motion of the wheel, and the jar 
and vibration caused by a roughness in the track is correspond- 
ingly reduced. 

The details of applying these principles are varied, but in 
general it is done as follows: 

(a) American and ten wheeled types. Drivers on each side 
form a system. The center -bearing pilot-truck is the third 
point of support. The method is illustrated in Fig. 196. 

(b) Mogul and consolidation types. The front pair of drivers 
is connected with the two-wheeled pilot-truck (as illustrated 
in Fig. 197) to form one system. The remaining drivers on 
each side are each formed into a system. 

The device of equalizers is an American invention. Until 
recently it has not been used on foreign locomotives. The 
necessity for its use becomes less as the track is maintained 
with greater perfection and is more free from sharp curves. A 
locomotive not equipped with this device would deteriorate 
very rapidly on the comparatively rough tracks which are 
usually found on light-traffic roads. It is still an open ques- 



§ 412, 



EOLLING-STOCK. 



447 



■G 



lO 



■ii ^ 



a 



.^N 



^yy^^-%^- 



Oi 




tion to what extent the neglect of this device is responsible for 
the statistical fact that average freight-train loads on foreign 



448 RAILROAD CONSTRUCTION. § 413. 

trains are less in proportion to the weight on the drivers than 
is the case with American practice. The recent increasing use 
of this device on foreign heavy freight locomotives is perhaps 
an acknowledgment of this principle. 

413. Counterbalancing. At very high velocities the cen- 
trifugal force developed by the weight of the rotating parts 
becomes a quantity which cannot be safely neglected. These 
rotating parts include the crank-pin, the crank-pin boss, the 
side rod, and that part of the weight of the connecting-rod 
which may be considered as rotating about the center of the 
crank-driver. As a numerical illustration, a driving-wheel 
62" in diameter, running 60 miles per hour, will revolve 325 
times per minute. The weights are: 

Crank-pin 110 lbs. 

boss 150 *' 

One-half side rod 240 " 

Back end of connecting-rod 190 *' 

Total . 690 lbs. 

If the stroke is 24", the radius of rotation is 12", or 1 foot. Then 
Gv^ 690X471^2X3252 ^^__^ ,, 
-^= 32.2X1X6Q2 -24821 lbs., 

which is half as much again as the weight on a driver, 16000 lbs. 
Therefore if no counterbalancing were used, the pressure be- 
tween the drivers and the rail would always be less (at any 
velocity) when the crank-pin was at its highest point. At a 
velocity of about 48 miles per hour the pressure would become 
zero, and at higher velocities the wheel would actually be 
thrown from the rail. As an additional objection, when the 
crank-pin was at the lowest point, the rail pressure would be 
increased (velocity 60 miles per hour) from 16000 lbs. to nearly 
41000 lbs., an objectionably high pressure. These injurious 
effects are neutralized by '^counterbalancing.'' Since all of 
the above-mentioned weights can be considered as concen- 
trated at the center of the crank-pin, if a sufficient weight is so 
placed in the drivers that the center of gravity of the eccentric 
weight is diametrically opposite to the crank-pin, this centrifu- 
gal force can be wholly balanced. This is done by filling up 
a portion of the space between the spokes. If the center of 
gravity of the counterbalancing weight is 20" from the center, 
then, since the crank-pin radius is 12", the required weight 
would be 690 X^ = 414 lbs. 



,w' 



§ 413. ROLLING-STOCK. 449 

In addition to the effect of these revolving parts there is 
the effect of the sudden acceleration and retardation of the 
reciprocating parts. In the engine above considered the weights 
of these reciprocating parts will be: 

Front end of connecting-rod 150 lbs. 

Cross-head '. . . . 174 ^' 

Piston and piston-rod 300 ' ' 

Total 624 lbs. 

Assume as before that the reciprocating parts may be con- 
sidered as concentrated at one point, the point P of the dia- 
gram in Fig. 198. Since the ^ ^ 

motion of P is horizontal ,^''' 

only, the force required to / /^ 

overcome its inertia at any ^^^ j l_ 

point will exactly equal p ' ***"*A^_Vy 

the horizontal component of \S^ 

the force required to over- ^^-^^ ^^y 

come the inertia of an equal 

^^' \.4r «+ e «^,r^i,r;„« ,-^ ■^^^- 198. — Action of Co: nterbalance. 
weight at o revoivmg m 

a circular path. Then evidently the horizontal component of 
the force required to keep W in the circular path will exactly 
balance the force required to overcome the inertia of P. 0£ 
course W=P. But a smaller weight W\ whose weight is 
inversely proportional to its radius of rotation, wdll evidently 
accomplish the same result. In the above numerical case, if 
the center of gravity of the counterweights is 20'' from the 
center, the required weight to completely counterbalance 
the reciprocating parts would be 624 X if = 374.4 lbs. This 
counterweight need not be all placed on the driver carrying 
the main crank-pin, but can be (and is) distributed among all 
the drivers. Suppose it were divided between the two drivers 
in the above case. At 60 miles pet hour such a counterweight 
would produce an additional pressure of 11211 lbs. when the 
counterw^eight was down, or a lifting force of the same amount 
when the counterweight was up. Although this is not suffi- 
cient to lift the driver from the rail, it would produce an objec- 
tionably high pressure on the rail (over 27000 lbs.), thus inducing 
just what it w^as desired to avoid on account of the eccentric 
rotating parts. Therefore a compromise must be made. Only 
a portion (one half to three fourths) of the weight of the recip- 
rocating parts is balanced. Since the effect of the rotating 



450 RAILROAD CONSTRUCTION. §413. 

weights is to cause variable pressure on the rail, while the effect 
of the reciprocating parts is to cause a horizontal wobbling or 
"nosing" of the locomotive, it is impossible to balance both. 
Enough counterweight is introduced to partially neutrahze the 
effect of the reciprocating parts, still leaving some tendency 
to horizontal wobbling, while the counterweights which were 
introduced to reduce the wobbling cause some variation of 
pressure. By using hollow piston-rods of steel, ribbed cross- 
heads, and connecting- and side-rods with an I section, the 
weight of the reciprocating parts may be greatly lessened with- 
out reducing their strength, and with a decrease in weight the 
effect of the unbalanced reciprocating parts and of the ''excess 
balance" (that used to balance the reciprocating parts) is 
largely reduced. 

Current practice is somewhat variable on three features: 

(a) The proportion of the weight of the connecting-rod which 
should be considered as revolving weight. 

(6) The proportion of the total reciprocating weight that 
should be balanced. 

(c) The distribution among the drivers of the counterweight 
to balance the reciprocating parts. 

» An exact theoretical analysis of (a) shows that it is a func- 
tion of the weights and dimensions of the reciprocating parts. 
The weight which may be considered as revolving equals * 




in which r = radius of the crank, Z = length of connecting-rod, 
/c = distance of center of gyration from wrist-pin, d = distance 
of center of gravity from wrist-pin, TFi= weight of connecting- 
rod in pounds, and ^2 = weight of piston, piston-rod, and cross- 
head in pounds; all dimensions in feet. An apphcation of this 
formula will show that for the dimensions of usual practice, 
from 51 to 57% of the weight of the connecting-rod should bd 
considered as revolving weight. 

The principal rules which have been formulated for counter- 
balancing may be stated as follows: 

1. Each wheel should be balanced correctly for the revolving 

parts connected wit h it. ^_ 

* R. A. Parke, in R. R. Gazette, Feb. 23, 1894. 



§414. 



ROLLING-STOCK. 



451 



2. In addition J introduce counterbalance sufficient for 50% 
of the weight of the reciprocating parts for ordinary engines, 
increasing this to 75% when the reciprocating parts are exces- 
sively heavy (as in compound locomotives) or when the engine 
is light and unable to withstand much lateral strain or when 
the wheel-base is short. 

3. Consider the weight of the connecting-rod as J revolving 
and i reciprocating when it is over 8 feet long; when shorter 
than 8 feet, consider -^^ of the weight as revolving and -^-^ as 
reciprocating. 

4. The part of the weight of the connecting-rod considered 
as revolving should be entirely balanced in the crank-driver wheel . 

5. The ^'excess balance" should be divided equally among 
the drivers. 

6. Place the counterbalance as near the rim of the wheel 
as possible and also as near the outside 
of the wheel as possible in order that 
the center of gravity shall be as near 
as possible opposite the center of 
gravity of the rods, etc., wliich are all 
outside of even the plane of the face 
of the wheel. 

In Fig. 199 is shown a section of a 
locomotive driver with the cavities in 
the casting for the accommodation of 
the lead which is used for the counter- 
balance weight. Incidentally several 
other features and dimensions are shown 
in the illustration. 

414. Mutual relations of the boiler power, tractive power, 
and cylinder power for various types. The design of a locomo- 
tive includes three distinct features which are varied in their 
mutual relations according to the work which the engine is 
expected to do. 

(a) The boiler power. This is limited by the rate at which 
steam may be generated in a boiler of admissible size and weight. 
Engines which are designed to haul very fast trains which are 
:comparatively light must be equipped with very large grates and 
heating surfaces so that steam may be developed with great 
rapidity in order to keep up with the very rapid consumption. 




Fig. 199. — Section of 
Locomotive-driver. 



452 RAILROAD CONSTRUCTION. §414. 

Engines for very heavy freight work are run at very much 
lower velocity and at a lower piston speed in spite of the fact 
that more strokes are required to cover a given distance and 
the demand on the boiler for rapid steam production is not 
as great as with high-speed passenger-engines. The capacity of 
a boiler to produce steam is therefore limited by the limiting 
weight of the general type of engine required. Although im- 
provements may be and have been made in the design of fire- 
boxes so as to increase the steam-producing capacity without 
adding proportionately to the weight, yet there is a more or less 
definite limit to the boiler power of an engine of given weight. 

(b) The tractive power. This is limited by the possible driver 
adhesion. The absolute limit of tractive adhesion between a 
steel-tired wheel and a steel rail is about one-third of the pressure, 
but not more than one-fourth of the weight on the drivers can 
be depended on for adhesion and wet rails will often reduce 
this to one fifth and even less. The tractive power is therefore 
absolutely limited by the practicable weight of the engine. In 
some designs, when the maximum tractive power is desired, not 
only is the entire weight of the boiler and running gear thrown 
on the drivers, but even the tank and fuel-box are loaded on. 
Such designs are generally employed in switching-engines (or 
on engines designed for use on abnormally heavy mountain 
grades) in which the maximum tractive power is required, but 
in which there is no great tax on the boiler for rapid steam pro- 
duction (the speed being always very low), and the boiler and 
fire-box, which furnish the great bulk of the weight of an engine, 
are therefore comparatively light, and the requisite weight for 
traction must, therefore, be obtained by loading the drivers 
as much as possible. On the other hand, engines of the highest 
speed cannot possibly produce steam fast enough to maintain 
the required speed unless the load be cut down to a compara- 
tively small amount. The tractive power required for this 
comparatively small load will be but a small part of the weight 
of the engine, and therefore engines of this class have but a 
small proportion of their weight on the drivers; generally 
have but two driving-axles and sometimes but one. 

(c) Cylinder power. The running gear forms a mechanism 
which is simply a means of transforming the energy of the boiler 
into tractive force and its power is unlimited, within the prac- 
tical conditions of the problem. The power of the running 



§414. 



ROLLING-STOCK. 



453 



gear depends on the steam pressure, on the area of the piston, 
oil the diameter of the drivers, and on the ratio of crank-pin 
radius to wheel radius, or of stroke to driver diameter. It 
is always possible to increase one or more of these elements 
by a relatively small increase of expenditure until the cylinders 
are able to make the drivers slip, assuming a sufficiently great 
resistance. Since the power of the engine is limited by the 
power of its weakest feature, and since the running gear is the 
most easily controlled feature, the power of the running gear 
(or the '^ cylinder power'') is always made somewhat excessive 
on all well-designed engines. It indicates a badly designed 
engine if it is stalled and unable to move its drivers, the steam 
pressure being normal. If it is attempted to use a freight- 
engine on fast passenger service, it will probably fail to attain 
the desired speed on account of the steam pressure falling. 
The tractive power and cylinder power are superabundant, but 
the boiler' cannot make steam as fast as it is needed for high 
speed, especially when the drivers are small. The practical 
result would be a comparatively low speed kept up with a forced 
fire. If it is attempted to use a high-speed passenger-engine 
on heavy freight service, the logical result is a slipping of the 
drivers until the load is reduced. The boiler power and cylinder 
power are ample, but the weight on the drivers is so small that 
the tractive power is only sufficient to draw a comparatively 
small load. 

These relations between boiler, cylinder, and tractive power 
are illustrated in the following comparative figures referring 
to a fast passenger-engine, a heavy freight-engine, and a switch- 
ing-engine. The weights of the passenger- and freight-engines 
are about the same, but the passenger-engine has only 74% * 





Cylinders. 


Total 
Wght. 


Wt. on 
Driv'rs 


Heat- 
ing 
Sur- 
face, 

sq. ft. 


Grate 
area 

sq. ft. 


Steam 
Pres- 
sure in 
Boiler. 


Stroke. 


Kind. 


Diam. 
Driver. 


Fast passenger . 
Heavy freight . 
Switcher 


19''X24'' 
20''X24" 
19''X24" 


126700 
128700 
109000 


81500 
112600 
109000 


1831.8 
1498.3 
1498.0 


26.2 
31.5 

22.8 


180 
140 
160 





* Computed from Eq. 137. 



454 KAILROAD CONSTRUCTION. § 415. 

of the tractive power of the freight. But the passenger-engine 
has 22% more heating-surface and can generate steam much 
faster; it makes less than two- thirds as many strokes in cover- 
ing a given distance, but it runs at perhaps twice the speed 
and probably consumes steam much faster. The switch- 
engine is lighter in total weight, but the tractive power is a little 
greater than the freight and much greater than the passenger- 
engine. While the heating-surfaces of the freight- and switch- 
ing engines are practically identical, the grate area of the switcher 
is much less; its speed is always low and there is but little neces- 
sity for rapid steam development. 

While these figures show the general tendency for the relative 
proportions, and in this respect may be considered as typical, 
there are large variations. The recent enormous increase in 
the dead weight of passenger-trains has necessitated greater 
tractive power. This has been provided sometimes by using 
the " Pacific " type, which combines rapid steaming capacity 
and great tractive power. On the other hand, the demand for 
fast-freight service, and the possibility of safely operating such 
trains by the use of air-brakes, has required that heavy freight- 
engines shall be run at comparatively high speeds, and that 
requires the rapid production of steam, large grate areas and 
heating surfaces. But in spite of these variations, the normal 
standard for passenger service is a four-driver engine carrying 
about two-thirds of the weight of the engine on the drivers, I 
which are very large; the normal standard for freight work is v 
an 8-driver engine with perhaps 90% of the weight on the 
drivers, which are small, but which must have the pony truck 
for such speed as it uses; and finally the normal standard for 
switching service has all the weight on the drivers and has com- 
paratively low steam-producing capacity. 

415. Life of Locomotives. The life of locomotives (as a 
whole) may be taken as about 800000 miles or about 22 to 24 
years. While its life should be and is considered as the period 
between its construction and its final consignment to the scrap 
pile, parts of the locomotive may have been renewed more 
than once. The boiler and fire-box are especially subject to 
renewal. The mileage life is much longer than formerl}^ This 
is due partly to better design and partly to the custom of 
drawing the fires less frequently and thereby avoiding some 
of the destructive strains caused by extreme alternations of 



§ 416. ROLLING-STOCK. 455 

heat and cold. Recent statistics give the average annual 
mileage on twenty-three leading roads to be 41000 miles. 

CARS. 

416. Capacity and size of cars. The capacity of freight-cars 
has been enormously increased of late years. In 1870 the usual 
live-load capacity for a box-car was about 20000 lbs. In 1916, 
out of 58299 box cars owned by the Pennsylvania R. R., 32923 
or 56% had a capacity of 100000 or over; 49597 or 85% had a 
capacity 70000 or over; only 555, less than 1%, had a capacity 
of less than 60000 lbs., and the most of these were refrigerator 
cars or cars for special service. The Norfolk & Western R. R. 
had (in 1916), 750 gondola drop-bottom coal cars, each with a 
nominal capacity of 180000 lbs.; their length is 46 feet lOf 
inches, and the extreme width 10 feet 4 J inches. These cars 
are carried on six-wheel trucks. The usual width of freight- 
cars is about 9 to 10 feet, while parlor-cars and sleepers are 
generally 10 feet wide and sometimes 11 feet. The highest 
point of a train is usually the smokestack of the locomotive, 
which is generally 15 feet above the rails and occasionally over 
16 feet. A sleeping-car usually has the highest point of the 
car about 14 feet above the rails. Box-cars are usually about 
8 feet high (above the sills), with a total height of 13 to 14 feet. 
Some furniture and automobile cars, whose unit live load per 
cubic foot of space is not high, have a total height of over 15 feet. 
The average length of freight cars, as required in the design of 
freight yards, is now considered to be 42 feet; the allowance for 
each car was formerly 40 feet. The P. R. R. standards vary 
between 38 feet 1 inch and 44 feet 6 inches in length. Day 
coaches have an extreme length varying from 45 to 80 feet. An 
80-foot all-steel coach weighs about 118000 lbs. and has a seating 
capacity of 88. Allowing the high average weight of 150 lbs., 
the maximum hve load would be 13200 lbs., a little over 11% of 
the dead load, which shows that the tractive force required to 
haul the car will be almost constant, whether the car is full or 
empty. A dining-car may weigh 150000 lbs. and a sleeper even 
more. The weight of the 25 or 30 passengers it may carry is 
hardly worth considering in comparison. 

417. Stresses to which car-frames are subjected. A car 
is structurally a truss, supported at points at some distance 
from the ends and subjected to transverse stress. There is, 



456 RAILROAD CONSTRUCTLON. § 418. 

therefore, a change of flexure at two points between the trucks* 
Besides this stress the floor is subjected to compression when 
the cars are suddenly stopped and to tension when in ordinary 
motion, the tension being greater as the train resistance is 
geater and as the car is nearer the engine. The shcxjks, jars, 
and sudden strains to which the car-frames are subjected are ii 
very much harder on them than the mere static strains due tof | 
their maximum loads if the loads were quiescent. Consequently 
any calculations based on the static loads are practically value- 
less, except as a very rough guide, and previous experience 
must be relied on in designing car bodies. As evidence of the 
increasing demand for strength in car-frames, it has been re- 
cently observed that freight-cars, built some years ago and 
built almost entirely of wood, are requiring repairs of wooden 
parts w^hich have been crushed in service, the wood being per- 
fectly sound as regards decay. 
I 418. The use of metal. The use of metal in car construction 




Fig. 201. 

is very rapidly increasing. The demand for greater strength 
in car-frames has grown until the wooden framing has become 
so heavy that it is found possible to make steel frames and 
trucks at a small additional cost, the steel frames being twice 
as strong and yet reducing the dead weight of the car about 
5000 lbs., a consideration of no small value, especially on roads 
ha\^ng heavy grades. Another reason for the increasing use 
of metal is the great reduction in the price of rolled or pressed 




IIOO.OOO-LB. Box Cab. 




Steel Coal Cab. 




Wooden Box Car; Steel Framb..^ 
Fig. 200. — Some Heavy Freioht Car§, 
{To face page 456.) 



§ 419. ROLLING-STOCK. 457 

steel, while the cost of wood is possibly higher than before. 
The advocates of the use of steel advise steel floors, sides, etc. 
For box-cars a wooden, floor has advantages. For ore and 
coal-cars an all-metal construction has advantages. (Fig. 200.) 
In Germany, where steel frames have been almost exclusively 
in use for many years, they have not yet been able to determine 
the normal age limit of such frames; none have yet worn outc 
The Hfe is estimated at 50 to 80 years 

Brake beams are also best made of metal rather than wood, 
as was formerly done. Metal brake-beams are generally used on 
cars having air-brakes, as a wooden beam must be excessively 
large and heavy in order to have sufficient rigidity. 

Truck-frames (see Fig. 201), which were formerly made prin- 
cipally of wood, are now largely made of pressed steel. It makes 
a reduction in weight of about 3000 lbs. per car. The increased 
durability is still an uncertain quantity. 

419. Draft gear. The enormous increase in the weight and 
live load capacities of rolling stock have necessitated a corre- 
sponding development in draft gear. Even within recent years, 
*^ coal- jimmies," carrying a few tons have been made up into 
trains by dropping a chain of three big links over hooks on the 
ends of the cars. But the great stresses due to present loadings 
would tear such hooks from the cars or tear the cars apart if 
such cars were used in the make-up of long heavy trains as now 
operated. The next stage in the development of draft gear was 
the invention of the '^spring coupler," by which the energy due 
to a sudden tensile jerk or the impact of compression may be 
absorbed by heavy springs and gradually imparted to the car 
body. Such devices, for which there are many designs, seemed 
to answer the purpose for cars of 25 to 40 tons capacity. The 
use of 100,000-pound steel cars soon proved the inadequacy of 
even spring couplers. The friction-draft gear was then in- 
vented. The general principle of such a gear^ is that, when 
acting at or near its maximum capacity, it harmlessly trans- 
forms into heat the excessive energy developed by jerks or 
compression. There are several different designs of such gear, 
but the general principle underlying all of them may be illus- 
trated by a description of the Westinghouse draft gear. The 
gear employs springs which have sufficient stiffness to act as 
ordinary spring-couplers for the ordinary pushing and pulling 
of train operations. Sections of the gear are shown in Fig. 202, 



1. 



458 



RAILKOAD CONSTRUCTION, 



§419. 




a- ^ 




§ 420. ROLLING-STOCK. 459 

while the method of its application to the framing of a car of 
the pressed steel type is shown in Fig. 203, a and h. When 
the draft gear is in tension the coupler, which is rigidly attached 
to By is drawn to the left, drawing the follower Z with it. Com- 
pression is then exerted through the gear mechanism to the 
follower A which, being restrained by the shoulders RRj against 
which it presses, causes the gear to absorb the compression. 
The coil-spring C forces the eight wedges n against the eight 
corresponding segments E. The great compression of these 
surfaces against the outer shell produces a friction which retards 
the compression of the gear. The total possible movement of 
the gear, as determined by an official test, was 2.42 inches, when 
the maximum stress was 180,000 pounds. The work done in 
producing this stress amounted to 18,399 foot-pounds. Of this 
total energy 16,666 foot-pounds, or over 90%, represents the 
amount of energy absorbed and dissipated as heat by the 
frictional gear. The remaining 10% is given back by the 
recoil. The main release spring K is used for returning the 
segments and friction strips to their normal position after the 
force to close them has been removed. It also gives additional 
capacity to the entire mechanism. The auxiliary spring L 
releases the wedge Z), while the release pin M releases the pres- 
sure of the auxiliary spring L against the wedge during fric- 
tional operation. If we omit from the above design the fric- 
tional features and consider only the two followers A and Z, 
separated by the springs C and K, acting as one spring, we have 
the essential elements of a spring-draft gear. In fact, this 
gear acts exactly like a spring-draft gear for all ordinary service, 
the frictional device only acting during severe tension and com- 
pression. 

420. Gauge of wheels and form of wheel-tread. — In Fig. 204 
is shown the standard adopted by the Master Car Builders' 
Association at their twentieth annual convention. Note the 
normal position of the gauge-line on the wheel-tread. In 
Fig. 118, § 267, the relation of rail to wheel-tread is shoT\Ti 
on a smaller scale. It should be noted that there is no definite 
position where the wheel-flange is absolutely ^^chock-a-block'' 
against the rail. As the pressure increases the wheel mounts 
a little higher on the rail until a point is soon reached when the 
resistance is too great for it to mount still higher. By this 
means is avoided the shock of unyielding impact when the car 



460 



RAILROAD CONSTRUCTION, 



§420. 




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§ 421. ROLLING-STOCK. 461 

sways from side to side. When the gauge between the inner 
faces of the wheels is greater or less than the limits given in 
the figure, the interchange rules of the Master Car Builders' 
Association authorize a road to refuse to accept a car from 
another road for transportation. At junction points of rail- 
roads inspectors are detailed to see that this rule (as well as 
many others) is complied with in respect to all cars offered 
for transfer. 

TRAIN-BRAKES. 

421. Introduction. Owing to the very general misappre- 
hension that exists regarding the nature and intensity of the 
action of brakes, a complete analysis of the problem is con- 
sidered justifiable. This misapprehension is illustrated by the 
common notion (and even practice) that the effectiveness of 
braking a car is proportional to the brake pressure, and there- 
fore a brakeman is frequently seen using a bar to obtain a 
greater leverage on the brake-wheel and using his utmost 
strength to obtain the maximum pull on the brake-chain while 
the car is skidding along with locked wheels. 

When a vehicle is mo\dng on a track with a considerable 
velocity, the mass of the vehicle possesses kinetic energy of 
translation and the wheels possess kinetic energy of rotation. 
To stop the vehicle, this energy must be destroyed. The 
rotary kinetic energy will vary from about 4 to 8% of tlie 
kinetic energy of translation, according to the car loading 
(see § 435). On steam railroads brake action is obtained by 
pressing brake-shoes against car-wheel treads. As the brake- 
shoe pressure increases, the brake-shoes retard with increasing 
force the rotary action of the wheels. As long as the wheels 
do not slip or ^'skid" on the rails, the adhesion of the rails 
forces them to rotate with a circumferential velocity equal to 
the train velocity. The retarding action of the brake-shoe 
checks first the rotative kinetic energy (which is small), and 
the remainder develops a tendency for the wheel to slip on the 
rail. Since the rotative kinetic energy is such a small per- 
centage of the total, it will hereafter be ignored, except as 
specifically stated, and it will be assumed for simplicity that 
the only work of the brakes is to overcome the kinetic energy 
of translation. The possible effect of grade in assisting or 
preventing retardation, and the effect of all other track resist- 



462 



KAILEOAD CONSTKUCTION. 



§421. 




Fig. 204. — M. C. B. Standard Wheel-tread and Axle. 



§ 422. ROLLING-STOCK. 463 

ances, is also ignored. The amount of the developed force 
which retards the train movement is limited to the possible 
adhesion or static friction between the wheel and the rail. 
When the friction between the brake-shoe and the wheel ex- 
ceeds the adhesion between the wheel and the rail, the wheel 
skids, and then the friction between the w^heel and the rail 
at once drops to a much less quantity. It must therefore be 
remembered at the outset that the retarding action of brake- 
shoes on wheels as a means of stopping a train is absolutely 
limited by the possible static friction between the braked 
wheels and the rails. 

422. Laws of friction as applied to this problem. Much of 
the misapprehension regarding this problem arises from a very 
common and widespread misstatement of the general laws of 
friction. It is frequently stated that friction is independent 
of the velocity and of the unit of pressure. The first of these 
so-called laws is not even approximately true. A very exhaus- 
tive series of tests were made by Capt. Douglas Galton on the 
Brighton Railway in England in 1878 and 1879, and by M. 
George Marie on the Paris and Lyons Railway in 1879, w^ith 
trains which ^were specially fitted with train-brakes and w^ith 
dynagraphs of various kinds to measure the action of the 
brakes. Experience proved that variations in the condition of 
the rails (w^et or dry), and numerous irregularities incident to 
measuring the forces acting on a heavy body moving with a 
high velocity, were such as to give somewhat discordant re- 
sults, even when the conditions were made as nearly identical' 
as possible. But the tests were carried so far and so persist- 
ently that the general laws stated below were demonstrated 
beyond question, and even the numerical constants were deter- 
mined as closely as they may be practically utilized. These 
laws may be briefly stated as follows: 

(a) The coefficient of friction between cast-iron brake-blocks 
and steel tires is about .3 when the wheels are "just mov- 
ing''; it drops to about .16 when the velocity is about 30 miles 
per hour, and is less than .10 when the velocity is 60 miles per 
hour. These figures fluctuate considerably with the condition 
i of the rails, wet or dry. 

(6) The coefficient of friction is greatest w^hen the brakes 
1 are first applied; it then reduces very rapidly, decreasing 
. nearly one third after the brakes have been applied 10 seconds, 



464 RAILROAD CONSTRUCTION. § 422. 

and dropping to nearly one half in the course of 20 seconds. 
Although the general truth of this law was established beyond 
question, the tests to demonstrate the law of the variation of 
friction with time of application were too few to determine* 
accurately the numerical constants. 

(c) The friction of skidded wheels on rails is always very 
much less than the adhesion when the wheel is rolling on the 
rail^sometimes less than one third as much. 

{d) An analysis of the tests all pointed to a law that the 
friction developed does not increase as rapidly as the intensity 
of pressure increases, but this may hardly be considered as 
an estabHshed law. 

(e) The adhesion between the wheel and the rail appears to 
be independent of velocity. The adhesion here means the force 
that must be developed before the wheel will slip on the rail. 

The practical effect of these laws is shown by the following ; 
observed phenomena: \ 

(a) When the brakes are first applied (the velocity being 
very high), a brake pressure far in excess of the weight on the 
wheel (even three or four times as much) may be applied with- 
out skidding the wheel. This is partly due to the fact that 
the wheel has a very high rotative kinetic energy (which varies 
as the square of the velocity, and which must be overcome 
first), but it is chiefly due to the fact that the coefficient of 
friction at the higher velocity is very small (at 60 miles per 
hour it is about .07), while the adhesion between the wheel and 
the rail is independent of the velocity. 

(b) As the velocity decreases the brake pressure must be 
decreased or the wheels will skid. Although the friction de- 
creases with the time required to stop and increases with the 
reduction of speed, and these two effects tend to neutralize 
each other, yet unless the stop is very slow, the increase in 
friction due to reduction of speed is much greater than the 
decrease due to time, and therefore the brake pressure must 
not be greater than the weight on the wheel, unless momentarily 
while the speed is still very high. 

(c) The adhesion between wheels and rails varies from .20 
to .25 and over when the rail is dry. When wet and slippery 
it may fall to .18 or even .15. The use of sand will always 
raise it above .20, and on a dry rail, when the sand is not blown 
away by wind, it may raise it to .35 or even .40. 



§ 423, ROLLING-STOCK. 465 

(d) Experiments were made tdth an automatic valve by 
which the brake-shoe pressure against the wheel should be 
reduced as the friction increased, but since (1) the essential 
requirement is that the friction produced by the brake-shoes 
shall not exceed the adhesion between rail and wheel, and 
since (2) the rail-wheel adhesion is a very variable quantity, 
depending on whether the rail is wet or dry, it has been found 
impracticable to use such a valve, and that the best plan is to 
leave it to the engineer to vary the pressure, if necessary, by the 
use of the brake-valve. 

MECHANISM OF BRAKES. 

423. Hand- brakes. The old style of brakes consists of brake- 
shoes of some type which are pressed against the wheel-treads 
by means of a brake-beam, which is operated by means of a 
hand-windlass and chain operating a set of levers. It is desir- 
able that brakes shall not be set so tightly that the wheels 
shall be locked, and then slide over the track, producing 
flat places on them, which are veiy destructive to the 
rolling-stock and track afterward, on account of the impact 
occasioned at each revolution. With air-brakes the maximimi 
pressure of the brake-shoes can be quite carefully regulated, 
and they are so designed that the maximum pressure exerted 
by any pair of brake-shoes on the wheels of any axle shall not 
exceed a certain per cent, of the weight carried by that axle 
when the car is empty, 90% being the figure usually adopted 
for passenger-cars and 70% for freight-cars. Consider the 
case of a freight-car of 100000 lbs. capacity, weighing 33100 lbs., 
or 8275 lbs. on an axle, and equipped with a hand-brake which 
operates the levers and brake-beams, which are sketched in 
Fig. 205. The dead weight on an axle is 8275 lbs.; 70% of 
this is 5792 lbs., which is the maximum allowable pressure 
per brake-beam, or 2896 lbs. per brake-shoe. With the dimen- 
sions shown, such a pressure will be produced by a pull of about 
1158 lbs. on the brake-chain. The power gained by the brake- 
wheel is not equal to the ratio of the brake-wheel diameter 
to the diameter of the shaft, about which the brake-chain 
"vvands, which is about 16 to IJ. The ratio of the circumfer- 
ence of the brake-wheel to the length of chain woimd up by 
one complete turn would be a closer figure. The loss of eSi- 



466 



RAILROAD CONSTRUCTION. 



§424. 



ciency in such a clumsy mechanism also reduces the effective 
ratio. Assuming the effective ratio as G : 1 it would require a 
pull of 193 lbs. at the circumference of the brake-wheel to 
exert 1158 lbs. pull on the brake-chain, or 5792 lbs. pressure 
on the wheels at B, and even this will not lock the wheels when 
the car is empty, much less when it is loaded. Note that the 
pressures at A and B are unequal. This is somewhat objec- 
tionable, but it is unavoidable w^ith this simple form of brake- 
beam. More complicated forms to avoid this are sometimes 
used. Hand-brakes are, of course, cheapest in first cost, and 
even with the best of automatic brakes, additional mechanism 
to operate the brakes by hand in an emergency is always pro- 
vided, but their slow operation w^hen a quick stop is desired 
makes it exceedingly dangerous to attempt to run a train at 
high speed unless some automatic brake directly under the 
control of the engineer is at hand. The great increase in the 




Fig. 205. — Sketch of Mechanism of Hand-brake. 



average velocity of trains during recent j^ears has only been 
rendered possible by the invention of automatic brakes. 

424. "Straight" air-brakes. The essential constructive fea- 
tures of this form of brake are (1) an air-pump on the engine, 
operated by steam, which compresses air into a reservoir on 
the engine; (2) a '' brake-pipe" running from the reservoir 
to the rear of the engine and pipes running under each car, 
the pipes having flexible connections at the ends of the cars 
and engine; (3) a cylinder and piston under each car which 



§ 425. ROLLING-STOCK. 467 

operates the brakes by a system of levers, the cyHnder being 
connected to the brake-pipe. The reservoir on the engine 
holds compressed air at about 45 lbs. pressure. To operate the 
brakes, a valve on the engine is opened which allows the com- 
pressed air to flow from the reservoir through the brake-pipe 
to each cylinder, moving the piston, which thereby moves the 
levers and appHes the brakes. The dejects of this system are 
many: (1) With a long train, considerable time is required for 
the air to flow from the reservoir on the engine to the rear cars, 
and for an emergency-stop even this delay would often be 
fatal; (2) if the train breaks in two, the rear portion is not 
pro\dded with pow^r for operating the brakes, and a dangerous 
collision would often be the result; (3) if an air-pipe coupling 
bursts under any car, the whole system becomes absolutely 
helpless, and as such a thing might happen during some emer- 
gency, the accident would then be especially fatal. 

This form of brake has almost, if not entirely, passed out of 
use. It is here briefly described in order to show the logical 
development of the form which is now in almost universal use, 
the automatic. 

425. Automatic air-brakes. The above defects have been 
overcome by a method which may be briefly stated as follows: 
A reservoir for compressed air is placed under each car and the 
tender; whenever the pressure in these reservoirs is reduced 
for any reason, it is automatically replenished from the main 
reservoir on the engine; whenever the pressure in the brake- 
pipe is reduced for any cause (opening a valve at any point of 
its length, parting of the train, or bursting of a pipe or coupler), 
valves are automatically moved under each car to operate the 
piston and put on the brakes. All the brakes on the train are 
thus applied almost simultaneously. K the train breaks in two, 
both sections will at once have all the brakes applied automati- 
cally ; if a coupling or pipe bursts, the brakes are at once applied 
and attention is thereby attracted to the defect; if an emer- 
gency should arise, such that the conductor desires to stop 
the train instantly T\dthout even taking time to signal to the 
engineer, he can do so by opening a valve placed on each car, 
which admits air to the train-pipe, which will set the brakes 
on the whole train, and the engineer, being able to discover 
instantly what had occurred, would shut off steam and do 
whatever else was necessary to stop the train as quickly as pos- 



468 RAILROAD CONSTRUCTION. § 426. 

sible. The most important and essential detail of this system 
is the '^automatic triple valve" placed under each car. Quot- 
ing from the Westinghouse Air-brake Company's Instruction 
Book, ^'A moderate reduction of air pressure in the train-pipe 
causes the greater pressure remaining stored in the auxiliary 
reservoir to force the piston of the triple valve and its slide- 
valve to a position which will allow the air in the auxiliary 
reservoir to pass directly into the brake-cylinder and apply the 
brake. A sudden or violent reduction of the air in the train- 
pipe produces the same effect, and in addition causes supple- 
mental valves in the triple valve to be opened, permitting the 
pressure from the train-pipe to also enter the brake-cylinder, 
augmenting the pressure derived from the auxiliary reservoir 
about 20%, producing practically instantaneous action of the 
brakes to their highest efficiency throughout the entire train. 
When the pressure in the brake-pipe is again restored to an 
amount in excess of that remaining in the auxiliary reservoir, 
the piston- and slide-valves are forced in the opposite direction 
to their normal position, opening communication from the train- 
pipe to the auxiliary reservoir, and permitting the air in the 
brake-cylinder to escape to the atmosphere, thus releasing the 
brakes. If the engineer wishes to apply the brake, he moves 
the handle of the engineer's brake-valve to the right, which 
first closes a port, retaining the pressure in the main reservoir, 
and then permits a portion of the air in the train-pipe to escape. 
To release the brakes, he moves the handle to the extreme 
left, which allows the air in ^ the main reservoir to flow freely 
into the brake-pipe, restoring the pressure therein." 

426. Tests to measure the efficiency of brakes. Let v repre- 
sent the velocity of a train in feet per second; W, its weight; 
F, the retarding force due to the bi akes ; d, the distance in feet 
required to make a stop ; and g, the acceleration of gravity 
(32.16 feet per square second); then the kinetic energy pos- 
sessed by the train (disregarding for the present the rotative 

kinetic energy of the wheels) =-^ — . The work done in stop- 

ping the train =i^c?. ,\Fd==-^r~, The ratio of the retarding 
force to the weight, 

Z-ZL-oissH" 



§ 427. ROLLING-STOCK. 4G9 

In order to compare tests made imder varying conditions, the 
ratio F-i-W should be corrected for the effect of grade ( + or — ), 
if any, and also for the proportion of the weight of the train 
which is on braked wheels. For example, a train weighed 
146076 lbs., the proportion on braked wheels was 67%, speed 
60 feet per second, length of stop 450 feet, track level. Sub- 
stituting these values in the above formula, we find (F-^W) 
= .124. This value is really unduly favorable, since the ordi- 
nary track resistance helps to stop the train. This has a value 
of from 6 to 20 lbs. per ton, averaging say 10 lbs. per ton dur- 
ing the stop, or .005 of the weight. Since the effect of this is 
small and is nearly constant for all trains, it may be ignored 
in comparative tests. The grade in this case was level, and 
therefore grade had no effect. But since only 67% of the 
weight was on braked wheels, the ratio, on the basis of all the 
wheels braked, or of the weight reduced to that actually on 
the braked wheels, is 0.124 -r- .67 = 0.185. This was called 
a '' good " stop, although as high a ratio as 0.200 has been 
obtained. 

427. Brake-shoes. Brake-shoes were formerly made of 
wrought iron, but when it was discovered that cast-iron shoes 
would answer the purpose, the use of wr ought-iron shoes was 
abandoned, since the cast-iron shoes are so much cheaper. A 
* cheap practice is to form the brake-shoe and its head in one 
piece, which is cheaper in first cost, but when the wearing-sur- 
face is too far gone for further use, the whole casting must be 
renewed. The ^'Christie" shoe, ' adopted by the Master Car 
Builders' Association as standard, has a separate shoe which 
is fastened to the head by means of a wrought-iron key. The 
shoe is beveled ^ in a width of 3f " to fit the coned wheel. 
This is a greater bevel than the standard coning of a car-wheel. 
It is perhaps done to allow for some bending of the brake- 
beam and also so that the maximum pressure (and wear) should 
come on the outside of the tread, rather than next to the flange, 
where it might tend to produce sharp flanges. By concen- 
trating the brake-shoe wear on the outer side of the tread, the 
wear on the tread is more nearly equalized, since the rail wears 
the wheel-tread chiefly near the flange. This same idea is 
developed still further in the ^'flange-shoes,'' which have a 
curved form to fit the wheel-flange and which bear on the 
wheel on the flange and on the outside of the tread. It is 



470 RAILROAD CONSTRUCTION. §427. 

claimed that by this means the standard form of the tread is 
better preserved than when the wear is entirely on the tread. 
The Congdon brake-shoe is one of a type in which wrought- 
iron pieces are inserted in the face of a cast-iron shoe. It is 
claimed that these increase the life of the shoe. 



CHAPTER XVI. 
TRAIN RESISTANCE. 

428. Classification of the various forms. The various resist- 
ances which must be overcome, by the power of the locomotive 
may be classified as follows : 

(a) Resistances internal to the locomotive, which include fric- 
tion of the valve-gear, piston- and connecting-rods, journal 
friction of the drivers; also all the loss due to radiation, con- 
densation, friction of the steam in the passages, etc. In short, 
these resistances are the sum-total of the losses by which the 
power at the circumference of the drivers is less than the power 
developed by the boiler. 

(h) Velocity resistances, which include the atmospheric resist- 
ances on the ends and sides; oscillation and concussion resist- 
ances, due to uneven track, etc. 

(c) Wheel resistances, which include the rolling friction be- 
tween the wheels and the rails of all the wheels (including the 
drivers) ; also the journal friction of all the axles, except those 
of the drivers. 

(d) Grade and curve resistances, which include those resist- 
ances which are due to grade and to curves, and which are not 
found on a straight and level track. 

(e) Brake resistances. As shown later, brakes consume 
power and to the extent of their use increase the energy to 
be developed by the locomotive. 

(/) Inertia resistances. The resistance due to inertia is not 
generally considered as a train resistance because the energy 
which is stored up in the train as kinetic energy may be util- 
ized in overcoming future resistances. But in a discussion 
of the demands on the tractive power of the engine, 'one of the 
chief items is the energy required to rapidly give to a starting 
train its normal velocity. This is especially true of suburban 
trains, which must acquire speed very quickly in order that 

471 



472 RAILROAD CONSTRUCTION. § 429. 

their general average speed between termini may be even reason- 
ably fast. 

429. Resistance internal to the locomotive. These are re- 
sistances which do not tax the adhesion of the drivers to the 
rails, and hence are frequently considered as not being a part 
of the train resistance properly so called. If the engine were 
considered as lifted from the rails and made to drive a belt 
placed around the drivers, then all the power that reached the 
belt would be the power that is ordinarily available for adhe- 
sion, while the remainder would be that consumed internally 
by the engine. The power developed by an engine may be 
obtained by taking indicator diagrams which show the actual 
steam pressure in a cylinder at any part of a stroke. From 
such a diagram the average steam pressure is easily obtained, 
and this average pressure, multiplied by the length of the stroke 
and by the net area of the piston, gives the energy developed 
by one half-stroke of. one piston. Four times this product 
divided by 550 times the time in seconds required for one stroke 
gives the ^'indicated horse-power^' Even this calculation 
gives merely the power behind the piston, which is several per 
cent, greater than the power w^hich reaches the circumference 
of the drivers, owing to the friction of the piston, piston-rod, 
cross-head, connecting-rod bearings, and driving-wheel jour- 
nals. (See § 411, Chapter XV.) By measuring the amount 
of water used and turned into steam, and by noting the boiler 
pressure, the energy possessed by the steam used is readily 
computed. The indicator diagrams will show the amount of 
steam that has been effective in producing power at the cylin- 
ders. The steam accounted for by the diagrams will ordinarily 
amount to 80 or 85% of the steam developed by the boiler, 
and the other 15 or 20% represents the loss of energy due to 
radiation, condensation, etc. 

Locomotive resistance has been estimated and tabulated by a 
Committee of the Amer. Rwy. Eng. Assoc, and the results are 
given in Table XXIX, which is taken from the Manual of that 
Association. As a numerical illustration, what is the computed 
resistance for a Mikado locomotive of which the total weight of 
engine and tender is 315,000 lbs. of which 153,200 lbs. is carried 
on the drivers, at a velocity of 6 miles per hour? In this case. 
Item A = (18.7X76.6) + (80X4) =1432 lbs. The weight carried 
on the engine and tender trucks = 315,000 -153,200 = 161,800 



§430. 



TRAIN RESISTANCE. 



473 



=80.9 tons. Item B = (2.6 X80.9) + (20 X6) = 330 lbs. Item C 
is comparatively insignificant at this low velocity. From the 
table, we read 9 lbs. Then the sum of A, B, and C = 1771 lbs., 
which must be subtracted from a computed tractive effort to 
obtain the estimated draw-bar pull. 

TABLE XXIX. LOCOMOTIVE RESISTANCES. 

Total Locomotive Resistance =A +jB +C, in which 
A = resistance between cylinder and rim of drivers, and in pounds 

= 18.7!r+80A^ 

in which T =tons weight 6n drivers, and 
N = number of driving axles; 

B = resistance of engine and tender trucks, and in pounds 
= 2.QT+20N • 

in which T =tons weight on engine and tender trucks 
and N = number of truck axles; 

C :=head end or " air " resistance, and in*pounds 
=:.0027U 

in which V = velocity in miles per hour, and 
A =end area of locomotive. 

On the basis that the end area averages 125 square feet, the formula 
becomes C =0.25V'. The number of pounds air resistance for various ve- 
locities is as given below. 



Vel.j Res. 


Vel.j Res. 


Vel. 


Res. 


Vel. 


Res. 


Vel. 


Res. 


Vel. 


Res. 


1 


0.25 


8 lie.oo 


15 


56 


22 


121 


29 


210 


36 


324 


2 


1.00 


9 '20.25, 


1 16 


64 


23 


132 


30 


225 


37 


342 


3 


2.25| 


10 


25.00 


1 17 


72 


24 


144 


31 


240 


38 


361 


4 


4.001 


11 


30 


18 


81 


25 


156 


32 


256 


39 


380 


5 


6.25 


12 


36 


19 


90 


26 


169 


33 


272 


40 


400 


6 


9.00 


13 


42 


20 


100 


27 


182 


34 


289 


50 


625 


7 


12.25j 


14 


49 


21 


110 


28 


196 


35 


306 


60 


900 



Draw-bar pull on level tangent equals the cylinder tractive power less 
the sum of the engine resistances. 

At low speeds, the adhesion of the drivers should be considered and avail- 
able draw-bar pull should never be estimated greater than 30% of weight 
on drivers at starting with use of sand, 25% of weight on drivers at running 
speeds. 



Taken from Table 7 in " Economics 
Rwy. Eng. Assoc, 1915 edition. 



section of Manual of the Amer. 



430. Velocity resistance, (a) Atmospheric. This consists of 
the head and tail resistances and the side resistance. The head 



474 RAILROAD CONSTRUCTION. §431. 

and tail resistances are nearly constant for all trains of given 
velocity, varying but slightly with the varying cross-sections 
of engines and cars. The side resistance varies with the length 
of the train and the character of the carfe, box-cars or flats, etc. 
Vestibuling cars has a considerable effect in reducing this side 
resistance by preventing much of the eddying of air-currents 
between the cars, although this is one of the least of the advan- 
tages of vestibuhng. Atmospheric resistance is generally 
assumed to vary as the square of the velocity, and although 
this may be nearly true, it has been experimentally demon- 
strated to be at least inaccurate. Values for head resistance 
are given in Table XXIX, which are probably accurate enough 
for all practical purposes, especially at ordinary freight train 
velocities. A freight-train composed partly of flat-cars and 
partly of box-cars will encounter considerably more atmospheric 
resistance than one made exclusively of either kind, other things 
being equal. The definite information on this subject is very 
unsatisfactory, but this is possibly due to the fact that it is of 
little practical importance to know just how much such resistance 
amounts to. 

(6) Oscillatory and concussive. These resistances are con- 
sidered to vary as the square of the velocity. Probably this 
is nearly, if not quite, correct on the general principle that such 
resistances are a succession of impacts and the force of impacts 
varies as the square of the velocity. These impacts are due to 
the defects of the track, and even though it were possible to 
make a precise determination of the amount of this resistance 
in any particular case, the value obtained would only be true 
for that particular piece of track and for the particular degree 
of excellence or defect which the track then possessed. The 
general improvement of track maintenance during late years 
has had a large influence in increasing the possible train-load 
by decreasing the train resistance. The expenditure of money 
to improve track will give a road a large advantage over a 
competing road with a poorer track, by reducing train resist- 
ance, and thus reducing the cost of handling traffic. 

431. Wheel resistances, (a) Rolling friction of the wheels. 
To determine experimentally the rolling friction of wheels, 
apart from all journal friction, is a very difficult matter and 
has never been satisfactorily accomplished. Theory as well 
as practice shows that the higher and the more perfect the 



i 



§ 431. TRAIN RESISTANCE. 475 

elasticity of the wheel and the surface, the less will be the roll- 
ing friction. But the determination, if made, would be of 
theoretical interest only. 

The combined effect of rolling friction and journal friction 
is determinable with comparative ease. From the nature of 
the case no great reduction of the rolling friction by any device 
is possible. It is only a very insignificant part of the total 
train resistance. 

(b) Journal friction of the axles. This form of resistance has 
been studied quite extensively by means of the measurement 
of the force required to turn an axle in its bearings under 
various conditions of pressure, speed, extent of lubrication, 
and temperature. The following laws have been fairly well 
established: (1) The coefficient of friction increases as the pres- 
sure diminishes; (2) it is higher at very slow speeds, gradually 
diminishing to a minimum at a speed corresponding to a train 
velocity of about 10 miles per hour, then slowly increasing 
with the speed; it is very dependent on the perfection of the 
lubrication, it being reduced to one sixth or one tenth, when the 
axle is lubricated by a bath of oil rather than by a mere pad 
or wad of waste on one side of the journal; (3) it is much lower 
at higher temperature, and vice versa. The practical effect of 
these laws is shown by the observed facts that (1) loaded cars 
have a less resistance per ton than unloaded cars, the figures 
being, for speeds of about 10 miles per hour, approximately: 

For passenger- and loaded freight-cars . . 4 lbs. per ton 

" empty freight-cars 8 " " '^ 

'' street-cars 10 " " " 

" freight-trucks without load 14 " " " 

(2) When starting a train, the resistances are about 20 lbs. 
per ton, notwithstanding the fact that the velocity resistance* 
are practically zero; at about 2 miles per hour it will drop to 
10 lbs. per ton and above 10 miles per hour it may drop to 
4 lbs. per ton if the cars are in good condition. (3) The re- 
sistance could probably be materially lowered if some practicable 
form of journal-box could be devised which would give a more 
perfect lubrication. (4) It is observed that freight-train loads 
must be cut dowTi in winter by about 10 or 15% of the loads 
that the same engine can haul over the same track in summer. 
This is due partly to the extra roughness and inelasticity of the 



476 



RAILROAD CONSTRUCTION. 



§432, 



track in winter, and partly to increased radiation from the 
engine wasting some energy, but this will not account for al] 
of the loss, and the effect, which is probably due largely to the 
lower temperature of the journal-boxes, is very marked and 
costly. It has been suggested that a jacketing of the journal- 
boxes, which would prevent rapid radiation of heat and enable 
them to retain some of the heat developed by friction, would 
result in a saving amply repaying the cost of the device. 

Roller journals for cars have been frequently suggested, and 
experiments have been made with them. It is found that they 
are very effective at low velocities, greatly reducing the start- 
ing resistance, which is very high with the ordinary forms of 
journals. But the advantages disappear as the velocity in- 
creases. The advantages also decrease as the load is increased, 
so that with heavily loaded cars the gain is small. The excess 
of cost for construction and maintenance has been found to be 
more than the gain from power saved. 

432. Grade resistance. The amount of this may be com- 
puted with mathematical exactness. Assume that the bail 
or cylinder (see Fig. 206) is being drawn up the plane. If W 




Fig. 206. 

is the weight, N the normal pressure against the rail, and G 

the force required to hold it or to draw it up the plane with 

uniform velocity, the rolling ♦resistances being considered zero 

or considered as provided for by other forces, then 

Wh 
G:W::h:d, or G=~^; 

but for all ordinary railroad grades, d=c to within a tenth of 

Wh * 

1%, i.e., G = = TF X rate of grade. In order that the student 

c 

may appreciate the exact amoimt of this approximation the per- 
centage of slope distance to its horizontal projection is given in 
the following tabular form: 



§432. 



TRAIN RESISTANCE. 



477 



Grade in per cent. 



Slope dist. 
hor. dist. 



xioo. 



100.005 



100.020 



100.045 



100.080 



100.125 





Grade in per cent. 


6 


7 


8 


9 


10 


Slonedist.^jpQ 

hor. disL. 


100.180 


100.245 


100.319 


100.404 


100.499 



This shows also the error on various grades of measuring with 
the tape on the ground rather than held horizontally. Since 
almost all railroad grades are less than 2% (where the error 
is but .02 of 1%), and anything in excess of 4% is unheard 
of for normal construction, the error in the approximation 
is generally too small for practical consideration. 

If the rate of grade is 1 : 100, G = WXTi-^y i-^., 6^=20 lbs. 
per ton ; .'.for any per cent, of grade, G = (20 X per cent, of grade) 
pounds per ton. When moving up a grade this force G is to 
be overcome in addition to all the other resistances. When 
moving do^Ti a grade, the force G assists the motion and may 
be more than sufficient to move the train at its highest allow- 
able velocity. The force required to move a train on a level 
track at ordinary freight-train speeds (say 20 miles per hour) 
is about 7 lbs. per ton. A down grade of /^ of 1% will fur- 
nish the same power; therefore on a dowTi grade of 0.35%, a 
freight-train would move indefinitely at about 20 miles per hour. 
If the grade w^ere higher and the train w^ere allowed to gain 
speed freely, the speed would increase until the resistance at 
that speed would equal W times the rate of grade, when the 
velocity would become uniform and remain so as long as the 
conditions were constant. If this speed w^as higher than a 
safe permissible speed, brakes must be applied and power 
wasted. The fact that one terminal of a road is considerably 
higher than the other does not necessarily imply that the extra 
power needed to overcome the difference of elevation is a 
total waste of energy, especially if the maximum grades are 
so low that brakes will never need to be applied to reduce a 
dangerously high velocity, for although more power must be 



478 RAILROAD CONSTRUCTION. § 433. 

used in ascending the grades, there is a considerable saving of 
power in descending the grades. The amount of this saving 
will be discussed more fully in Chapter XXIII. 

433. Curve resistance. Some of the principal laws will be 
here given without elaboration. A more detailed discussion 
will be given in Chapter XXII. 

(a) While the total curve resistance increases as the degree 
of curve increases, the resistance yer degree of curve is much 
greater for easy curves than for sharp curves; e.gr., the resist- 
ance on the excessiveh^ sharp curves (radius 90 feet) of the 
elevated roads of New York City is very much less per degree 
of curve than that on curves of 1° to 5°. (Jb) Curve resistance 
increases with the velocity, (c) The total resistance on a 
curve depends on the central angle rather than on the radius; 
I.e., two curves of the same central angle but of different radius 
would cause about the same total curve resistance. This is 
partly explained by the fact that the longitudinal slipping will 
be the same in each case. (See § 395, Chapter XV.) In each 
case also the trucks must be twisted around and the wheels 
slipped laterally on the rails by the same amount A°. (See 
§ 396, Chapter XV.) 

434. Brake resistances. If a down grade is excessively steep 
so that brakes must be applied to prevent the train acquiring 
a dangerous velocity, the energ}^ consumed is hopelessly lost 
without any compensation. When trains are required to make 
frequent stops and yet maintain a high average speed, consid- 
erable power is consumed by the application of brakes in stop- 
ping. All the energy which is thus turned into heat is hope- 
lessly lost, and in addition a very considerable amount of steam 
is drawn from the boiler to operate the air-brakes, which con- 
sume the power already developed. It can be easily demonstrated 
that engines drawing trains in suburban service, making fre- 
quent stops, and yet developing high speed between stop^, will 
consume a very large proportion of the total power developed 
by the use of brakes. Note the double loss. The brakes con- 
sume power already developed and stored in the train as kinetic 
or potential energy, while the operation of the brakes requires 
additional steam power from the engine. 

435. Inertia resistance. The two forms of train resistance 
«vhich under some circumstances are the greatest resistances 
to be overcome by the engine are the grade and inertia resist- 



§435. TRAIN RESISTANCE. 479 

ances, and fortunately both of these resistances may be com- 
puted with mathematical precision. The problem may be 
stated as follows: What constant force P (in addition to the 
forces required to overcome the various frictional resistances, 
etc.) will be required to impart to a body a velocity of v feet 
per second in a distance of s feet? The required number of 
foot-pounds of energy is evidently Ps, But this work imparts 

a kinetic energy which may be expressed by -h— . Equating 

^9 

these values, we have Ps^-^^—, or 

^=2^ (104) 

The force required to increase the velocity from Vj to v- may 

W 
likewise be stated as P=^ — W—'^i^)- Substituting in the 

formula the values TF=2000 lbs. (one ton), gr =32.16, and 5 = 
5280 feet (one mile), we have 

P = m58SW-v\^). 

Multiplying by (5280 -^ 3600) ^ to change the unit of velocity 
to miles per hour, we have 

P = . 01267(72' -"^^1^)^ 

But this formula must be modified on account of the rotative 
kinetic energy which must be imparted to the wheels of the cars. 
The precise additional percentage depends on the particular 
design of the cars and their loading and also on the design of 
the locomotive. Consider as an example a box-car, 60000 lbs. 
capacity, weighing 33000 lbs. The wheels have a diameter 
of 36" and their radius of gyration is about 13'''. Each wheel 
weighs 700 lbs. The rotative kinetic energy of each wheel is 
4877 ft. -lbs. when the velocity is 20 miles per hour, and for 
the eight wheels it is 39016 ft.-lbs. For greater precision 
(really needless) we may add 192 ft.-lbs. as the rotative kinetic 
energy of the axles. When the car is full}^ loaded (weight 
93000 lbs.) the kinetic energy of translation is 1,244,340 ft.-lbs.; 
when empty (weight 33000 lbs.) the energy is 441540 ft.-lbs^ 
The rotative kinetic energy thus adds (for this particular 
car) 3.15% (when the car is loaded) and 8.9% (when the car 
is empty) to the kinetic energy of translation. The kinetic 



480 RAILROAD CONSTRUCTION. § 435. 

energy which is similariy added, owing to the rotation of the 
wheels and axles of the locomotive, might be similarly com- 
puted. For one type of locomotive it has been figured at about 
8%. The variations in design, and particularly the fluctua- 
tions of loading, render useless any great precision in these 
computations. For a train of ''empties'' the figure would be 
high, probably 8 to 9%; for a fully loaded train it will not 
much exceed 3%. Wellington considered that 6% is a good 
average value to use (actually used 6.14% for "ease of compu- 
tation"), but considering (a) the increasing proportion of live 
load to dead load in modern car design, (h) the greater care 
now used to make up full train-loads, and (c) the fact that 
full train-loads are the critical loads, it would appear that 5% 
is a better average for the conditions of modern practice. Even 
this figure allows something for the higher percentage for the 
locomotive and something for a few empties in the train. There- 
fore, adding 5% to the coefficient in the above equation, we 
have the true equation 

P = .0133 (722 -7i2), (105) 

in which V^ and Vi are the higher and lower velocities respec- 
tively, in miles per hour^ and P is the force required per ton to 
impart that difference of velocity in a distance of one mile. If 
more convenient, the formula may be used thus: 

Pi = -(722-712), (106) 

s 

in which s is the distance in feet and Pi is the corresponding 
force. 

As a numerical illustration, the force required per ton to 
impart a kinetic energy due to a velocity of 20 miles per hour 
in a distance of 1000 feet will equal 

^ 70(400-0) _„ 

Pi = — ^- =28 lbs., 

1000 

tyhich is the equivalent (see.§ 432) of a 1.4% grade. Since 
the velocity enters the formula as 72, while the distance enters 
Duly in the first power, it follows that it will require four times 



§ 436. TRAIN RESISTANCE. 481 

the force to produce twice the velocity in the same distance, or 
that with the same force it will require four times the distance 
to attain twice the velocity. 

As another numerical illustration, if a train is to increase its 
speed from 15 miles per hour to 60 miles per hour in a distance 
of 2000 feet, the force required (in addition to all the other 
resistances) will be 

„ 70.224(3600-225) , , ^ r:mu + 

^' ^ ^00 = 1 18.50 lbs. per ton. 

This is equivalent to a 5.9% grade and shows at once that it 
would be impossible unless there were a very heavy down 
grade, or that the train was very light and the engine very 
powerful. 

436. Dynamometer tests. These are made by putting a 
'^ dynamometer-car '^ between the engine and the cars to be 
tested. Suitable mechanism makes an automatic record of 
the force w^hich is transmitted through the dynamometer at 
any instant, and also a record of the velocity at any instant. 
Gne of the practical difficulties is the accurate determination 
of the velocity at any instant when the velocity is fluctuating. 
When the velocity is decreasing, the kinetic energy of the train 
is being turned into work and the force transmitted through the 
dynamometer is less than the amount of the resistance which 
is actually being overcome. On the other hand, when the 
velocity is increasing, the dynamometer indicates a larger 
force than that required to overcome the resistances, but the 
excess force is being stored up in the train as kinetic energy. 
Grade has a similar effect, and the force indicated by the dy-- 
namometer may be greater or less than that required at the 
given velocity on a level by the force which is derived from, 
or is turned into, potential energy. Therefore the resistance 
indicated b}^ the dynamometer of a train will not be that on a 
level track at uniform velocity, unless the track is actually 
level and the velocity really uniform. 

Dynamometer tests under other circumstances are there- 
fore of no value unless it is possible to determine the true 
velocity at any instant and its rate of change, and also to de- 
termine the grade. Of course, the grade is easily found. An 
allowance for an increase or decrease of kinetic or potential 
energy must therefore be made before it is possible to 



482 



KAILROAD CONSTRUCTION. 



§437. 



know how much force is being spent on the ordinary resist- 
ances. 

437. Gravity or " drop " tests. Dynamometer tests require 
the use of a dynamometer which is capable of measuring a 
force of several thousands of pounds, and which therefore 
cannot determine such values with a close percentage of accu- 
racy, especially if the force is small. A drop test utihzes the 
force of gravity which may be measured with mathematical 
accuracy. The general method is to select a stretch of track 
which has a uniform grade of about 0.7% and which is prefer- 
ably straight for two or three miles. On such a grade cars 
with running gear in good condition may be started by a push. 
The velocity will gradually increase until at some velocity, 
depending on the resistances encountered, the cars will move 
uniformly. The only work requiring extreme care with this 
method is the determination of the velocity. If the velocity 
is fluctuating, as it is during the time when it is of the greatest 
importance to know the velocity, it is not sufficient to deter- 
mine the time required to run some long measured distance, 
for the average velocity thus obtained would probably differ 




I 



Fig. 207. — Loss in Velocity-head. 



considerably from the velocity at the beginning and end of that 
space. If the train consists of five cars or more, the velocity 
may be determined electrically (as described by Wellington 
in his "Economic Location," etc., p. 793 et seq.) from the 
automatic record made on a chronograph of the passage of the 
first wheel and the last, the chronograph also recording auto- 



§ 438. TRAIN RESISTANCE. 483 

matically the ticks of a clock beating feeconds. From this the 
exact time of the passage of the first and last wheels of the 
train of cars may be determined to the tenth or twentieth of a 
second. 

Velocity -head. From theoretical mechanics we know that 
if a body descends through any path by the action of gravity, 
and is unaffected by friction, its velocity at any point in the 
direction of the path of motion is V = \/2gh. If the body is 
retarded by resistances, its velocity at any point will be less 
than this. If AM, Fig. 207, represents any grade (exaggerated 
of course), then BJ, CK, etc., represent the actual fall at any 

point. Let BF represent the fall /ij, determined from hi = ~, 

in which v^ is the actual observed velocity at J, Then /i^=the 
velocity-head consumed by the resistances between A and ./. 
If the train continues to K, the corresponding /lo is CG) the 
remaining fall GK consists of GN {=JFj which is the velocity- 
head lost back of /) and NK, the velocity-head lost between J 
and K. At some velocity (Vn) on any grade, the velocity 
will not further increase and the line AFGHI will then be hori- 
zontal and at a distance (hn) =EI below A , . , E, The grade 
AM is the ''grade of repose" for that velocity (Fn); i.e., it is 
the grade that would just permit the train to move indefinitely 
at the velocity Vn- The broken line AFGHI should really be 
a curve, and the grade of repose at any point is the angle between 
AM and the tangent to that curve at the given point. The 
"grade of repose" by its definition gives the total resistance 
of the train at the particular velocity, or multiplying the grade 
of repose in per cent, by 20 gives the pounds per ton of resist- 
ance. Thus being able to determine the total resistance in 
pounds per ton at any velocity, the variation of total resistance 
with velocity may be determined, and then by varying the 
resistances, using different kinds of cars, empty and loaded, 
box-cars and flats, the resistances of the different kinds at 
various velocities may be determined. 

438. Formulae for train resistance. These are generally given 
in one of the forms 



R = aV+c, . . . (1) 
R = bV' + c, ... (2) 
J?=aF+6F2 + c, , . (3) 



(107) 



484 RAILROAD CONSTRUCTION. § 438. 

in which R is the resistance in pounds per ton, a and h are coeffi- 
cients to be determined, V is the velocity in miles per hour, and 
c is a constant, also to be determined. These formulae disregard 
grade and curve resistances, inertia resistance and the active 
resistance (or assistance) of windy as distinct from mere atmos- 
pheric resistance. In short, they are supposed to give the re- 
sistance of a train moving at a uniform velocity over a straight 
and level track, there being no appreciable wind. 

The various formulae are sometimes based directly on experi- 
ments made by the proposer of the formula ; sometimes they are 
deduced from a mere study of the results of one or more series 
of tests made by others. Unfortunately for either method, no 
one investigator has ever been able to make tests which are so 
thorough and made under such a wide range of conditions that 
his results may be considered as conclusive, while a student of 
the tests of others is handicapped by a lack of knowledge of 
precise conditions, which, if fully understood, would perhaps 
permit some reconciliation of the very discordant figures which 
are reported. As already intimated, the condition of the 
rolling stock, the unit weight on the axles, the lubrication of the 
axles, the length of the train in relation to its weight and the 
condition of the track, which involves the weight of rail, spacing 
and size of ties, tamping of ties, etc., all have their influence in 
modifying the apparent resistance. There is also good reason to 
believe that the effect of grade, curvature, and changing velocity 
has not been properly allowed for in deducing many of the 
formulae. In view of all these considerations, it may be con- 
sidered as demonstrated that no one formula, and especially a 
simple formula, will represent the resistance for all conditions. 
But, since some of the calculations of railroad economics are 
absolutely dependent on the law of tractive resistance, some 
law must be deduced with sufficient accuracy for the purpose. 
Fortunately several of the formulae are amply accurate for such 
purposes. A report of a committee of the A. R. E. & M. W. 
Assoc. (1907) quoted sixty-one different formulae which have been 
suggested. Some of- these are chiefly of historical value, since 
they were deduced from tests made many years ago with track 
and rolling stock very dissimilar from those in use at the present 
time. Such formulae will therefore be omitted. For con- 
venience of comparison, all formulae will be changed (if neces- 
sary) from the original statement of them so that they give the 



§438. 



TRAIN RESISTANCE. 



485 



resistance per ton of 2000 pounds. The coefficients of V and 
V^ will be given decimally. Other notation occasionally used 
is as follows: 

< = weight of train in tons of 2000 pounds; 
L = length of train in feet ; 
n = number of cars in train; 
A = area of front of train in square feet. 

(a) Formulae of the first class: R = aV + c. Among those 
most commonly used are the following: 



Engineering News, 
Baldwin locomotive, 
New York Central, 

Henderson, 



72 = 0.257 + 2.0 . . 
7^ = 0.177 + 3.0 . . 
i2 = 0.117+1.8, . . 

22=0.257+^- +0.5. 



(108) 
(109) 
(110) 
(111) 



Although Henderson's formula is in a class by itself, on account 
of the extra term, and although it is not applicable to general 
use, when the character of the trains cannot be estimated, it 
is perhaps more accurate than the others. It is apparently not 
intended for use at very low velocities. 

(b) Formulae of the second class: R = bV^-\-c: . 



Crawford, 
Wolff, 
Henderson, 
Forney, 



R = 0.0Q2UV^ + 2.5 (112) 

^ = 0.0035772 + 2.7 ..... (113) 

72=0.004617^ + 3.0 ..... (114) 

72 = 0.0058572 + 4.0 ..... (115) 



Wel- 
ling- 
ton 



^772 
72 = 0.005672 + ^^^V- +3.9 (for loaded flat cars) 



72 = 0.007572 



t 

.6472 
t 



+ 3.9 (for loaded box cars) 



5772 
72 = 0.008372 + ^—- +6.0 (for empty flat cars) 
t 

6472 
72 = 0.010672 + ^-— +6.0 (for empty box cars) 
z 



, (116) 



Notice in formulae (150) the additioaal jour.nal resistance 
(indicated by the constant term) for unloaded cars. The second 



486 RAILROAD .CONSTRUCTION. § 438. 

term evidently indicates the atmosplieric resistance. The first 
term allows for the oscillatory resistances. Assuming the con- 
stant term and the coefficients to have been correctly deter- 
mined, these formulae should be better than the others, since 
a choice of formulae can be made depending on the conditions. 
A train consisting partly of box-cars and partly of flat-cars 
will have a higher resistance than is shown by any of the above 
formulae (and not a mean value), on account of the increased 
atmospheric resistance acting on the irregular form of the train, 
(c) Formulse of the third class: R='aV-{'bV^+c: 

W.N.Smith, 72=0.177+ >5:52?MZ!+3.0;. . . . (117) 
VonBorries, 72=0.047^-0.001672+3.0; .... (118) 
Lundie, 72=0.247+^4^+4.0; (119) 

Sprague, i?=0.177+^;^^+4.0. (120) 



Although several formulae have been proposed which involve 
the area of the front of the train in order to allow more definitely 
for the atmospheric resistance, only one of these (117) has been 
quoted. In applying this formula, the proper value to choose 
for A is somewhat indefinite, since the shape of the front of the 
train will make a considerable difference in the atmospheric 
resistance encountered. The area is called 125 square feet in 
§ 429. In the comparison of the formulae given below, A will 
be assumed as 125 square feet. In order to compare these resist- 
ances, the values of R for the various speeds of 10, 20, 30, 40, 
50 and 60 miles per hour will be computed by these formulae on 
the basis of a train of twelve cars, having a length of 480 feet, 
and a weight of 600 tons. Therefore in applying the formula, 
t = 600, L = 480, n = 12, and A = 125. In order to apply formula 
(116) to this case, it will be assumed that this train consists of 
loaded box-cars, and therefore we must apply the second of that 
group of formulae. Computing the resistance according to 
these several formulae, we may tabulate the results as given 
below; 



439. 



TRAIN RESISTANCE. 



487 



Formula. 


Velocity in miles per hour. 




10 


20 


30 


40 


50 


60 


108 
109 
110 
111 

112 
113 
114 
115 
116 

117 
118 
119 
120 


2700 
2800 
1747 
2400 

1628 
1834 
2077 
2751 
2854 

2845 
2136 
4320 
3453 


4200 
3800 
2413 
3900 

2014 
2477 
2906 
3804 
4396 

3940 
2664 

7200 
4573 


5700 
4800 
3080 
5400 

2656 
3548 
4289 
5559 
6966 

5085 

3384 

11040 

5760 


7200 
5800 
3747 
6900 

3554 
5047 
6226 
8116 
10564 

6280 

4296 

15840 

7013 


8700 
6800 
4413 
8400 

4710 

6975 

8715 

11175 

15188 

7525 

5400 

19440 

8333 


10200 
7800 
5080 
9900 

6122 

9331 

11746 

15036 

20844 

8820 

6696 

28080 

9720 



Although there is a fair agrqement among the results fop 
ordinary velocities, it should be said, in fairness to the proposers 
of the various formulae, that some of them evidently were not 
designed for use at high velocities such as 60 miles per hour. 

Another method of comparing formulae is to plot them on 
cross-section paper, using velocities as abscissae and resistances 
as ordinates. For general use this method may only be applied 
to formulae which do not involve the weight, length or area of 
the train nor the number of cars. All of the above formulae 
have thus been plotted on Plate IX, with the exception of Nos. 
Ill, 116, 117, 119 and 120. 

439. American Railway Engineering Association Formula. 
The Economics Committee of the Amer. Rwy. Eng. Assoc, after 
considering all published formulae, on the basis of some elaborate 
tests with freight trains, developed the fact that the resistance 
of freight trains is so nearly constant between the velocities of 
7 and 35 miles per hour that it may be so considered in compara- 
tive calculations. A formula was then developed which is 
independent of velocity and which has the form, following the 
previous notation, R = at+ hn, 

in which a and h have several values, depending on the tempera- 
ture. These formulae may be grouped as follows: 



Rating 


Temp. (F) 


Formula 


A 
B 

C 
D 


above 35° 
35° to 20° 
20° to 0° 
below 0° 


R=2.2t-\-122n 
R==S.0t+137n 
R=4:.0t+153n 
R^dAt+Uln 



(121) 



488 RAILROAD CONSTRUCTION. §439. 

Applying the data of the above numerical case — 12 cars, 600 
tons, ^ = 600 and n = 12, we would have i? = 2784 lbs. for A rating 
or for a temperature above 35° F. This means an average resist- 
ance of 4.64 lbs. per ton. If we draw in Plate IX, a horizontal 
line at the height 4.64 from the 7 vertical to the 35 vertical, it 
will represent the velocity curve for this train. The line, which 
is straight and not curved, intersects every curve shown in that 
diagram. And so, although the formula is utterly different from 
those previously given, there is a rough agreement at freight- 
train velocities. 



§439, 



TRAIN RESISTANCE. 



489 



OK 


































PLA 


TE 


X. 


TRA 


N F 


ESli 


dTAI 


MCE 




1 








• 










































































/ 






^ 




^ 


















/ 




























/ 


1 












' 












1 


(^ 


/ 
























1 




K 
























/i 


/ 


/ 




a 
2-45- 


















/ 


) 


/^ 


<^ 1 




















/ 


> 


^ 


A 




«3 

d 


















/ 


^ 


/ 






m 
o 
















/ 


^""Z 


(<> 


/. 


^ 




"5 
















^ 


A 


f > 


^ 






iesistan^ 














/ 




y 


/ 




/^ 














A 


/ 


A 


f/ 




/ 


/ 
















// 


^ 




# 


^ 


/ 


V 














/; 


^ 


/ 


/ 


^' 


c^ 




^ 












J 


/^ 


^ 


A 




pi 


i>^ 














> 


^ 


/ 


^y" 


x^ 


<^ 


















r 




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'^A 


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w^ 


















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-^ 


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( 


1 


1 


) 


2 

•Ve 


loclt; 


r in miles v 


4 
erho 


p 

ur 


T 


60 1 



CHAPTER XVII 

COST OF RAILROADS. 

440. General considerations. Although there are many ele- 
ments in the cost of railroads which are roughly constant per 
mile of road, yet the published reports of the cost of railroads 
differ very widely. The variation in the figures is due to several 
tauses. (a) Economy requires that a road shall be operated 
and placed on an earning basis as soon as possible. Therefore 
the reported cost of a road during the first few years of its 
existence is somewhat less than that reported later. This is 
well illustrated when a long series of consecutive reports from 
an old-established road is available; nearly every year there 
will be shown an addition to the previous figiires. And this 
is as it should be. The magnificent road-beds of some old 
roads cannot be the creation of a single season. It takes many 
years to produce such settled perfect structures, (b) A large 
part of the variation is due to a neglect to charge up " permanent 
improvements'' as additions to the cost of the road. For the 
first few years of the life of a road a great deal of work is done 
which is in reality a completion of the work of construction, 
and yet the cost of it is buried under the item '^maintenance 
of way." For example, a long wooden trestle is replaced by 
an earth embankment and a culvert. Since the original trestle 
is to be considered a temporary structure, the excess of the 
cost of the permanent structure over that of the temporarj^ 
structure should evidentty be considered as an addition to the 
cost of the road. But if the filling-in was done slowly, a few 
train-loads at a time, and the work scattered over many years, 
the cost of operating the "mud-train" has perhaps been buried 
under '^ maintenance" charges, (c) The reports from which 
many of the following figures were taken have not always 
analyzed the items of cost with the same detail as has been 
here attempted, and to that is probably due many of the ^^aria- 
tions and apparent discrepancies. 

490 



§ 441. COST OF RAILROADS. 491 

The various items of cost will be classified as follows: 

1. Preliminary financiering. 

2. Surveys and engineering expenses. 

3. Land and land damages. 

4. Clearing and grubbing. 

5. Earthwork. 

6. Bridges, trestles, and culverts 

7. Trackwork. 

8. Buildings and miscellaneous structures. 

9. Interest on construction. 
10. Telegraph line. 

441. Item I. PRELIMINARY FINANCIERING. The COst of this 
preliminary work is exceedingly variable. The work includes 
the clerical and legal work of organization, printing, engraving 
of stocks and bonds, and (sometimes the most expensive of ^11) 
the securing of a charter. This sometimes requires special 
legislative enactments, or may sometimes be secured from a 
State railroad commission. It has been estimated that about 
2% of the railway capital of Great Britain has been spent in 
Parliamentary expenses over the charters. These expenses 
are usually but a small percentage of the total cost of the enter- 
prise, but for important lines the gross cost is large, while the 
amount of money thus spent by organizations which have 
never succeeded in constructing their roads is, in the aggregate, 
an enormous amount, although it is of course not ascertainable 
by any investigator. 

Another occasional feature of the financing of a road must be 
kept in mind. The promoters of a railroad enterprise frequently 
endeavor to limit their own personal expenditures to the purely 
preliminary expenses as mentioned above. The project, after 
having been surveyed, mapped, and written up in a glowing 
'^prospectus," is submitted to capitalists, in the endeavor to 
have them furnish money for construction, the money to be 
secured by bonds. If the project will stand it, the amount of 
the bond issue is made suflScient to pay the entire cost of the 
road, even with a discount of perhaps 15%. The bond issue 
may also provide for a very generous commission to the broker 
who is the intermediary between the promoters and the capi- 
talists. The bond issue may even provide for repaying the 
promoters for their preliminary expenses. Frequently a con- 
siderable proportion of the capital stock goes to the capitalists 



492 . RAILROAD CONSTRUCTION. § 442. 

who take the bonds, the promoters retaining only such propor- 
tion as may be agreed upon. In such a case, the capital stock 
is *'pure velvet,'' and costs nothing. Its future value, whatever 
it may be, is so much clear profit. The effect of such a financial 
policy is to burden the project with a capitalization which is 
far in excess of the actual cost of constructing the road. Com- 
paratively few projects will stand such over-capitalization. 
The apparent financial failure of many railroads, which have 
gone into the hands of receivers is due to their inability to 
make returns on an over-capitalization rather than because 
they could not earn enough to pay the legitimate cost of their 
construction. These features of financiering are really foreign 
to the engineer's work, but he should know that many projects 
which would return a handsome profit on an investment amount- 
ing only to the legitimate cost, will be rejected by capitalists 
because it is apparent that there is not enough * Wei vet" 
in it. 

442. Item 2. Surveys aistd Engineering Expenses. The 
comparison of a large number of itemized reports on the cost 
of construction shows that the cost of the " engineering '' will 
average about 2% of the total cost of construction. This in- 
cludes the cost of surveys and the cost of laying out and super- 
intending the constructive work. The cost of mere surveying 
up to the time when construction actually commences has 
been variously quoted at $60, $75, and even $300 per mile. 
The lower figures generally refer to the hasty, ill-considered work 
which was formerly common and which has resulted in so much 
badly located road, much of which has been reconstructed, 
when improvements are practicable. See the introductory par- 
agraphs of Chapter I. Except when the topography limits the 
location to one very obvious route, a thorough survey may cost 
about $300 per mile. In the estimate given at the end of this 
chapter the cost of ^^ engineering and office expenses '^ is given 
at 5% of the cost of the construction work. The item then 
includes the cost of the very considerable amount of clerical 
work and superintendence incident to the. expenditure of such a 
large sum of money. 

443. Item 3. Land and Land Damages. The cost of this 
item varies from the extreme, in which not only the land for 
right-of-way but also grants of public land adjoining the road 
are given to the corporation as a subsidy, to the other extreme 



§444. COST OF RAILROADS. 493 

where the right-of-way can only be obtained at exorbitant 
prices. The width required is variable, depending on the 
width that may be needed for deep cuts or high fills, or the 
extra land required for yards, stations, etc. A strip of land 
1 mile long and 8.25 feet wide contains precisely 1 acre. An 
average width of 4 rods (66 feet), therefore, requires 8 acres per 
mile. On the Boston & Albany Railroad the expenditure 
assigned to ^'land and land damages" averages over $25000 
per mile. Of course this includes some especially expensive 
land for terminals and stations in large cities. Less than $300 
per mile was assigned to this item by an unimportant 18-mile 
road. 

444. Item 4. CLEARING AND GRUBBING. The cost of this 
may vdry from zero to 100% for miles at a time, but as an 
average figure it may be taken as about 3 acres per mile at a 
cost of say $50 per acre. The possibility of obtaining valuable 
timber, which may be utilized for trestles, ties, or othervNdse, 
and the value of which may not only repay the cost of clearing 
and grubbing, but also some of the cost of the land, should not 
be forgotten. 

445. Item 5. EARTHWORK. This item also includes rock- 
work. The methods of estimating the cost of earthwork and 
rockwork have been discussed in Chapter III. The percentage 
of this item to the total cost is very variable. On a western 
prairie it might not be more than 5 to 10%. On a road through 
the mountains it \\dll run up to 20 or 25%, and even more. 
The item also includes tunneHng, which on some roads is a 
heavy item. 

446. Item 6. BRIDGES, TRESTLES, AND CULVERTS. This item 
will usually amount to 5 or 6% of the total cost of the road. 
In special cases, where extensive trestling is necessary, or 
several large bridges are required, the percentage will be much 
higher. On the Other hand, a road whose route avoids the 
watercourses may have very little except minor culverts > On 
the Boston & Albany the cost is given as $5860 per mile; on 
the Adirondack Railroad, $2845 per mile. Considering their 
relative character (double and single track), these figures are 
relatively what we might expect. 

447. Item 7. Trackwork. ' This item will be considered as 
including everything above subgrade, except as otherwise 
itemized. 



494 



KAILROAD CONSTRUCTION. 



§447. 



(a) Ballast. With an average width, for single track, of 
10 feet and an average depth of 15 inches, 2444 cubic yards of 
ballast will be required. The Pennsylvania Railroad estimate is 
2500 yards of gravel per mile of single track. At an estimate 
of 60 c. per yard, this costs $1500 per mile. Broken-stone 
ballast must be filled out over the ends of the ties and there- 
fore more is required; 2800 cubic yards of broken stone at 
$1.25 per yard in place will cost $3500 per mile. 

(b) Ties. Ties cost anywhere from 80 c. do\^Ti to 35 c. and 
even 25 c. At an average figure of 50 c, 2640 ties per mile 
will cost $1320 per mile of single track. The cheaper ties are 
usually smaller and more must be used per mile, and this tends 
to compensate the difference in cost. 

The following tabular form is convenient for reference : 



TABLE XXX. NUMBER OF CROSS-TIES PER MILE. 



Number per 


Average spacing 


Number 


33' rail. 


center to center. 


per mile. 


22 


18.0 inches 


3520 


21 


18.9 *' 


3360 


20 


19.8 •* 


3200 


19 


20.9 '• 


3040 


18 


22.0 ** 


2880 


17 


23.3 ** 


2720 


16 


24.75 *' 


2560 


15 


26.4 •* 


2400 


14 


28.3 ** 


2240 


13 


30.5 •* 


2080 



(c) Rails. The total weight of the rails used per mile may 
best be seen by the tabular form. 

A convenient and useful rule to remember is that the number 
of long tons (2240 lbs.) per mile of single track equals the weight 
of the rail per yard times V-' The rule is exact. For example, 
there are 3520 yards of rail in a mile of single track; at 70 lbs. 
per yard this equals 246400 lbs,, or 110 long tons (exactly); 
but 70 XV- = 110. 

Any calculation of the required weight of rail for a given 
weight of rolling-stock necessarily depends on the assumptions 
which are made regarding the support which the rails receive 
from the ties. This depends not only on the width and spacing 
of the ties (which are determinable), but also on the support 
which the ties receive from the ballast, which is not only very 
uncertain but variable. No general rule can therefore claim 



§447. 



COST OF RAILROADS. 



495 



TABLE XXXI. TONS PER MILE (wiTH COST) OF RAILS OF VARIOUS 

WEIGHTS. 





Tons 








Tons 






Weight 


(2240 lb.) 


Cost at 


Cost at 


Weight 


(22401b.) 


Cost at 


Cost at 


in lbs. 


per mile 


$26 per 


$30 per 


in lbs. 


per mile 


$26 per 


$30 per 


per yd. 


of single 
track. 


ton. 


ton. 


per yd. 


of single 
track. 


ton. 


ton. 


8 


12.571 


$326.86 


$377.14 


65 


102 . 143 


$2655.71 


$3064.29 


10 


15.714 


408 . 57 


471.43 


66 


103.714 


2696.57 


3111.43 


12 


18.857 


490.29 


565.71 


67 


105 . 286 


2737.43 


3158.59 


14 


22.000 


572.00 


660.00 


68 


106.857 


2778.29 


3205 . 79 


16 


25.143 


653.71 


754 . 20 


70 


110.000 


2860 . 00 


3300.00 


20 


31.429 


817.14 


942.86 


71 


111.571 


2900.86 


3347.14 


25 


39 . 286 


1021.43 


1178.57 


72 


113.143 


2941.71 


3394.29 


30 


47.143 


1225.71 


1414.29 


73 


114.714 


2982.57 


3441.43 


35 


55 . 000 


1430.00 


1650.00 


75 


117.857 


3064 . 29 


3535.71 


40 


62.857 


1634.29 


1885.71 


78 


122.571 


3186.86 


3677.14 


45 


70.714 


1838.57 


2121.43 


80 


125.714 


3268.57 


3771.43 


48 


75.429 


1961.14 


2262.86 


82 


128.857 


3350.29 


3865.71 


50 


78.571 


2042 . 86 


2357.14 


85 


133.571 


3472 . 86 


4007.14 


52 


81.714 


2124.57 


2451.43 


88 


138.286 


3595.43 


4148.57 


56 


88.000 


2288.00 


2640 . 00 


90 


141.429 


3677.14 


4242.86 


57 


89.571 


2328.86 


2687.14 


92 


144.571 


3758.86 


4337.14 


60 


94.286 


2451.43 


2828.57 


95 


149.286 


3881.43 


4478.57 


61 


95.857 


2492 . 29 


2875.71 


98 


154 . 000 


4004.00 


4620.00 


63 


99.000 


2574.00 


2970.00 


100 


157.143 


4085 . 71 


4714.29 



About two per cent. (2%) extra should be allowed for waste in cutting. 

any degree of precision, but the following is given by the Bald- 
win Locomotive Works : " Each ten pounds weight per yard of 
ordinary steel rail, properly supported by cross-ties (not less 
than 14 per 30-foot rail), is capable of sustaining a safe load 
per wheel of 3000 pounds." For example, a Mikado loco- 
motive with 153200 lbs. on 8 drivers has a load of 19150 lbs. 
per wheel. This divided by 3000 gives 6.38. According to the 
rule, the rails for such a locomotive should weigh at least 63.8 
lbs. per yard. 

On the basis of 33-foot lengths, and 10% shorter lengths, vary- 
ing by even feet down to 27 feet (see § 274, 8), the average 
length, assuming an equal number each of the shorter length 
rails would be 32.65 feet. Calculating similarly for 30-ft. rails, 
with 10% shorts to 24 feet, the average length would be 29.65 
feet. 60-ft. rails, used extensively for electric roads, with 10% 
shorts to 40 feet, will have average length of 58.95 feet. 

(d) Splice-bars, track-bolts, and spikes. These are usually 
sold by the pound, except the patented forms of rail-joints, 
which are sold by the pair. In any case they are subject to 
market fluctuations in price. As an approximate value the 
following prices are quoted: Splice-bars, 1.35 cents per pound; 



496 



EAILROAD CONSTRUCTION. 



§447, 



track-bolts, 2.4 cents; spikes, 1.75 cents. The weight of the 
splice-bars will depend on the precise pattern adopted — its 
cross-section and length. 

In Table XXXII are quoted from a catalogue of the Illinois 
Steel Co. the weights per foot of sections of angle-bars which 
they recommend for various weights of rail and which are de- 
signed to fit standard A. S. C. E. rail sections of those weights. 
The net weight of the angle-bars may be approximated by 
subtracting about 2.5% to 4% from the gross weight to allow 
for the bolt-holes. A deduction of 2.5% is usually about 
right for the heavier sections. Their recommendations regard- 
ing lengths of angle-bars do not include those for rails heavier 
than 50 pounds per yard. On the basis of a length of 23 inches 
for four-hole splices and of 33 inches for six-hole splices, the 
weights of splice-bars have been computed for the several 
styles of splices for heavier rails, allowing 2.5% for the holes. 
The lengths recommended for track bolts are those which will 
allow about i inch for the nutlock and for margin, except for 
the lighter rails. 



TABLE XXXII. — SPLICE-BARS FOR VARIOUS WEIGHTS OP RAILS. 



Weight 
of 
rail. 


Length 
angle- bar. 


Weight 

per 

foot. 


Weight 

of 

pair. 


Proper 

size of 

track-bolt. 


Proper size 
of spikes. 


30 


21" 


4.49 


15.1 


2i"Xf" 


4" xr 


35 


21" 


4.7 


15.9 


21" Xt" 


4rxr 


40 


21" 


6.54 


18.8 


3 "Xf" 


5 "xr 


45 


21" 


6.3 


21.5 


3 "Xf" 


5rxA" 


50 


21" 


6.97 


23.4 


3rXf" 


5|"xS" 


55 


24" 


7.5 


29.2 


3f"X|" 


5i"Xi^" 


60 


24" 


8.4 


32.8 


3f"Xf" 


5rx# 


65 


124" 


9.2 


35.9 


4 "Xf" 


5rx,%" 


\32" 


9.6 


49.9 


4i"X|-" 


5rx3%" 


70 


./ 24" 


9.0 


35.1 


4 "Xf" 


5rxA" 


32" 


10.0 


52.0 


4 "Xf" 


5i"X^/' 


15 


/24" 


10.68 


42.6 


4i"Xf" 


5i"XT^" 


132" 


11.9 


61.9 


4 "Xf" 


5rxT^" 


80 


J 24" 


10.61 


42.3 


4V'Xi" 


5rxT^" 


\32" 


14.65 


76.2 


4i"Xr 


srxiV 


85 


32'' 


12.4 


64.5 


4>"XF 


5rXA"orr 


90 


32" 


13.5 


70.2 


4f"Xf" 
4f"X|" 


5rX^"or|" 


95 


32" 


14.7 


76.4 


5i"Xi^6"^or|" 


100 


32" 


15.78 


82.1 


A3_f/ y 7.// 


6rXA"or|" 



(e) Track-laying. Much depends on the force of men em- 
ployed and the use of systematic methods; $528 per mile is 
the estimate employed by the Pennsylvania Railroad. $500 per 
mile is the estimate given in § 451. 



§447. 



COST OF RAILROADS. 



497 



TABLE XXXIII. RAILROAD SPIKES. 







Ties 24" between cen- 






Average 


ters, 4 spikes per tie, 


Suitable 


Size meas- 


number 


number per mile. 


weight of 


ured under 
head. 


per keg of 
200 pounds 




rail. 








■ 


Pounds. 


Kegs. 




srxr 


275 


7680 


38.40 


90 to 100 


5rxi^" 


375 


5632 


28.16 


45 ** 100 


5" XA" 
5" Xi" 


400 


5280 


26.40 


40 •* 56 


450 


4692 


23.46 


40 


4rxr 


530 


3984 


19.92 


35 


4" X¥' 

4rxi^" 


600 


3520 


17.60 


30 


680 


3104 


15.52 


25 to 30 



TABLE XXXIV. — TRACK-BOLTS, 
Average number in a keg of 200 pounds. 



Size of 


Square 


Hexagonal 


Suitable 


bolt. 


nut. 


nut. 


rail. 


3" xr' 


366 


395 


40 pound 


3" xr 


250 


270 




3rxr' 


243 


261 




3rxf" 


236 


253 


50 


sf'x¥' 


229 


244 


55 to 60 


4„ ><|r/ 


222 


236 


65 ** 70 


4rxf" 


215 


228 


75 


s^'xi" 


170 


180 




Sf'Xi" 


165 


175 




4- Xi" 


161 


170 




4rxr 


157 


165 


80 


4:i"X¥' 


153 


160 


85 


41" xr 


149 


156 


90 



TABLE XXXV. RAIL- JOINTS AND TRACK-BOLTS. 

OP TRACK. 



NUMBER PER MILE 





Average 


Number of 


Number 


of bolts. 


Length of 


length of 


rails or 






rail. 


rail. 


complete 






Feet. 


Feet. 


joints. 


4-bolt. 


6-bolt. 


All 30 


30 


352 


1408 


2112 


30-24 


• 29.65 


356.2 


1425 


2137 


All 33 


33 


320 


1280 


1920 


33-27 


32.65 


323.4 


1294 


1941 


All 60 


60 


176 


704 


1056 


60-40 


58.95 


179.1 


717 


1075 



498 RAILROAD CONSTRUCTION. § 448. 

448. Item 8. Buildings and Miscellaneous Structures. Ex- 
cept for rough and preliminary estimates, these items must 
be individually estimated according to the circumstances. The 
subitems include depots, engine-houses, repair-shops, water- 
stations, section- and tool-houses, besides a large variety of 
smaller buildings. The structures include turn-tables, cattle- 
guards, fencing, road-crossings, overhead bridges, etc. The 
detailed estimate, given in § 451, illustrates the cost of these 
smaller items. 

449. Item 9. Interest on Construction. The amount 
of capital that must be spent on a railroad before it has begun 
to earn anything is so very large that the interest on the cost 
during the period of construction is a very considerable item. The 
amount that must be charged to this head depends on the cur- 
rent rate of money on the time required for construction and 
on the abilit}^ of the capitalists to retain their capital where 
it will be earning something until it is actually needed to pay 
the company's obligations. Of course, it is not necessary to 
have the entire capital needed for construction on hand when 
construction commences. Assuming money to be worth 6%, 
that the work of construction will require one year, that the 
money may be retained where it will earn something for an 
average period of six months after construction commences, 
or, in other words, it will be out of circulation six months before 
the road is opened for traffic and begins to earn its way, then 
we may charge 3% on the total cost of construction. 

. 450. Item 10. TELEGRAPH LINES. This evidently depends 
on the scale of the road and the magnitude of the business to 
be operated. In the following estimate it is given as $200 
per mile, which evidently is intended to apply to the business 
of a small road. 

451. Detailed estimate of the cost of a line of road. The fol- 
lowing estimate was given in the Engineering News of Dec. 27, 
1900, of the cost of the Duluth, St. Cloud, Glencoe & Mankato 
Railroad, 157.2 miles long. 

The estimate is exactly as copied from the Engineering N'ews, 
There are some numerical discrepancies. Item 26 should evi- 
dently be based on the sum of the first 25 items, and item 27 
on the sum of the first 26. The figures in parentheses ( ) are 
deduced from the figures given. 



§451. 



COST OF RAILROADS. 499 



1. Right-of-way; 1905.3 acres (12.12 acres per mile) @ $100 per 

acre $190530 

2. Clearing and grubbing. 144 acres (0.916 acre per mile) @ $50 

per acre 7200 

3. Earth excavation. 1907590 cu. yds. (12135 cu. yds. per mile) 

@ 15 c ; 286138 

4. Rock excavation. 5100 cu. yds. (32.44 cu. yds. per mile) @ 80 c. 4080 
( Wooden-box culverts. 508300 ft. B.M. @ $30 per M. . $15249 

I Iron-pipe culverts. 879840 lbs. @ 3c. per lb 26395 41644 

J Pile trestling. 4600 lin. ft. @ 35 c. per lin. ft 1610 

■ 1 Timber trestling. 509300 ft. B.M. @ $30 per M 15279 16889 

„ \ Bridge masonry: 5520 cu. yds. @ $8 per cu. yd 44160 

'• ] Bridges, iron, 100 spans, 2000000 lbs. @ 4 c. per lb. . . 80000 124160 

8. Cattle-guards 8750 

9. Ties (2640 per mHe) . 419813 (159.02 miles) @ 35 c 146935 

10. Rails (70 lbs. per yd.): 110 tons per mile^ 17492.2 tons (159.02 

miles @$26 384797 

11. Rail sidings (70 lbs. per yd.) : 110 tons per mile, 3300 tons 

(30 miles @ $26 85800 

12. Switch timbers and ties 3300 

13. Spikes: 5920 lbs. per mile. 1107040 (187 m.) @ 1.75. c. per lb. 19373 

14. Splice-bars: 2635776 lbs. @ 1.35 c. per lb 35583 

15. Track-bolts (2 to joint (?)): 188458.3 lbs. @ 2.4 c. per lb 4520 

16. Track-laying. 187.2 miles @ $500 per mile.- 93600 

17. Ballasting: 2152 cu. yds. per mile, 402854 (187.2 m.) @ 60 c. . 241712 

18. Turn-out and switch furnishings 6450 

19. Road-crossings, 68040 ft. B.M. @ $30 per M 2041 

20. Section and tool-houses, 16 @ $800 12800 

21. Water-stations 15000 

22. Turn-tables, 6 @ $800 4800 

23. Depots, grounds, and repair-shops 78000 

24. Terminal grounds and special land damages 150000 

25. Fencing, 314 miles ($150 per mile) 47100 

26. Engineering and office expenses (5% of $1984458) 99222 

27. Interest on construction (3% of $2083680) 62510 

28. Rolling-stock ($5000 per mile) 786000 

29. Telegraph line: 157 miles @ $200 per mile 31400 

$3060340 
Average cost per mile ready for operation, $19467. 
Approximate cost of 130 miles from St. Cloud to Duluth, estimated at 

$23000 per mile. 
Approximate cost of entire line from Albert Lea to Duluth, 287.2 miles, 

$6050340 ($21060 per mile). 



CHAPTER XVIII. 

THE POWER OF A LOCOMOTIVE. 

452c Pounds of steam produced. The power that can be 
developed by a locomotive depends very greatly on the quality 
of the coal burned and the design of the locomotive must corre- 
spond to the general kind or quality of coal to be used. A 
British thermal luiit (symbolized as B.t.u.), is the quantity of 
heat required to raise the temperature of 1 lb. of pure water 
1° F., when the water is at or near its maximum density at 39.1° 
F. When it is said that a certain grade of coal has 14000 
B.t.u. it means that the heat in 1 lb. of that coal will raise the 
temperature of 14000 lbs. of water 1°, or, approximately, 100 lbs. 
of water 140°. But, although ip only requires 180.9 heat units 
to heat water from 32° to 212°, it requires 965.7 more heat units 
to change it from water at 212° to steam at 212°. It requires 
only 53.6 more heat units to change it from steam at 212° to 
steam at 387.6° or with a pressure of 200 lbs. per square inch. 

A study of locomotive tests made at the St. Louis Exposition 
resulted in the compilation of Table XXXVI, which is copied 
from the Proceedings of the American Railway Engineering 
Association, and is now included as Table I, in the " Economics " 
section of their Manual. It was found that the steam produced 
per square foot of heating surface is very nearly proportional to 
the coal burned per square foot of heating surface. The results 
are purposely made about 5% below the results obtained in the 
St. Louis tests to allow for ordinary working conditions. 

453. Numerical example. The theory developed in this 
chapter will be illustrated numerically by applying it to a Mikado 
type of locomotive whose dimensions are as follows: 



Cylinder diam. 22" 

Cylinder stroke 28" 

Driving wheel diam. 57" 

Boiler pressure 185 lbs. 

Fire-box length 102|" 

Fire-box width 65|" 

Grate area 46 . 8 sq. f t. 



Weight, driving wheels. 153,200 lbs. 

engine alone 196,100 lbs. 

engine and tender. . . . 31 5,000 Jbs. 
Heating surface, fire-box 

and tubes 2565 sq. ft. 

superheating surface . • 550 sq. ft. 



500 



I 



§453. 



THE POWER OF A LOCOMOTIVE, 



501 



TABLE XXXVI. — AVERAGE EVAPORATION IN LOCOMOTIVE BOILERS 
BURNING BITUMINOUS AND SIMILAR COALS OF VARIOUS QUAL- 
ITIES, AND FOR VARIOUS QUANTITIES CONSUMED PER SQUARE 
FOOT OF HEATING SURFACE PER HOUR. 
(Based on feed water at 60° Fahrenheit, and boiler pressure 200 pounds) 





Steam per pound of coal of given thermal value 


Coal per square 






(lb.) 






foot of heating 












surface per hour 














(lb.) 


15,000 


14,000 


13,000 


12,000 


11,000 


10,000 




B.t.u. 


B.t.u. 


B.t.u. 


B.t.u. 


B.t.u. 


B.t.u, 


0.8 


7.86 


7.34 


6.81 


6.29 


5.76 


5.24 


0.9 


7.58 


7.07 


6.57 


6.06 


5.56 


5.05 


1.0 


7.31 


6.82 


6.34 


5.85 


5.36 


4.87 


1.1 


7.06 


6.59 


6.12 


5.65 


5.18 


4.71 


1.2 


6.82 


6.37 


5.91 


5.46 


5.00 


4.55 


1.3 


6.59 


6.15 


5.71 


5.27 


4.83 


4.39 


1.4 


6.37 


5.95 


5.52 


5.10 


4.67 


4.25 


1.5 


6.17 


5.76 


5.35 


4.94 


4.52 


4.11 


1.6 


5.97 


5.57 


5.18 


4.78 


4.38 


3.98 


1.7 


5.79 


5.40 


5.02 


4.63 


4.25 


3.86 


1.8 


5.61 


5.24 


4.86 


4.49 


4.12 


3.74 


1.9 


5.44 


5.08 


4.71 


4.35 


3.99 


3.63 


2.0 


5.27 


4.92 


4.57 


4.22 


3.86 


3.51 


2.1 


5.12 


4.78 


4.44 


4.10 


3.75 


3.41 


2.2 


4.97 


4.64 


4.31 


3.98 


3.64 


3.31 


2.3 


4.83 


4.51 


4.19 


3.86 


3.54 


3.22 


2.4 


4.69 


4.38 


4.07 


3.75 


3.44 


3.13 


2.5 


4.56 


4.26 


3.95 


3.65 


3 . 34 


3.04 


2.6 


4.44 


4.14 


3.84 


3.55 


3.25 


2.96 


2.7 


4.32 


4.03 


3.74 


3.46 


3.17 


2.88 


2.8 


4.21 


3.93 


3.64 


3.37 


3.09 


2.80 


2.9 


4.10 


3.83 


3.55 


3.28 


3.01 


2.73 


3.0 


3.99 


3.73 


3.46 


3.19 


2.93 


2.66 



The quantity of steam evaporated for intermediate quantities or qualities 
of coal can be found by interpolation. 

On bad-water districts deduct the following from tabular quantities: 

For each Ye inch of accumulated scale 10 per cent 

For each grain per U. S. gallon of foaming salts 

in the average feed water 1 per cent 

Assume that this locomotive is using coal whose air-dried 
mine samples tested 13000 B.t.u.; then the average run-of-car 
coal would have about 90% of this or 11700 B.t.u. On the 
basis that a fireman can handle 4000 lbs. of coal per hour and 
maintain such work throughout his run, the coal may be fed at 
the rate of (4000 -^ 2565) =1.56 lbs. per hour per square foot of 
heating surface. Interpolating in Table XXXVI for 1.56 and 
11700 we find that the pounds of steam per pound of coal would 
be 4.72. The tests at St. Louis showed that a reduction in 



502 RAILROAD CONSTRUCTION. § 454. 

boiler pressure increased very slightly the amount of steam pro- 
duced, but that this amount was only 0.5% greater when the 
pressure was 160 lbs. instead of 200 lbs. The effect of variation 
of pressure can therefore be ordinarily ignored. In this case 
it might add 0.2% or make the figure 4.73. Considering that 
a superheater adds from 15 to 25% to the efficiency, we will 
assume the average of 20% and say that 0.80 lb. of the super- 
heated steam produced may be considered as having the same 
volume and pressure as 1 lb. of saturated steam. Then the 
amount of steam developed by 1 lb. of coal would be the equiva- 
lent of 4.73 -^0.80 = 5.91 lbs. Then the equivalent amount of 
steam developed per hour equals 5.91 X 4000 = 23640 lbs. 

454. Weight of steam per stroke at full cut-off. This may be 
computed most easily by utilizing Table XXXVII, which is 
also taken (but somewhat amplified), from the Proceedings of 
the American Railway Engineering Association, and is now 
included as Table 2 in the '^Economics" section of their Manual. 
The weight of steam per foot of stroke for 22 ins. diameter and 
185 lbs. gauge pressure is 1.161 lbs. and for a stroke of 28 ins. 
(2| ft.) it is 2.709 lbs. For a complete revolution of the drivers 
it is 4X2.709 = 10.836 lbs. Since the engine can develop the 
equivalent of 23640 lbs. of steam per hour and will use 10.836 lbs. 
at one revolution, it can run at a speed of 23640-7-10.836 = 2182 
revolutions per hour, or 36.36 revolutions per minute, at full 
stroke and maintain full boiler pressure. The drivers are 57 
ins. in diameter and, therefore, have a circumference of (57-f-12) 
X3.1416 = 14.923 ft. The maximum engine speed for full 
stroke is 36.36X14.923 = 542.6 ft. per minute. Multiplying by 
60 and dividing by 5280, or dividing by 88, we have 6.167 miles 
per hour as the maximum speed at which full stroke can be main- 
tained, which is the value M for these conditions. 

455. Pounds of steam and per cent, of cut-off for multiples of 
M velocity. In Table XXXVIII, also taken from the Pro- 
ceedings of the American Railway Engineering Association and 
now included at Table 4 in the '' Economics" section of the Man- 
ual, are given the pounds of steam per indicated horse-power 
hour for simple and for compound locomotives for various 
velocities, which are multiples of M, the maximum velocity 
at which the locomotive can use steam at full stroke and yet the 
boiler can maintain steam at full pressure. The table is com- 
puted on the basis of 200 lbs. gauge pressure, but factors are 



§ 455. 



THE POWER OF A LOCOMOTIVE. 



503 



TABLE XXXVII. — WEIGHT OF STEAM USED IN ONE FOOT OF STROKE 
IN LOCOMOTIVE CYLINDERS. 

(Cylinder diameter is for high-pressure cylinders in compound locomotives) 





Weight of steam per foot of stroke for 


various 


gauge 




] 


pressures. 






Diameter 
















of cylinder 


220 lbs. 


210 lbs. 


200 lbs. 


190 lbs. 


180 lbs. 


170 lbs. 


160 lbs. 


(inches) 


per 


per 


per 


per 


per 


per 


per 




sq. in. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 


sq. in. 




(lb.) 


(lb.) 


(lb.) 


(lb.) 


(lb.) 


(lb.) 


(lb.) 


12 


0.405 


0.389 


0.370 


0.354 


0.337 


0.321 


0.304 


13 


0.475 


0.456 


0.435 


0.415 


0.396 


0.376 


0.357 


14 


0.551 


0.529 


0.504 


0.482 


0.459 


0.436 


0.414 


15 


0.633 


0.607 


0.579 


0.553 


0.527 


0.501 


0.476 


15i 


0.675 


0.649 


0.618 


0.590 


0.562 


0.535 


0.508 


16 


0.720 


0.691 


0.658 


0.629 


0.599 


0.570 


0.541 


17 


0.812 


0.780 


0.744 


0.710 


0.676 


0.643 


0.611 


18 


0.911 


0.875 


0.834 


0.796 


0.759 


0.722 


0.685 


181 


0.962 


0.924 


0.881 


0.841 


0.801 


0.762 


0.724 


19 


1.015 


0.975 


0.928 


0.887 


0.845 


0.804 


0.763 


191 


1.069 


1.027 


0.978 


0.934 


0.890 


0.847 


0.804 


20 


1.125 


1.080 


1.029 


0.983 


0.936 


0.891 


0.836 


201 


1.181 


1.134 


1.081 


1.032 


0.984 


0.936 


0.888 


21 


1.240 


1.191 


1.134 


1.083 


1.032 


0.982 


0.932 


22 


1.361 


1.307 


1.245 


1.189 


1.133 


1.078 


1 023 


23 


1.487 


1.428 


1.361 


1.300 


1.238 


1.178 


1.118 


24 


1.620 


1.555 


1.482 


1.416 


1.348 


1.283 


1.218 


25 


1.758 


1.688 


1.608 


1.536 


1.462 


1.392 


1.322 


26 


1.901 


1.825 


1.739 


1.661 


1.582 


1.506 


1.430 


27 


2.050 


1.968 


1.875 


1.792 


1.706 


1.624 


1.542 


28 


2.204 


2.117 


2.017 


1.926 


1.835 


1.745 


1.657 



For weight of steam used per revolution of drivers at full cut-off: 
Multiply the tabular quantity by four times the length of stroke in feet 

for simple and four-cylinder compounds. For two-cylinder compounds 

multiply by two times the length of stroke. 



given for other pressures. For example, continuing the above 
numerical problem, the pounds of steam per i.h.p.-hour, for a 
simple locomotive, at M velocity, and at 200 lbs. pressure, taken 
from Table XXXVIII, is 38.30; for 185 lbs. pressure we must 
multiply by the factor 1.0095, which makes the quantity 38.66. 
Dividing this into 23640, the steam produced per hour, we have 

611.5, the i.h.p. at M velocity. Multiplying this by 33000, 
the foot-pounds per minute in one horse-power, and dividing by 

542.6, the velocity in feet per minute, we have 37190, the cylinder 
tractive power in pounds, when burning 4000 lbs. of coal per 
hour and running at 6.167 m.p.h. . , 



504 



RAILROAD CONSTRUCTION. 



§456. 



TABLE XXXVIII. — MAXIMUM CUT-OFF AND POUNDS OF STEAM PER 

I.H.P.-HOUR FOR VARIOUS MULTIPLES OF M, 

(M is maximum velocity in miles per hour at full cut-off, with boiler 
pressure at 200 pounds per square inch) 



Velocity 


Cut-off 
per cent 


Pounds steam per 
I.H.P.-hour 


Velocity 


Cut-off 
per cent 


Pounds steam per 
I.H.P.-hour 


Simple 


Com- 
pound 


Simple 


Com- 
pound 


1.0 M 

1.1 " 

1.2 " 

1.3 " 

1.4 " 


Full 
94.4 
89.1 
84.3 
79.7 


38.30 
36.46 
34.89 
33.56 
32.41 


25.80 
24.36 
23.24 
22.35 
21.65 


2.9 M 
3.0 " 
3.2 " 
3.4 " 
3.6 " 


38.5 
37.0 
34.2 
31.8 
29.8 


24.37 
24.22 
24.00 
23.85 
23.80 


21.04 
21.21 
21.57 
21.93 
22.27 


1.5 " 

1.6 " 

1.7 " 

1.8 " 

1.9 " 


75.4 
71.4 
67.7 
64.3 
61.0 


31.40 
30.49 
29.67 
28.93 
28.25 


21.14 
20.77 
20.52 • 
20.40 
20.40 


3.8 " 
4.0 " 
4.25 " 
4.50 " 
4.75 " 


28.0 
26.4 
24.7 
23.3 
22.1 


23.80 
23.87 
24.05 
24.24 
24.44 


22.57 
22.85 
23.22 
23.56 
23.85 


2.0 •' 

2.1 •♦ 

2.2 •• 

2.3 " 

2.4 •' 


58.0 
55.2 
52.6 
50.1 

47.8 


27.62 
27.05 
26.52 
26.06 
25.67 


20.40 
20.40 
20.40 
20.40 
20.40 


5.0 " 
5.5 " 
6.0 " 
6.5 " 
7.0 " 


21.1 
19.5 
18.4 
17.6 
17.1 


24.64 
24.98 
25.20 
25.45 
25.60 


24.15 
24.70 


2.5 " 

2.6 " 

2.7 '* 

2.8 " 


45.7 
43.7 
41.8 
40.1 


25.32 
25.02 
24.76 
24.54 


20.47 
20.60 
20.73 

20.88 


7.5 " 
8.0 " 
9.0 " 


16.7 
16.4 
16.1 


25.70 
25.80 
25.90 




For steam per i 
centages of values 


.h.p.-hour 
given in 


for other 
table: 


boiler pressure take 


the following per- 


160 
170 


lb.. 103. ( 
lb., 102.] 


)% 1 
L% 1 


180 lb. 
190 lb. 


, 101.3% 
. 100.6% 


2 
2 


10 lb., 99 
00 lb., 99 


.5% 
.2% 



456. Draw-bar Pull. To obtain the draw-bar pull we must 
deduct the engine resistance. These have already been dis- 
cussed in § 429 and the numerical value of the resistance of this 
same locomotive has been there computed to be about 1771 lbs. 
Subtracting this from 37190 we have 35419 lbs., the estimated 
draw-bar pull for that speed and coal consumption. 

457. Effect of increasing the rate of coal consumption. To 
note the effect of increasing the rate of coal consumption, the 
problem may be again worked through on the basis that the rate 
of coal consumption is increased, even temporarily, from 4000 
lbs. to 5000 lbs. per hour. The steam developed per pound of 
coal is reduced from 5.91 to 5.23, but the total steam produced 
per hour is increased from 23640 to 26150. The increased ca- 
pacity comes through a loss of efficiency. The increased steam 



§457. 



THE POWER OF A LOCOMOTIVE. 



505 



production raises the velocity at which full stroke may be main- 
tained from 6.167 m.p.h to 6.820 m.p.h and the i.h.p. from 
611.5 to 676.4. But the computed cylinder tractive power is 
practically identical, the numerical computation of 37190 being 
only changed to 37189. But these cyhnder tractive powers 
are each computed for the ^' M '' velocities, the maximum ve- 
locities at which full stroke can be maintained, and " ikf " is 
higher with increased coal consumption. For a real comparison, 
the figures must be reduced to the same velocity, e.g., the work- 
ing velocity of 10 m.p.h. 10-^6.167 = 1.621, the multiple for the 
original problem. For 5000 lbs. of coal per hour, M velocity is 



TABLE XXXIX 



•PER CENT CYLINDER TRACTIVE 
VARIOUS MULTIPLES OF M. 



POWER FOR 



(M is maximum velocity in miles per hour at which boiler pressure can be 
maintained with full cut-off) 



Veloc- 
ity 


Per cent 
(Com- 


Per cent 
(Sim- 


Veloc- 
ity 


Per cent 
(Com- 


Per cent 
(Sim- . 


Veloc- 
ity 


Per cent 
(Com- 


Per cent 
(Sim- 


pound) 


ple) 


pound) 


pie) 


pound) 


ple) 


Start 


135.00 


106.00 


3.6 M 


32.40 


44.75 


6.4 M 




23.59 


0.5 M 


103 . 00 


103.00 


3.7 " 


31.25 


43.56 


6.5 " 




23.18 


1.0 " 


100.00 


100.00 


3.8 " 


30.10 


42.39 


6.6 •' 




22.79 


1.1 " 


96.28 


95.57 


3.9 " 


29.14 


41.24 


6.7 " 




22.42 


1.2 " 


92.55 


91.53 


4.0 " 


28.24 


40.10 


6.8 " 




22.06 


1.3 " 


88.83 


87.83 


4.1 " 


27.38 


39.00 


6.9 " 




21.71 


1.4 " 


85.12 


84.46 


4.2 " 


26.56 


37.96 


7.0 " 




21.38 


1.5 " 


81.40 


81.37 


4.3 " 


25.77 


36.97 


7.1 " 




21.06 


1.6 " 


77.68 


78.55 


4.4 " 


25.03 


36.03 


7.2 " 




20.75 


1.7 " 


73.96 


75.97 


4.5 " 


24.34 


35.13 


7.3 " 




20.45 


1.8 •• 


70.25 


73.60 


4.6 " 


23.69 


34.26 


7.4 " 




20.16 


1.9 " 


66.54 


71.41 


4.7 " 


23.07 


33.41 


7.5 " 




19.88 


2.0 " 


63.21 


69.37 


4.8 " 


22.48 


32.59 


7.6 " 




19.61 


2.1 " 


60.20 


67.47 


4.9 " 


21.92 


31.82 


7.7 '' 




19.34 


2.2 " 


57.48 


65.67 


5.0 " 


21.38 


31.11 


7.8 " 




19.08 


2.3 •• 


54.97 


63.94 


5.1 •' 


20.87 


30.42 


7.9 " 




18.82 


2.4 " 


52.68 


62.22 


5.2 " 


20.37 


29.75 


8.0 *• 




18.57 


2.5 " 


50.42 


60.55 


5.3 •• 


19.89 


29.10 


8.1 " 




18.33 


2.6 " 


48.16 


58.92 


5.4 " 


19.43 


28.48 


8.2 •* 




18.09 


2.7 " 


46.08 


57.33 


5.5 " 


18.99 


27.87 


8.3 '' 




17.86 


2.8 " 


44.10 


55.78 


5.6 " 




27.33 


8.4 " 




17.64 


2.9 " 


42.29 


54.26 


5.7 " 




26.81 


8.5 ♦' 




17.43 


3.0 " 


40.57 


52.78 


5.8 " 




26.30 


8.6 " 




17.22 


3.1 ♦• 


38.95 


51.33 


5.9 " 




25.81 


8.7 •* 




17.01 


3.2 " 


37.42 


49.91 


6.0 " 




25.34 


8.8 ♦' 




16.82 


3.3 " 


35.98 


48.55 


6.1 " 




24.88 


8.9 •' 




16.63 


3.4 " 


34.66 


47.24 


6.2 " 




24.44 


9.0 '* 




16.45 


3.5 " 


33.53 


45.97 


6.3 *• 




24.01 









* Table 5 in "Economics 
Engineering Association. 



Section of Manual of American Railway 



506 RAILROAD CONSTRUCTION. § 458. 

6.820 m.p.h., and the multiple Is 1.466. From Table XXXIX 
we find that the percentages of cylinder tractive power for simple 
engines for these two multiples of M are 78.01 and 82.42, respec- 
tively. The higher value is 105.7% of the lower, which shows 
that, in this case, adding 25% to the rate of coal consumption 
adds only 5.7 to the cylinder tractive power at 10 m.p.h. 

458. Effect of using a better quality of coal. As another 
instructive variation of the same problem, assume that the coal 
has effective B.t.u. of 13000, instead of only 11700. It will be 
found that steam will be produced more rapidly, the M velocity 
is 6.867 m.p.h. and the horsepower at that velocity is 680.3, 
but the cy Under power is computed to be 37191 lbs., which is 
again almost identical with the previous values, although the M 
velocity is still higher. The multiple for 10 m.p.h. is 1.456 and 
by Table XXXIX the per cent, of cylinder tractive power is 
82.73, which is an increase of 6% over 78.01%, showing that the 
increase in effective B.t.u. from 11700 to 13000 adds 6% to the 
cylinder tractive power at 10 m.p.h. 

459. Check with approximate rule. Applying Eq. 103 to 
the above data on the basis that the " effective steam pressure '' 
is 85% of the gauge pressure (185) or 157 lbs., we will have 

22^X157X28 

Tractive force = = 37327 lbs. 

57 

This agrees with the more precise value (37190) computed above 
to within one-half of one per cent. This rule is more simple as a 
method of obtaining merely the maximum tractive power at 
slow velocities, but the previous method, although longer, is 
preferable, since it computes the critical velocity M, and also 
the tractive force at higher velocities. 

460. Tractive Force at Higher Velocities. At higher velocities 
than M, the cyhnder power falls off quite rapidly, since the steam 
is cut off at part stroke and is used expansively. The proper 
per cent of cut-off for any given velocity and the number of 
pounds of steam per i.h.p. are shown in Table XXXVIII, in 
which is give the per cent of cylinder tractive power for multi- 
ples of M. The table shows, for example, that, for simple 
engines, the cylinder tractive power is 69.37% of its value for 
full stroke when the velocity is 2M and that when the velocity 
is increased to 5M the tractive power is reduced to 31.11%. 



I 



§ 460. 



THE POWER OF A LOCOMOTIVE. 



507 



Applying this to the above numerical problem, when Af = 6.167 
m.p.h., the cylinder tractive power is reduced to 31.11% of 
37190, or 11570 lbs., but, since the velocity is five times as great, 
the horse-power developed is 31.11%X5 = 1.55 times as great. 
It should be noted that Table XXXIX shows a slight excess of 
tractive power (6% when starting), for the simple engine. This is 
due to the fact that with very low velocities the cylinder pressure 
more nearly equals the full boiler pressure and there is not the 
usual reduction of about 15%. Also, compound locomotives are 
operated with all the cylinders using full-pressure steam, which 
increases their effectiveness at starting about 35%, although at 
some loss in economy of steam due to compounding. But since 
the starting resistances are so much greater than the resistances 
above 5 miles per hour, the extra assistance is very timely. 

Any competent locomotive designer will, of course, make a 
design such that there is a proper relation between cyhnder 
power and tractive adhesion. In the above case, 106% of 
37190 = 39421 lbs., which is 25.7% of the weight on the drivers, 
and this is just about the ratio of adhesion which may be ex- 
pected > 



Velocity. 


Cylinder tractive, 
power 


Locomo- 
tive resist- 
ance 
pounds. 


Draw-bar 

pull, 
pounds 


Multiples 
of M. 


Miles 
I)er hour. 


Per cent. 


Pounds. 


0.0 
1.0 
1.2 
1.5 
2.0 
3.0 
4.0 
5.0 
6.0 


0.000 

6.167 

7.400 

9.250 

12.334 

18.501 

24 . 668 

30.835 

37.002 


106.00 
100.00 
91.53 
81.37 
69.37 
52.78 
40.10 
31.11 
25.34 


39421 
37190 
34040 
30261 
25799 
19629 
14913 
11570 
9424 


1762 
1771 
1776 
1783 
1800 
1847 
1913 
1999 
2104 


37659 
35419 
32264 
28478 
23999 
17782 
13000 
9571 
7320 



A graphical illustration of the variation in tractive power and 
velocity may be obtained by computing first and setting down 
in tabular form the multiple values of ikf (6.167) ; the percentages 
taken from Table XXXIX, for each multiple of M; the products 
of each percentage times the tractive force (37190), for M veloc- 
ity; the locomotive resistance, from Table XXIX, for each 
velocity; and the net draw-bar pull for each velocity. These 
several values for cylinder tractive power and for draw-bar pull 
may be plotted as shown in Fig. 208. 



508 



RAILROAD CONSTRUCTION, 



461. 



The student should realize that the above values represent 
the maximum draw-bar pull which the locomotive can produce, 
provided the fire-box is fed with 4000 lbs. of coal per hour. These 
draw-bar pulls as given will overcome the resistance of a train of 
some definite weight, at uniform speed, along a straight level 
track, at the several velocities given. A less weight of train 
will be drawn somewhat faster; or, it will travel at the same 
speed by using less coal or by throttling the steam and, perhaps, 
wasting it at the blow-off. A heavier train could not maintain 
such speed. While the values given are approximately correct, 
a variation in the quality of the coal, or in the condition of the 



'm,m\ 


F 


"-- 


;e: 






■ ■■ 






"*■ 


■^ " 


■ ~ 




"" 


■■ 


" 




■■ ' 


■"T 


■ "" 




- 


— 


■— 


— " 


— 


-" 


■ 


w ' 






:^k 
















































1 






'-.^ 


fe 




























1 


























r 


s< 


















































% 30,000 










\v 




























































^s 




























































s 


si 














































^ 














V^s 


>^^A- 


























































^^> 


,^s^ 


(^ 










































,^20,000 
















^. > 


> 


r< 


. 




















































^'^rr 


7. 


P' 


'^ 


i'Q 


c. 


































tS 


















<>c 


^^ 


w 




^« 






^ 
























§ 






















p 






■ 


V 


^ 


^ 






















fc 
























r 




" 


■ux 


"». 




















1 10,000 


































■- 


-^ 










^ 


















































^ -^ 










"Hi 


) 






fi 
















































' 


-tj 








J5 




























































^ 
















































































1 












- 




1__ 

























10 



15 20 25 

Velocity-miles per hour 



30 



35 



10 



Fig. 208. — Tractive Power, Mikado Locomotive. 



track, or in the firing, or in the management by the engineman, 
will alter the results materially, and they should not be relied 
on to give an accurate measure of what can and will be accom- 
plished at all times. But the method is useful and dependable 
in comparing two types of engines, or, for comparing the oper- 
ating results of light trains at faster speed or heavier trains at 
slower speed, using the same engine, or, as shown later, of com- 
paring the operating results of using a certain type of engine on 
two grades and thus estimating the value of reducing the higher 
grade. 

461. Effect of Grade on Tractive Power. The effect of grade 
on tractive power is best shown by some numerical computations 
whose results are plotted in Fig. 209. The cylinder tractive 
power was computed for three engines of greatly different total 
weight and power, but which had driving-axle loads nearly 
identical (about 50750 lbs.), and, therefore, by the Baldwin 



§461. 



THE POWER OF A LOCOMOTIVE. 



509 



Locomotive Works rule, given in § 268, could all be operated on 
the same kind of track. Using the rule, J X 50750 4- 300 = 84.5, 
which means that the rails should weigh at least 85 lbs. per yard. 
Making computations for these locomotives, using 12000 B.t.u. 
coal, similar to those already detailed in §§ 453 et seq,y it was 
foimd that the cy Under tractive powers of the Pacific, Mikado, 
and Mallet locomotives were 29718, 33575, 49095 lbs., respec- 
tively, when the velocity was uniformly 10 m.p.h. and the loco- 
motives each burned 4000 lbs. of coal per hour. The several 
engine resistances at 10 m.p.h. are easily computed from Table 
XXIX and are tabulated below. 



Engine characteristics 
(At velocity F =10 m.p.h.) 


Pacific 

4-6-2 

(lb.) 


Mikrdo 

2-8-2 

(lb.) 


Mallet 

2-8-8-2 

(lb.) 


Cylinder tractive power 

Engine resistance on level 

Draw-bar pull on level 

Draw-bar pull on 3% grade. . , . 


29,718 . 

2,205 
27,513 
15,213 


33,575 

2,648 

30,927 

18,207 


49,095 

4,864 

44,231 

25.631 



The net values, or the draw-bar pulls, are plotted on the left- 
hand vertical fine of Fig. 209, and in each case are the left-hand 
ends of the sohd Hnes which show the tractive powers of the 
locomotives. On a 3% grade the grade resistances for the loco- 
motives equal 60 lbs. per ton, and are 12300, 12720 and 18600 
lbs., respectively. This reduces the effective draw-bar pull ap- 
proximately 40% in each case. Since this reduction varies 
uniformly with the grade, we may plot the three values, 15213, 
18207 and 25631, on the 3% vertical Hne and draw straight 
lines which represent in each case the tractive power of the 
locomotive at 10 m.p.h. and on any grade within that range. 

Assume trains of cars, all averaging 50 tons per car and vary- 
ing from 10 cars weighing 500 tons to 50 cars weighing 2500 tons. 
The resistances at 10 m.p.h on a level grade are given by Eq. 121, 
and may be plotted on the left-hand vertical Hne of Fig. 209. 
Grade adds resistance proportional to the grade. For example, 
on a 0.7% grade the grade resistance per ton is 14 lbs. and for 
2500 tons is 35000 lbs. Adding this to 11580, the tractive resist- 
ance, we have 46580, which we plot on the 0.7% vertical hne. 
It is indicated by a small circle. Joining the two points gives 
the resistance Hne for 2500 tons hauled at 10 m.p.h. The circles 
on the other Hnes indicate similar computations. The inter- 



510 



RAILROAD CONSTRUCTION. 



§ 461. 



sections of these resistance lines with the Hnes of tractive power 
indicate the relative power of each locomotive. For example, 
the 1000-ton train can be hauled by the Pacific locomotive at 
10 m.p.h. up a 0.96% grade, but a Mikado can do the same on a 
1.1% grade, while the Mallet can do it on a 1.52% grade. 



50,000 " HE"" ^ 


"z T : 


It i 1 




t ZL 








jf- rtftft _ /_j 2 - 


y...t. _: -J.. 


45,000 til 


1 ^ ^ 


s^ C ' _j 


I t 


- ^^ t - ^ t 




s I t 


t ^ ~ 




_ ^ : " _z 


40,000 -_--- I'^t- -r-t 


_ ^ _ _ _ __ 






ojD? 'X^s.t 


^ z 


f^fe/ ^/ |C 






z ^ 


35,000 - - :oib ^ Ti: '<§""§! 


^ _ _ _ __ 


^^r§ 'sfv,^ '^/ 


^^-;s. 


I'^i §f't/ ^j 


T^^fe ,« 


i^-T^^i^^i- 


_^^ ^, , ^^ 




'<^°/ % y 


•§ 30,000 " '^ ^sft: ■ 7 f y" 


4j2^ ^^i^ : 




^ I^ ::^(?'« 


■ g, ^, l](:^==2^ t~ ~' 


^ '^f^<^' 


^ "^-^^L I r^^^ 


r 1^/ ^v; 




HT!! — c^- 1 - 


^ 25,000 --^: "■=:^^_,"- ^^ 


^«^ ^^ 


S, Xj^ z ' fc^ ^ 


it52' Jk 


r tf ^v -, '-^^^ 


± ^2^i : 


S '^^ ^ Z V ^ 


I'tlr. />>* p 




--vV i "*>, 


fe 20,000 i-j tt t I 


y^^G^ >=*" ■ 


t 11-^ ^-, V- 


V -.=^^ ^^ 


O h^-i^ ^ y 


- ^^ 


J tt 2 ^"^ 


^■^ 






16,000 1 rlj f--i- -y-- 




1 1^1 2 u^ 




f-j^^ -> y 




</^^4 t / 








iO,000, 'Tl~^~ .^~ 




t^ L ^^ JI 




( ^-tj J 




1 ,■ ■- r r" 




_ rtrtA^t Z^' 




6,000 < 5:^ + 




' / / 




<!^ 








A -L. _ 





0.5 1.0 1.5 2.0 2.5 3.0 

Per cent of grade 

Fig. 209. — Curves Showing Effect of Grade on Tractive Power. 



All of these calculations were made on the basis of burning 
4000 lbs. of coal per hour, which, as before stated, is the prac- 
tical limit of what an ordinary fireman can be expected to do for 
an extended run. 

The description of the Mallet locomotive (built by the Bald- 
win Locomotive Works), stated that its tractive power is 91000 
lbs. A computation of its cylinder tractive power at M velocity, 
using 12000 B.t.u. coal, shows it to be 95389 lbs. Subtracting 
the engine resistance (4843 lbs.), we would have 90546 lbs., 
which is a very fair check, especially as the Baldwin Locomotive 
Works method of calculation is different. 



§462. THE POWER OF A LOCOMOTIVE. 511 

462. Acceleration-Speed curves. The time required for an 
engine of given weight and power to haul a train of known 
weight and resistance over a track with known grades and cur- 
vature is an important and necessary matter for an engineer to 
compute, since the saving in time has such a value as to justify 
constructive or operating changes which will reduce that time. 
Fig. 208 shows that the draw-bar pull is very much greater at 
very low velocities than at the moderate speed of even 15 m.p.h. 
In spite of the increased resistance at these low velocities the 
margin of power left for acceleration is also greater and the 
^' speed curve " is really a cm:ve and not a straight hne. Its 
general form may be most easily developed by a numerical 
example, especially as each case has its own special curve. 

Illustrative Example. The Mikado locomotive, whose char- 
acteristics have already been investigated in §§ 453 et seq., has 
draw-bar pulls at various velocities as shown in the tabular 
form in § 460, to which frequent reference must be made in this 
demonstration. Assume that this locomotive starts from rest on 
a 0.4% upgrade, hauling a train of 14 cars, each weighing 50 
tons, and a caboose weighing 10 tons. Then the normal level 
tractive resistance, by Eq. 121, equals 

jR = (2.2X710) + (122Xl5) =3392 lbs. 

The grade resistance of the cars will be 20X0.4X710 = 5680 lbs. 
The extra starting resistance will be considered as 6 lbs. per ton, 
or 4260 lbs. These three items total 13332 lbs. The average 
draw-bar pull of the locomotive at velocities between zero and M 
velocity, which is 6.167 m.p.h., is 4(37659+35419) =36539 lbs., 
but this must be diminished in this case by 20 X 0.4 X 157.5 = 1260 
lbs. for grade and by 157.5X6 = 945 lbs. for starting resistance, 
leaving a net draw-bar pull of 34334 lbs., excluding the force 
required for the acceleration of the locomotive. The net force 
available for acceleration of both the locomotive and the train 
is 34334-13332 = 21002 lbs., or prorated, is 21002 -^ (157.5 + 
710) =24.21 lbs. per ton. Transposing Eq. 106, with Fi = 0, 
F2 = 6.167, and P = 24.21 lbs., we have s = 70(38.03-0) ^24.21 
= 110 feet, the distance required to attain a velocity of 6.167 
m.p.h. 

While the velocity is increasing from 1.0 ikf to 1.2 M, the mean 
draw-bar pull is §(35419+32264) -1260 = 32582 lbs., less the 
accelerative resistance of the locomotive. Subtracting the 



512 RAILROAD CONSTRUCTION. § 462 

tractive and grade resistances of the cars, we have 32582—3392 
—5680 = 23510 lbs. Note that there is no longer any starting 
resistance. The accelerative force in pounds per ton is 23510 
-^ 867.5 = 27.10. The distance s required to increase the veloc- 
ity from 6.167 m.p.h. to 7.400 m.p.h., is 70(54.76-38.03)^ 
27.10 = 43 feet. Similarly the distances required to increase 
the velocity from 1.2 M to 1.5 M, from 1.5 iW to 2ikf, etc., are 
computed as in the accompanying tabular form. 

The corresponding distances and velocities have been plotted 
in Fig. 210. The velocity of 10 m.p.h. is acquired in a Httle over 
300 feet, but it requires 500 feet to acquire a velocity of 12.33 
m.p.h. and about 16000 feet to raise it to 29 m.p.h. The force, 
in pounds per ton, available for acceleration, is maximimi at low 
velocities, after the extra starting resistance is overcome. As 
the margin per ton for acceleration becomes less and less, the 
greater is the distance required to increase the velocity 1 mile 
per hour — especially through the last increments — up to the 
velocity at which the net draw-bar pull exactly equals the total 
car resistance and the velocity becomes uniform, which is later 
computed to be 4.78 M, There is an approximation in using 
average draw-bar pulls between the different velocities at which 
the draw-bar pull has been definitely computed, but the com- 
puted distances are practically correct up to 4 M velocity or 
24.67 m.p.h. But the computation for the distance required to 
increase the velocity from 4 M up to 4.78 M is far less accurate if 
the average draw-bar pull is used. The effective pull at 4 M 
velocity equals 13000 — 1260 = 11740, less the accelerative resist- 
ance of the locomotive. The tractive and grade resistance of 
the cars at this velocity is 3392+5680 = 9072. This leaves 
11740 — 9072 = 2668 lbs. available for acceleration of both loco- 
motive and cars. The reduction in tractive force between 4 M 
velocity and 5 M velocity (see § 460), is 13000-9571 =3429 lbs. 
By proportionate interpolation we would then say that the 
excess force available for acceleration would be exhausted at 
(2668 H- 3429) = .78 of the interval, or at a velocity of 4.78 M, 
or 29.48 m.p.h. The mean accelerative force is one-haK of 2668, 
or 1334 lbs., which is 1.53 lbs. per ton of train. The dis- 
tance, by an inversion of Eq. 106, is computed to be 11925 feet. 
Owing to the approximate equality of working force and resist- 
ance and the momentary variations in both, the precise point 
where the acceleration would cease and the velocity would 



§462. 



THE POWER OF A LOCOMOTIVE, 



513 



i 


• 


rt< Tt4 00 CO 00 i-H O 

^CO 


<M00ri<CO 

COOuOOi 

rH 


tn 

o 

a 

S 


Total 
from 
starti 

feet. 


OCOCOOtJ^OiO 
1-H lO Tj^ O 05 05 tH 
tH ,-( (M lO lO 1> 1> 

I— 1 


(Nrt<i-i<M 
COOit>iO 
(NO»0(M 
1-* CO CO 00 


Accel- 
eration, 
or re- 
tarda- 
tion, 

feet 


O CO CO T}H Tt^ CO lO 

T-H T}< a> lo 05 05 (N 

tH (NOt-<05 

i-HCOi-H 

r-l 


<N(NI>T~I 

CO CO 1> 00 
(NOO-^CO 
,-1 r-l CO 1-1 


o 
o 

1 

> 

g 


Net 

force 

per 

ton, 

lbs. 


t-HOOrf |>COCO 
(N i-H r-( CO rH 00 lO 


COI>C0(M 

Tt^l-HOOl-l 


rJH l> CO 00 <M lO rH 
(M(N(Nt-1tH 


TfHOCOO 


Differ- 
ence ef- 
fective 
for ac- 
celera- 
tion or 
retarda- 
tion, 
lbs. 


(M O 05 t> O 05 -^ 

O 1-1 CO O lO lO CO 
010005LOOCO 

rH CO O lO O lO 1-1 
(M(M(Mr-lTH 


lO 00 CO r-l 
(MOO CO 
1-1 


Car re- 
sistance 
tractive 

grade, 
plus 

start* 

lbs. 


(N C^ (M (M (M (M (M 

CO 05 05 05 Oi 05 05 

1— ( 

* 


20432 
20432 
20432 
20432 


Actual 
draw- 
bar pull, 
average, 

lbs. 


Tt< (N !-( 05 ^ i-H CO 

CO 00 r-l !>. CO CO O 
CO lO rH 05 CD 1-1 Tfl 

Tt* (N O TtH 05 Tt( O 

CO CO <M CSI T-< r-l ,-1 


7882 
11611 
17111 
20326 


Loco- 
motive 
resist- 
ance, 
grade 
plus 
start* 

lbs. 


lOOOOOOO 
O CO CO CO CD CO CO 
(M (N (M <N (N (M (M 

C<1 tH tH i-< rH ,-H J— 1 

* 


oooo 

00 00 00 00 
CO CO CO CO 


Mean 
draw- 
bar pull, 
level, 

lbs. 


C5 C<J tH 05 1-^ i-' CO 

CO "^ t^ CO 05 o CO 

lO 00 CO (N 00 CO CO 
CO CO O CO O "^ T-i 
CO CO CO (N (N 1-1 tH 


11662 
15391 
20891 
24106 


1 

•s 


Mean, 

feet per 

sec. 


(N lO (M CO T-i CO tH 

iO 05 (M 00 CO CO 1> 


i-HCOrHOi 

t>COCOOi 


rJH 05 (N »0 (N 1-1 05 
r-l 1-1 (N COCO 


Oii-i<Nl> 
COC0(NtH 


. o 

«3 ft 

1 


coo»ocoot^oo 

r-< Tj< (N CO lO CO ^ 


i>ocoi-i 

COiOCO(N 


CO 1> 05 <N 00 rt( C35 
r-lr-<(M(N 


TtHOOlMlN 
<NtHt-It-I 


OCOO>OCOOI> 
O 1-1 '^ <M CO lO CO 


OOt^OCO 

rtCOlOCO 


O CO t^ 05 (M 00 Tt< 
rH,-l(M 


Oi-^OOtM 

(M(Nr-lTH 


6 
o 

ft 


O TjH CO l> Oi CO 00 
O O 00 iO O i-t i-H 


Tt<00C0O5 
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rHrHi-KNCO 


CO CO l> 00 
Tj<CO(Mi-H 






.2 

*-+3 

1; 
c:> 
o 


.2 



514 



RAILROAD CONSTRUCTION. 



§463. 



actually become uniform would be be very uncertain. For- 
tunately the inaccuracy is of little or no practical importance 
and for the purposes of our calculations we may call this last 
interval 11925 feet, assuming that the grade is as long as 16715 
feet or 3.1 miles. If the 0.4% grade continued indefinitely the 
train would travel at this uniform velocity as long as the loco- 
motive operated on the basis assumed for this problem. Note 
that Fig. 210 would have to be extended to nearly three times its 













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^VEL0CITy^\2 


9i^48M.P.H. ON'0.4 


"ji GRADE^. 




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E OF UNI 


FORM VE 


L0C1TY( 12.2 


V M.P.H.) 0.N.1.2fl{) 


GBAOE " 


*- in 


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J.000 2000 3000 4000 

DISTANCES IN FEET 

Fig. 210. 



^5000 



6000 



present length before the time curve would reach and become 
tangent to the ^' line of uniform velocity." 

463. Retardation-speed curves. When, on account of grade 
resistance, the total of tractive and grade resistance is greater 
than the draw-bar pull, there is retardation. 

Illustrative Example. Continuing the numerical problem of 
§ 462, assume that, while moving up the 0.4% grade at a velocity 
of 4.78 M, or 29.48 m.p.h., the train reaches a grade of -f-1.2%. 
The grade resistance of the cars will be 20X1.2X710 = 17040 
lbs. The tractive resistance will be 3392 lbs., as before, making 
a total of 20432 lbs. Interpolating in the tabular form in § 460 
for the draw-bar pull at 4.78 M velocity, we find 10325; at 4 ikf 
it is 13000 and the mean is 11662; but from this must be sub- 
tracted 20X1.2X157.5 = 3780 for grade resistance of the loco- 
motive, leaving 7882 lbs. for the net draw-bar pull. The retard- 
ing force is 20432-7882 = 12550; or in pounds per ton of train, 
is 12,550 -^867.5 = 14.46. As before, using an inversion of 



§464. THE POWER OF A LOCOMOTIVE. 515 

Eq. 106, s = (29.482 -24.672)70 ^14.46 = 1262 feet, the distance 
at which the velocity would reduce to 4 M. As before, the other 
quantities may be computed and recorded, with less danger of 
confusion and error, by tabulating them, as given in § 462. 

The mean velocity, when retarding from 4.78 M to 4.0 M, 
reduced to feet per second, is as before 39.71 feet per second, and 
dividing this into the distance, 1262 feet, gives 32, the time in 
seconds. The quantities for the reduction in velocity from 
4 M to 3 M and from SM to2M are computed similarly. The 
level draw-bar pull for 1.5 M is 28478 (see § 460), and by sub- 
tracting 3780, we get 24698 lbs. the actual net pull on the grade. 
Similarly, the actual pull at 2 ikf is 20219 lbs. The increase from 

213 
20219 to 20432 is -—- = 4.7% of the interval from 20219 to 
4479 

24698 and 4.7% X. 5 = .02; therefore, the actual draw-bar pull 

just equals the resistance at 2.00 — .02 = 1. 98ikf, or 12.21 m.p.h. 

The deficiency of draw-bar puU at 2.0 M = 20,432 - 20219 = 213 

lbs. At 1.98 ikf the deficiency is zero and, therefore, the mean 

deficiency is one-haK of 213, or 106. Dividing this by 867.5, 

we have 0.122, which is the value of P in Eq. 106. Then 

s = (152.01 -149.08)70-^0.122 = 1681 ft. 

Velocities in miles per hour can be readily converted into 
velocities in feet per second by multiplying by 1.4667. Averag- 
ing the two velocities at the beginning and the end of each period 
gives the mean velocity; and dividing each of these into the 
distance for that period gives the time in seconds. 

464. Drifting. The tractive resistance of the cars of the 
problem just worked out is 3392 lbs. ; the locomotive resistance 
at 20 m.p.h. is 1862 lbs., or a total of 5254 lbs. Variation in 
velocity will affect this but Httle. Dividing by 867.5, the total 
weight in tons, we have 6.06 lbs., the resistance per ton, from 
which the equivalent rate of grade is 6.06 -i- 20 = .303%. This 
means practically that when this train is running down a grade 
which is over .303% it will run by gravity and steam may be 
shut off. If the grade is much greater than .303% the accelera- 
tion on the downgrade may become so great, if the grade is very 
long, that the velocity may become objectionably high. 

Illustrative Example. Assume that the hmiting safe velocity 
for freight trains, considering the condition of track and roUing 



516 RAILROAD CONSTRUCTION. § 464. 

stock, is 35 m.p.h.; assume that the train we have been consider- 
reaches a 0.4% downgrade at a velocity of 15 m.p.h. How far 
down the grade will it run with steam shut off, before the speed 
reaches 35 m.p.h. and brakes must be applied?- There is no 
question here of variable tractive power since the only motive 
power is gravity. The resistance is nearly independent of 
velocity and we will here assume it to be so and utilize Table 
XLII. At 15 m.p.h. the train has a velocity head of 7.90 feet. 
At 35 m.p.h. the velocity head is 43.01 feet. The train can, 
therefore, drop down the grade a vertical height of 43.01 — 7.90 
= 35.11 feet before the velocity reaches 35 m.p.h. On a 0.4% 
grade the distance required for such a fall is 35.1 l-r- .004 = 8777 
feet. The problem in § 462 assumed that the 0.4% grade is 
16715 feet or more, and this shows what will happen to the 
trains moving in the opposite direction. 

But it must not be thought that there is no loss of energy 
during drifting. Even though no steam is used in the cylinders, 
some is frequently wasted at the safety valve and more is used 
in operating brakes and in maintaining the brake air-reservoir 
at full pressure. But the greatest loss of heat is that due to 
radiation, especially in winter, in spite of all the jacketing devices 
to retain heat. Although the results of the numerous tests 
which have been made are quite variable, the following approxi- 
mate averages may be used: The loss due to radiation while 
standing may be figured at 120 lbs. of coal per hour per 1000 
square feet of heating surface; while drifting the loss will in- 
crease to 220 lbs. per hour. The amount of coal used for firing 
up will be about 510. This is based on the use of 12000 B.t.u. 
coal. The better the coal, the less will be used. 

Illustrative Example. The Mikado locomotive we have been 
considering has 2565 square feet of heating surface. It will then 
require about 2.565X510 = 1308 lbs. of coal to fire up. While 
drifting down the grade, referred to above, a distance of 8777 feet, 
the average velocity is J(15+35)=25 m.p.h. =36.67 ft. per sec. 
and the required time is 8777-^36.67 = 239 seconds = 3 min. 59 
sec. =.066 hour. The coal used while drifting down this short 
run would be 

220X2.565X.066 = 37 lbs. 

At this point brakes would need to be applied and the time 
spent in drifting beyond this point must be computed as an item 



§465. THE POWER OF A LOCOMOTIVE. 517 

in the total time spent on the run and also to compute the total 
amount of coal consumed while drifting. Although this item 
of 37 lbs. is relatively very small, its method of computation is 
typical of the computation of the several items to make up the 
total of coal consumed during a trip. 

465. Review of computed power of one locomotive. It was 
assumed that it started on a +0.4% grade with a load of 15 cars 
weighing 710 tons. After moving 16715 feet (assuming that 
the grade was that long), and doing it in 493 seconds, or 8 min- 
utes 13 seconds, the train acquired a velocity of 29.48 m.p.h. 
and the power of the locomotive would then be sufficient, when 
burning 4000 lbs. of coal per hour, to keep it moving up such a 
grade indefinitely at that velocity. In case the grade were not 
as long as 16715 feet, it would be necessary to compute the 
velocity where the rate of grade changed and make that the 
basis for the computation on the succeeding grade. But, 
assuming that the grade were as long as 16715 feet, or more, and 
that the velocity of 29.48 m.p.h. had been acquired, and that the 
train had rim at that speed for some distance — although this 
does not modify the problem — the train is assumed to reach a 
still steeper grade +1.2%. The velocity then begins to decrease 
and in a total distance of 8252 feet and a total time of 337 sec- 
onds, or 5 minutes 37 seconds, the velocity is reduced to 
12.21 m.p.h., at which velocity the locomotive is able to make 
steam fast enough to overcome the higher resistance on the 
steeper grade. From that point on, assuming that the 1.2% 
grade is longer than 8252 feet, the train would continue for 
the remaining length of that grade at the velocity of 12.21 
m.p.h. 

As before stated, precision in the above results depends on 
many factors (such as B.t.u. of coal used, or the actual consump- 
tion in pounds per hour), which are somewhat variable. Some- 
times the variation of these factors from the values used above is 
known; sometimes it is unknown and then the accuracy of the 
results is correspondingly uncertain. But whether accurately 
known or not, when this method is used, employing the best 
values for the factors which are obtainable, the method shows 
a valuable comparison of two proposed alinements or grades. 
In such a comparison, any error in the factors will affect both 
results nearly, if not quite, equally, and the cf^mparative results 
wiU still be substantially correct. 



518 RAILROAD CONSTRUCTION. § 466. 

466. Selection of route. The preceding articles may be 
utilized in comparing two routes. If one of the lines is already 
in operation, the engineer has the great advantage of being able 
to determine by test exactly what results may be obtained on 
that line and what factors should be used in computations. 

It is then only necessary to compute the quantities for the 
proposed new line. When both lines are *^ on paper " there is 
less certainty as to the accuracy of the results, except that the 
line which is shown to be most advantageous will probably con- 
tinue to be most advantageous even if the uncertain factors used 
in the comparison are somewhat changed. Using the methods 
outlined in §§ 462 to 464, there will be computed the behavior of 
an assumed type of locomotive, hauling one or more types of 
train load, and passing over tracks having definite grades and 
lengths. The effect of curves may be disregarded provided that 
the grades were properly compensated during original con- 
struction, and then the rate of grade for the entire length of 
straight and curved track may be taken as the rate on the 
straight track. If the rate of grade is actually uniform, even 
through the curves, then the lengths of curved track must 
be computed separately and on the basis of a rate of grade 
equal to the actual rate plus an allowance of .035% for each 
degree of curve. The behavior of a train from starting to 
stopping must be computed, making due allowance for each 
change in condition which will affect the hauling power of the 
locomotive. The locomotive is assumed to be working at the 
limit of its steaming capacity, except when drifting with steam 
shut off on a down grade, or when brakes are applied, either to 
prevent objectionably high velocity on a down grade or to make 
a stop. The action of brakes during a service stop (as distin- 
guished from an emergency stop), may be considered as a retard- 
ing force varying from 10% to 20% of the train weight. Un- 
fortunately brake action is so variable, being directly under the 
control of the locomotive engineer and varying from zero to 
the full braking power, that any computation of energy used in 
operating them or of the effect of the brakes is impracticable 
except on the basis of arbitrary assumptions such as the require- 
ment that the brakes are used in such a way that a train will be 
retarded at a specified rate. The performance of the locomotive 
over the entire division, the total time required, its velocity in 
critical places, etc., can be computed. In §§ 462 and 463 it 



§ 467 THE POWER OF A LOCOMOTIVE. 519 

was shown that the locomotive considered could haul the par- 
ticular train considered up a 0.4% grade at a velocity of 29.48 
m.p.h. and maintain such speed indefinitely; also that it could 
haul the same train up a 1.2% grade at 12.21 m.p.h. and main- 
tain its velocity indefinitely. This of course,, means, that a 
much heavier train could be hauled up the 0.4% grade and that a 
somewhat heavier train could be hauled up the 1.2% grade with- 
out being stalled, although the velocities in each case would be 
reduced. There are an infinite number of combinations, but 
there are usually some considerations which narrow the choice. 
Even after construction is complete these tables may be utiHzed 
in a study of the most economical combination of type of loco- 
motive and amount of train load for the track conditions as 
they may exist. 

467. Rating of locomotives. The maximum power of a loco- 
motive on any grade at M velocity is measured by its " rating.'' 

Let P = the tractive power of the locomotive, measured at 
the rim of the drivers; 
£7 = Weight of' engine and tender, in pounds; 
TF = Weight of cars behind tender, in pounds; 
r=rate of grade, or the ratio of vertical to horizontal; 
a = a constant, which as determined by tests = 2.2 lbs. 

per ton or .0011 lb. per pound of train; 
6 = a constant, which as determined by tests = 122 lbs. 
per ton. a and b are the same constants as are uspd 
in § 439. 
n = number of cars in train. 
Then P = (E-}-W) {r+a)+bn. 
Transforming, 

P h 

■E^^W+n (122) 



r-j-a r+a 

The right-hand side of this equation is called the ^'rating," A, 
and is the weight of the train behind the tender plus the number 
of cars times a quantity made up of two constants and the rate 
of grade. This quantity is independent of any special engine or 
train values and may be tabulated for various rates of grade, 
as given in Table XL. 

Examples. The Mikado locomotive considered in §§ 453, 
et seq., has a tractive power, measured at the rim of the drivers. 



520 



RAILROAD CONSTRUCTION. 



§467 



TABLE XL. LOCOMOTIVE RATING DISCOUNTS. 

VALUES OF C-T-(R+K) FOR VARIOUS GRADES, 
(la tons per car) 



? 




? 




nt) 


ij 


a 






B3|y( 


c^8 

11 




f^8 

la 


Is ! 




Is 




is 




as 

go 


O 


H 


o 


Eh 


O 


H 


O 


H 


o 


H 


Level 


55 


0.5 


10.0 1 


1.0 


5.5 


1.5 


3.8 


2.0 


2.88 


0.1 


29 


0.6 


8.5 


1.1 


5.0 


1.6 


3.6 


2.1 


2.75 


0.2 


20 


0.7 


7.5 


: 1.2 


4.6 


1.7 


3.4 


2.2 


2.63 


0.3 


14 


0.8 


6.7 


1.3 


4.3 


1.8 


3.2 


2.3 


2.52 


0.4 


12 


0.9 

1 


6.0 

i 


1.4 


4.0 


1.9 


3.0 


2.4 


2.42 



at M velocity, or 6.167 m.p.h., of 37190-1432 = 35758 lbs., 
which equals P; 1432 is the locomotive resistance between 
cylinder and rim of drivers, see § 429. The weight of engine 
and tender is 315000 lbs. What is its rating on a 1.2% grade? 
The value of r f or a 1.2% grade = .012; a = .0011 lb. per pound. 
Then 

315000 = 2,414,000 lbs. = 1207 tons, 



A = 



E 



r+a .012H-.0011 

which is the rating for that locomotive for Si 1.2% grade. But 
this does not mean 1207 tons of cars. Placing this equal to the 
right-hand side of Eq. 122, we have 

h 



1207 = W+n 



r-\-a 



The value of 
Then 



r+a 



for a 1.2% grade is given in Table XL as 4.6. 
TF = 1207-4.6n, 



which shows that the w^eight of train depends on the number of 
cars. Assume that n = 16. Then W = 1133.4 and the average 
weight per car is 70.8 tons. Assume that the cars are all 
" empties," weighing 18 tons each; then W = 18n, and 

71 = 1207-^(18+4.6) =53.4, 

which must be interpreted as 53 empty cars. 

In the above examples the pulling power P is determined on the 
basis of the locomotive working at the maximum velocity M at 



§467. THE POWER OF A LOCOMOTIVE. 521 

which it can maintain full stroke. See § 455. This represents 
practically the maximum power of the locomotive. The velocity 
M is usually from 4 to 7 miles per hour and is as low as should be 
allowed on maximum grades, since an attempt to utilize a shghtly 
higher tractive force at a somewhat lower velocity would prob- 
ably result in stalling the train if an imexpected resistance in 
the track slightly increased the normal resistance. 



- CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

468. Method of formation of railroad corporations. Many 
business enterprises, especially the smaller ones, are financed 
entirely by the use of money which is put into them directly 
in the form of stock or mere partnership interest. A railroad 
enterprise is frequently floated with a comparatively small 
financial expenditure on the part of the original promoters. 
The promoters become convinced that a railroad between A 
and B, passing through the intermediate towns of C and D, 
with others of less importance, will be a paying investment. 
They organize a company, have surveys made, obtain a charter, 
and then, being still better able (on account of the additional 
information obtained) to exploit the financial advantages of 
their scheme, they issue a prospectus and invite subscriptions 
to bonds. Sometimes a portion of these bonds are guaranteed, 
principal and interest, or perhaps the principal alone, by town- 
ships or by the national government. The cost of this pre- 
liminary work, although large in gross amount if the road is 
extensive, is yet but an insignificant proportion of the total 
amount involved. The proportionate amount that can be 
raised by means of bonds varies with the circumstances. In 
the early history of railroad building, when a road was pro- 
jected into a new country where the traffic possibilities were 
great and there was absolutely no competition, the financial 
success of the enterprise would seem so assured that no diffi- 
culty would be experienced in raising from the sale of bonds 
aU the money necessary to construct and equip the road. But 
the promoters (or stockholders) must furnish all money for the 
preliminary expenses, and must make up all deficiencies be- 
tween the proceeds of the sale of the bonds and the capital needed 
for construction. 

'^In theory, stocks represent the property of the responsible 
owners of the road, and bonds are an encumbrance on that 

522 



§ 469. PROMOTION OF RAILROAD PROJECTS. 523 

property. According to this theory, a railroad enterprise 
should begin with an issue of stock somewhere near the value 
of the property to be created and no more bonds should be 
issued than are absolutely necessary to complete the enter- 
prise. Now it is not denied that there are instances in which 
this theory is followed out. In New England, for example, 
as well as in some of the. Southern States, there are a few roads 
represented wholly by stock or very lightly mortgaged. But 
this theory does not^ conform to the general history of railway 
construction in the United States, nor is it supported by the 
figures that appear in the summary. The truth is, railroads 
are built on borrowed capital, and the amount of stock that is 
issued represents in the majority of cases the difference between 
the actual cost of the undertaking and the confidence of the 
public expressed by the amount of bonds it is willing to absorb 
in the ultimate success of the venture." * 

'^The same general law obtains and has always obtained 
throughout the world, that such properties (as railways) are 
always built on borrowed money up to the limit of what is 
regarded as the positive and certain minimum value. The 
risk only — the dubious margin which is dependent upon sagac- 
ity, skill, and good management — is assumed and held by the 
company proper who control and manage the property." f 

469. The two classes of financial interests — the security and 
profits of each. From the above it may be seen that stocks, 
bonds, car-trust obhgations, and even current liabilities repre- 
sent railroad capital. The issue of the bonds ^'was one means 
of collecting the capital necessary to create the property against 
which the mortgage lies." The variation between these inter- 
ests lies chiefly in the security and profits of each. The current 
liabilities are either discharged or, as frequently happens, they 
accumulate until they are funded and thus become a definite 
part of the railroad capital. 

The growth of this tendency is shown in the following tabular 
form (see next page) : 

The bonded interest has greater security than the stock, but 
less profit. The interest on the bonds must be paid before any 
money can be disbursed as dividends. If the bond interest 



* Henry C. Adamsj Statistician, U. S. Int. Con. Commission, 
t A. M. Wellington, Economic Theory of Railway Location 



524 



RAILROAD CONSTRtrCTION. 



§469. 



Capitalization of 


June 30, 1888. 


June 30. 1898. 


June 30, 1912. 


Railroads in the United 

States. 


Amount, 
millions. 


Per 
^ cent. 


Amount, 
millions. 


Per 
cent. 


Amount, 
millions. 


Per 

cent. 


Stocks 


3864 

3869 

396 


47.5 

47.6 

4.9 


5311 
5510 
1087 


44.6 
46. 3\ 
9.1/ 


8622 
11130 


43 7 


Funded debt 




Current liabilities, etc . . 


56.3 



is not paid, a receivership, and perhaps a foreclosure and sale 
of the road, is a probability, and in such case the stockholder's 
interests are frequently wiped out altogether. The bond- 
holder's real profit is frequently very different from his nomi- 
nal profit. He sometimes buys the bonds at a very considerable 
discount, which modifies the rate which the interest received 
bears to the amount really invested. Even the bondholder's 
security may suffer if his mortgage is a second (or fifth) mort- 
gage, and the foreclosure sale fails to net sufficient to satisfy 
all previous claims. 

On the other hand, the stockholder, who may have paid in 
but a small proportion of his subscription, mayy if the venture 
is successful, receive a dividend which equals 50 or 100% of the 
money actually paid in, or, as before stated, his entire holdings 
may be entirely wiped out by a foreclosure sale. When the 
road is a great success and the dividends very large, additional 
issues of stock are generally made, which are distributed to the 
stockholders in proportion to their holdings, either gratuitously 
or at rates which give the stockholders a large advantage over 
outsiders. This is the process known as '^watering." While 
it may sometimes be considered as a legitimate ^'salting down" 
of profits, it is frequently a cover for dishonest manipulation of 
the money market. 

For the twelve years between 1887 and 1899 about two thirds 
of all the railroad stock in the United States paid no dividends, 
while of those that paid dividends the average rate varied 
from 4.96 to 5.74%. The year from June 30, 1898, to June 30, 
1899, was the most prosperous year of the group, and yet nearly 
60% of all railroad stock paid no dividend, and the average 
rate paid by those which paid at all was 4.96%. The total 
amount distributed in dividends was greater than ever before, 
but the average rate is the least of the above group because many 
roads, which had passed their dividends for many previous 



§ 470. PROMOTION OF RAILROAD PROJECTS. 525 

years, distinguished themselves by declaring a dividend, even 
though small. During that same period but 13.35% of the 
stock paid over 6% interest. The total dividends paid amounted 
to but 2.01% of all the capital stock, while investments ordi- 
narily are expected to yield from 4 to 6% (or more) according 
to the risk. Of v course the effect of ^^ watering" stock is to 
decrease the nominal rate of dividends, but there is no dodging 
the fact that, watered or not, even in that year of ^^good times,'' 
about 60% of all the stock paid no dividends. Unfortunately 
there are no accurate statistics showing how much of the stock 
of railroads represents actual paid-in capital and how much 
is ''water." The great complication of railroad finances and 
the dishonest manipulation to which the finances of some rail- 
roads have been subjected would render such a computation 
practically worthless and hopelessly unreliable now. 

During the year ending June 30, 1898 (which may in general 
be considered as a sample), 15.82% of the funded debt paid no 
interest. About one third of the funded debt paid between 
4 and 5% interest, which is about the average which is paid. 

The income from railroads (both interest on bonds and divi- 
dends on stock) may be shown graphically by diagrams, such 
as are given in the annual reports of the Interstate Commerce 
Commission. They show that while railroad investments are 
occasionally very profitable, the average return is less than 
that of ordinary investments to the investors. The indirect 
value of railroads in building up a section of country is almost 
incalculable and is worth many times the cost of the roads. 
It is a discouraging fact that very few railroads (old enough to 
have a history) have escaped the experience of a receivership, 
with the usual financial loss to the then stockholders. But 
there is probably not a railroad in existence which, however 
much a financial failure in itself, has not profited the community 
more than its cost. 

470. The small margin between profit and loss to projectors. 
When a railroad is built entirely from the funds furnished by 
its promoters (or from the sale of stock) it will generally be a 
paying investment, although the rate of payment may be very 
small. The percentage of receipts that is demanded for actual 
operating expenses is usually about 67%. The remainder will 
usually pay a reasonable interest on the total capital involved. 
But the operating expenses are frequently 90 and even 100% of 



526 RAILROAD CONSTRUCTION. § 471. 

the gross receipts. In such cases even the bondholders do not 
get their due and the stockholders have absolutely nothing. 
Therefore the stockholder's interest is very speculative. A 
comparatively small change in the business done (as is illus- 
trated numerically in §472) will not only wipe out altogether the 
dividend — taken from the last small percentage of the total 
receipts and which may equal 50% or more of the capital stock 
actually 'paid in — but it may even endanger the bondholders' 
security and cause them to foreclose their mortgage. In such 
a case the stockholders' interest is usually entirely lost. It 
does not alter the essential character of the above-stated rela- 
tions that the stockholders sometimes protect themselves 
somewhat by buying bonds. By so doing they simply decrease 
their risk and also decrease the possible profit that might result 
from the investment of a given total amount of capital. 

471. Extent to which a railroad is a monopoly. It is a popu- 
lar fallacy that a railroad, when not subject to the direct com- 
petition of another road, has an absolute monopoly — that it 
controls ^^all the traffic there is" and that its income will be 
practically independent of the facilities afforded to the public. 
The growth of railroad traffic, like the use of the so-called 
necessities or luxuries of life, depends entirely on the supply 
and the cost (in money or effort) to obtain it. A large part of 
railroad traffic belongs to the unnecessary class — such as travel- 
ing for pleasure. Such traffic is very largely affected by mere 
matters of convenience, such as well-built stations, convenient 
terminals, smooth track, etc. The freight traffic is~ very largely 
dependent on the possibility of delivering manufactured articles 
or produce at the markets so that the total cost of production 
and transportation shall not exceed the total cost in that 
same market of similar articles obtained elsewhere. The crea- 
tion of facilities so that a factory or mine may successfully 
compete with other factories or mines will develop such traffic. 
The receipts from such a traffic may render it possible to still 
further develop facilities which will in return encourage further 
business. On the other hand, even the partial withdrawal of 
such facilities may render it impossible for the factory or mine 
to compete successfully with rivals; the traffic furnished by 
them is completely cut off and the railroad (and indirectly the 
whole community) suffers correspondingly. The ^^ strictly 
necessary" traffic is thus so small that few railroads could pay 



§472. 



PROMOTION OF RAILROAD PROJECTS. 



527 



their operating" expenses from it. The dividends of a road 
come from the last comparatively small percentage of its revenue, 
and such revenue comes from the '' unnecessary" traffic which 
must be coaxed and which is so easily affected by apparently 
insignificant ' ^ conveniences . ' ^ 

472. Profit resulting from an increase in business done; loss 
resulting from a decrease. In a subsequent chapter it will 
be shown that a large portion of the operating expenses are 
independent of small fluctuations in the business done and that 
the operating expenses are roughly two thirds of the gross 
revenue. Assume that by changes in the alinement the business 
obtained has been increased (or diminished) 10%. Assume for 
simplicity that the operating expenses on the revised track 
are the same as on the route originally planned; also that the 
cost of the track is the same and hence the fixed charges are 
assumed to be constant for all the cases considered. Assume 
the fixed charges to be 28%. The additional business, when 
carried in cars otherwise but partly filled will hardly increase 
the operating expenses by a measurable amount. When 
extra cars or extra trains are required, the cost mil increase 
up to about 60% of the average cost per train mile. We may 
say that 10% increase may in general be carried at a rate of 
40% of the average cost of the traffic. A reduction of 10% 
in traffic may be assumed to reduce expenses a similar amount. 
The effect of the change in business will therefore be as follows : 





Business increased 10%. 


Business decreased 10%. 


Operating exp. = 67 
Fixed charges =28 


67(1 + 10%X40%)= 69.68 
28.00 


67(1-10%X40%) = 


= 64.32 
.28.00 




Income 

Deficit 




95 
Total income. . . 100 


97.68 
Income 110.00 


92.32 

. 90.00 


Available for divi- 
dends 5 


Available for divi- 
dends 12.32 


. 2.32 



In the one case the increase in business, which may often 
be obtained by judicious changes in the alinement or even by 
better management without changing the alinement, more than 
doubles the amount available for dividends. In the other case 
the profits are gone, and -there is an absolute deficit. The 
above is a numerical illustration of the argument, previously 



528 RAILROAD CONSTRUCTION. § 473. 

stated, of the small margin between profit and loss to the original 
projectors. 

473. Estimation of probable volume of traffic and of probable 
growth. Since traffic and traffic facilities are mutually inter- 
dependent and since a large part of the normal traffic is merely 
potential until the road is built, it follows that the traffic of a 
road will not attain its normal volume until a considerable 
time after it is opened for operation. But the estimation even 
of this normal volume is a very uncertain problem. The esti- 
mate may be approached in three ways: 

1st. The actual gross revenue derived by all the railroads 
in that section of the country (as determined by State or U. S. 
Gov. reports) may be divided by the total population of the 
section and thus the average annual expenditure per head of 
population may be determined. A determination of this value 
for each one of a series of years will give an idea of the normal 
rate of growth of the traffic. Multiplying this annual contri- 
bution by the population which may be considered as tributary 
gives a valuation of the possible traffic. Such an estimate is 
unreliable (a) because the average annual contribution may not 
fit that particular locality, (6) because it is very difficult to 
correctly estimate the number of the true tributary population 
especially when other railroads encroach more or less into the 
territory. Since a rough value of this sort may be readily 
determined, it has its value as a check, if for nothing else. 

2d. The actual revenue obtained by some road whose 
circumstances are as nearly as possible identical with the road 
to be considered may be computed. The weak point consists 
in the assumption that the character of the two roads is identical 
or in incorrectly estimating the allowance to be made for ob- 
served differences. The method of course has its value as a 
check. 

3d. A laborious calculation may be made from an actual 
study of the route — determining the possible output of all 
factories, mines, etc., the amount of farm produce and of lumber 
that might be shipped, with an estimate of probable passenger 
traffic based on that of like towns similarly situated. This 
method is the best when it is properly done, but there is always 
the danger of leaving out sources of income — both existent 
and that to be developed by traffic facilities, or, on the other 
handj of overestimating the value of expected traffic. In the 



§473. 



PROMOTION OF RAILROAD PROJECTS. 



529 



following tabular form are shown the population, gross re- 
ceipts, receipts per head of population, mileage, earnings per 
mile of line operated, and mileage per 10,000 of population for 
the whole United States. It should be noted that the values 
are only averages, that individual variations are large, and that 
only a very rough dependence may be placed on them as applied 
to any particular case. 



Year. 


Population 
(estimated). 


Gross 
receipts. 


Receipts 
per head 
of popu- 
lation. 


Mileagef 


Earnings 

per mile 

of line 

operated. 


Mileage 
per 
10,000 

popula- 
tion. J 


1888... 
1889... 
1890... 
1891... 


60,100,000 

61,450,000 

*62,801,571 

64,150,000 


$910,621,220 

964,816,129 

1051,877,632 

1096,761,395 


$15.15 
15.81 
16.75 
17.10 


136,884 
153,385 
156,404 
161,275 


56653 
6290 
6725 
6801 


24.94 
25.67 
26.05 
26.28 


1892... 
1893... 
1894... 
1895... 
1896... 


65,500,000 
68,850,000 
68,200,000 
69,550,000 
70,900,000 


1171,407,343 
1220,751,874 
1073,361,797 
1075,371,462 
1150,169,376 


17.89 
18.26 
15.74 
15.46 
16.22 


162,397 
169,780 
175,691 
177,746 
181,983 


7213 
7190 
6109 
6050 
6320 


26.19 
26.40 
26.20 
25.97 

25.78 


1897... 
1898... 
1899... 
1900... 
1901. .. 


72,350,000 
73,600,000 
74,950,000 
*76,295,220 
77,863,000 


1122,089,773 
1247,325,621 
1313,610,118 
1487,044,814 
1588,526,037 


15.53 
16.95 
17.53 
19.49 
20.47 


183,284 
184,648 
187,535 
192,556 
195,562 


6122 
6755 
7005 
7722 
8123 


25.53 
25.32 
25.25 
25.44 
25.52 


1902... 
1903... 
1904... 
1905... 
1906... 


79,431,000 
80,998,000 
82,566,000 
84,134,000 
85,701,000 


1726,380,267 
1900,846,907 
1975,174,091 
2082,482,406 
2325,765,167 


21.88 
23.70 
24.23 
25.15 
27.65 


200,155 
205,314 
212,243 
216,974 
222,340 


8625 
9258 
9306 
9508 
10460 


25.76 
26.03 
26.34 
26.44 
26.78 


1907... 
1908... 
1909... 
1910... 
1911... 
1912... 


87,279,000 
88,837,000 
90,405,000 
*91,972,266 
93,572,266 
95,172,266 


2589,105,578 
2393,805,989 
2418,677,538 
2750,667,435 
2789,761,669 
2842,695,382 


29.63 
26.95 
26.71 
29.91 
29.81 
29.87 


227,455 
231,540 
234,800 
238,609 
244,476 
247,981 


11383 

10338 

10301 

11528 

11411 

11463 


26.38 
26.30 
26.20 
26.14 
26.10 
25.93 



♦Actual. t Excludes a small percentage not reporting "gross receipts.' 

$ Actual mileage. 



The probable growth in traffic, after the traffic has once 
attained its normal volume, is a small but almost certain quantity. 
In the above tabular form this is indicated by the gradual 
growth in ''receipts per head of population'^ from 1897 to 
1907. Then the sudden drop due to the panic of 1907 is clearly 
indicated, and also the gradual growth in the last few years. 
Even in England, where the population has been nearly station- 
ary for many years, the growth though small is unmistakable. 
On the other hand the growth in some of the Western States 



530 RAILROAD CONSTRUCTION. §474 

has been very large. For example, the gross earnings per head 
of population in the State of Iowa increased from $1.42 in 1862 
to $10.00 in 1870, and to $19.46 in 1884. 

There will seldom be any justification in building to accommo- 
date a larger business than what is ^'in sight.*' Even if it 
could be anticipated with certainty that a large increase in 
business would come in ten years, there are many reasons why 
it would be unwise to build on a scale larger than that required 
for the business to be immediately handled. Even though it 
may cost more in the future to provide the added accommo- 
dations {e.g. larger terminals, engine-houses, etc.), the extra 
expense will be nearly if not quite offset by the interest saved 
by avoiding the larger outlay for a period of years which may 
often prove much longer than was expected. A still more im- 
portant reason is the avoidance of uselessly sinking money at 
a time when every cent may be needed to insure the success 
of the enterprise as a whole. 

[ 474. Probable number of trains per day. Increase with 
growth of traffic. The number of passenger trains per day 
cannot be determined by dividing the total number of passengers 
estimated to be carried per day by the capacity of the cars 
that can be hauled by one engine. There are many small 
railroads, running three or four passenger trains per day each 
way, which do not carry as many passengers all told as are 
carried on one heavy train of a trunk line. But because the 
bulk of the passenger traffic, especially on such light-traffic 
roads, is " unnecessary" traffic (see § 471) and must be encouraged 
and coaxed, the trains must be run much more frequently 
than mere capacity requires. The minimum number of passen- 
ger trains per day on even the lightest-traffic road should be 
two. These need not necessarily be passenger trains exclusively. 
They may be mixed trains. 

The number required for freight service may be kept more 
nearly according to the actual" tonnage to be moved. At least 
one local freight will be required, and this is apt to be considerably 
within the capacity of the engine. Some very light-traffic 
roads have little else than local freight to handle, and on such 
there is less chance of economical management. Roads with 
heavy traffic can load up each engine quite accurately according 
to its hauling capacity and the resulting economy is great. Fluc- 
tuations in traffic are readily allowed for by adding on or drop- 



§ 475o PROMOTION OF RAILROAD PROJECTS. 531 

ping off one or more trains. Passenger trains must be run on 
regular schedule, full or empty. Freight trains 'are run by 
train-despatcher's orders. A few freight trains per day may be 
run on a nominal schedule, but all others will be run as extras. 
The criterion for an increase in the number of passenger trains 
is impossible to define by set rules. Since it should always 
come before it is absolutely demanded by the train capacity 
being overtaxed, it may be said in general terms that a train 
should be added when it is believed that the consequent in- 
crease in facilities will cause an increase in traffic the value of 
which will equal or exceed the added expense of the extra train. 
475. Effect on traffic of an increase in facilities. The term 
facilities here includes everything which facilitates the transport 
of articles from the door of the producer to the door of the 
consumer. As pointed out before, in many cases of freight 
transport, the reduction of facilities below a certain point will 
mean the entire loss of such traffic owing to local inability to 
successfully compete with more favored localities. Sometimes 
owing to a lack of facilities a railroad company feels compelled 
to pay the cartage or to make a corresponding reduction on 
what would normally be the freight rate. In competitive freight 
business such a method of procedure is a virtual necessity in 
order to retain even a respectable share of the business. Even 
though the railroad has no direct competitor, it must if possible 
enable its customers to meet their competitors on even terms. 
In passenger business the effect of facilities is perhaps even 
more marked. The pleasure travel will be largely cut down 
if not destroyed. 
476. Loss caused by inconvenient terminals and by stations 
far removed from business centers. This is but a special case 
of the subject discussed just in the preceding paragraph. The 
competition once existing between the West Shore and the 
New York Central was hopeless for the West Shore from the 
start. The possession of a terminal at the Grand Central 
Station gave the New York Central an advantage over the West 
Shore with its inconvenient terminal at Weehawken which 
could not be compensated by any obtainable advantage by 
the West Shore. This is especially true of the passenger busi- 
ness. The through freight business passing through or termi- 
nating at New York is handled so generall}^ by means of floats 
that the disadvantage in this respect is not so great. The 



532 RAILROAD CONSTRUCTION. § 746. 

enormous expenditure (roughly $10,000,000) made by the 
Pennsylvania R. R., on the Broad Street Station (and its ap- 
proaches) in Philadelphia, a large part of which was made in 
crossing the Schuylkill River and running to City Hall Square, 
rather than retain their terminal in West Philadelphia, is an 
illustration of the policy of a great road on such a question. 
The fact that the original plan and expenditure has been very 
largely increased since the first construction proves that the 
management has not only approved the original large outlay, 
but saw the wisdom of making a very large increase in the ex- 
penditure. 

The construction of great terminals is comparatively infrequent 
and seldom concerns the majority of engineers. But an engineer 
has frequently to consider the question of the location of a 
way station with reference to the business center of the town. 
The following points may (or may not) have to be considered, 
and the real question consists in striking a proper balance 
between conflicting considerations. 

(1) During the early history of a railroad enterprise it is 
especially needful to avoid or at least postpone all expenditures 
which are not demonstrably justifiable. 

(2) The ideal place for a railroad station is a location im- 
mediately contiguous to the business center of the town. The 
location of the station even one fourth of a mile from this may 
result in a loss of business. Increase this distance to one mile 
and the loss is very serious. Increase it to five miles and the 
loss approaches 100%. 

(3) The cost of the ideal location and the necessary right 
of way may be a very large sum of money for the new enterprise. 
On the other hand the increase in property values and in the 
general prosperity of the town, caused by the railroad itself, 
will so enhance the value of a more convenient location that its 
cost at some future time will generally be extravagant if not 
absolutely prohibitory. The original location is therefore under 
ordinary conditions a finality. 

(4) To some extent the railroad will cause a movement of 
the business center toward it, especially in the establishment 
of new business, factories, etc., but the disadvantages caused 
to business already established is permanent. 

(5) In any attempt to compute the loss resulting from a 
location at a given distance from the business center it must be 



§ 477. PROMOTION OF RAILROAD PROJECTS. 533. 

recognized that each problem is distinct in itself and that any 
change or growth in the business of the town changes the amonnt 
of this loss. 

The argument for locating the station at some distance from 
the center of the town may be based on (a) the cost of right 
of way, thus involving the question of a large initial outlay, 
(h) the cost of very expensive construction (e,g. bridges), 
again involving a large initial outlay, (c) the avoidance of ex- 
cessive grade into and out of the town. It sometimes happens 
that a railroad is following a line which would naturally cause 
it to pass at a considerable elevation above (rarely below) 
the town. In this case there is to be considered not only the 
possible greater initial cost, but the even more important increase 
in operating cost due to the introduction of a very heavy grade. 
The loss of business due to inconvenient location can only be 
guessed at. Wellington says that at a distance of one mile 
the loss would average 25%, with upper and lower limits of 
10 and 40%, depending on the keenness of the competition 
and other modifying circumstances. For each additional mile 
reduce 25% of the preceding value. While such estimates are 
grossly approximate, yet with the aid of sound judgment they 
are better than nothing and may be used to check gross errors. 

477. General principles which should govern the expenditure 
of money for railroad purposes. It will be shown later that 
the elimination of grade, curvature, and distance have a positive 
money value ; that the reduction of ruling grade is of far greater 
value; that the creation of facilities for the handling of a large 
traffic is of the highest importance and yet the added cost of 
these improvements is sometimes a large percentage of the 
cost of some road over which it would be physically possible 
to run trains between the termini. 

The subsequent chapters will be largely devoted to a discussion 
of the value of these details, but the general principles governing 
the expenditure of money for such purposes may be stated as 
follows: 

1. No money should be spent (beyond the unavoidable 
minimum) unless it may be shown that the addition is in itself 
a profitable investment. The additional sum may not wreck 
the enterprise and it may add something to the value of the 
road, but unless it adds more than the improvement costs it is 
not justifiable. 



534 KAILROAD CONSTRUCTION. § 478. 

^2. If it may be positively demonstrated that an improvement 
will be more valuable to the road than its cost, it should certainly 
be made even if the required capital is obtained with difFicidty. 
This is all the more necessary if the neglect to do so will per- 
manently hamper the road with an operating disadvantage 
which will only grow worse as the traffic increases. 

3. This last principle has two exceptions: (a) the cost of 
the improvement may wreck the whole enterprise and cause 
a total loss to the original investors. For, unless the original 
promoters can build the road and operate it until its stock 
has a market value and the road is beyond immediate danger 
of a receivership, they are apt to lose the most if not all of 
their investment; (h) an improvement w^hich is very costly 
although unquestionably wise may often be postponed by. means 
of a cheap temporary construction. Cases in point are found 
at many of the changes of alinement of the Pennsylvania R. R., 
the N. Y., N. H. & H. R. R., and many others. While some of 
the cases indicate faulty original construction, at many of the 
places the original construction was wise, considering the then 
scanty traffic, and now the improvement is wise considering 
the great traffic. 

478. Study of railroad economics — its nature and limitations. 
The multiplicity of the elements involved in most problems 
in railroad construction preclude the possibffity of a solution 
which is demonstrably perfect. Barring out the comparatively 
few cases in this country where it is difficult to obtain any 
practicable location, it may be said that a comparatively low 
order of talent will suffice to locate anywhere a railroad over 
which it is physically possible to run trains. It may be very 
badly located for obtaining business, the ruling grades may 
be excessive, the alinement may be very bad, and the road 
may be a hopeless financial failure, and yet trains can be run. 
Among the infinite number of possible locations of the road, 
the engineer must determine the route which will give the best 
railroad property for the least expenditure of money — the 
road whose earning capacity is so great that after paying the 
operating expenses and interest on the bonds the surplus avail- 
able for dividends or improvements is a maximum. 

An unfortunate part of the problem is that even the blunders 
are not always readily apparent nor their magnitude. A defec- 
tive dam or bridge will give way and every one realizes the 



§ 479. PROMOTION OF RAILROAD PROJECTS. 535 

failure, but a badly located railroad affects chiefly the finances 
of the enterprise by a series of leaks which are only perceptible 
and demonstrable by an expert, and even he can only say that 
certain changes would probably have a certain financial value. 
479. Outline of the engineer's duties. The engineer must 
realize at the outset the nature and value of the conflicting 
interests which are involved in variable amount in each possi- 
ble route. 

(a) The maximum of business must be obtained, and yet it may 
happen that some of the business may only be obtained by an 
extravagant expenditure in building the line or by building a 
line very expensive to operate. 

(b) The ruling grades should be kept low, and yet this may 
require a sacrifice in business obtained and also mxiy cost more 
than it is worth. 

(c) The alinement should be made as favorable as possible; 
favorable alinement reduces the future operating expenses, 
but it may require a very large immediate outlay. 

(d) The total cost must be kept within the amount at which 
the earnings will make it a profitable investment. 

(e) The road must be completed and operated until the 
'^ normal" traffic is obtained and the road is self-supporting 
without exhausting the capital obtainable by the projectors ; 
for no matter how valuable the property may ultimately be- 
come, the projectors will lose nearly, if not quite, all they have 
invested if they lose control of the enterprise before it becomes 
a paying investment. 

Each new route suggested makes a new combination of the 
above conflicting elements. The engineer must select a route 
by first eliminating all lines which are manifestly impracticable 
and then gradually narrowing the choice to the best routes 
whose advantages are so nearly equal that a closer detailed 
comparison is necessary. 

The ruling grade and the details of alinement have a large 
influence on the operating expenses. A large part of this course 
of instruction therefore consists of a study of operating expenses 
under average normal conditions, and then a study of the effect 
on operating expenses of given changes in the alinement. 



i 



CHAPTER XX. 

OPERATING EXPENSES. 

480. Distribution of gross revenue. When a railroad com- 
prises but one single property, owned and operated by itself, 
the distribution of the gross revenue is a comparatively simple 
matter. The operating expenses then absorb about two thirds 
of the gross revenue; the fixed charges (chiefly the interest on 
the bonds) require about 25 or 30% more, leaving perhaps 3 
to 8% (more or less) available for dividends. The report on 
the Fitchburg R. R. for 1898 shows the following: 

Operating expenses $5,083,571 69. 1% 

Fixed charges 1,567,640 21 .3% 

Available for dividends, surplus, or per- 
manent improvements 708,259 9 . 6% 

Total revenue $7,359,470 100.0% 

But the financial statements of a large majority of the railroad 
corporations are by no means so simple. The great consolida- 
tions and reorganizations of recent years have been effected 
by an exceedingly complicated system of leases and sub-leases, 
purchases, "mergers," etc., whose forms are various. Railroads 
in their corporate capacity frequently own stocks and bonds 
of other corporations (railroad properties and otherwise) and 
receive, as part of their income, the dividends (or bond interest) 
from the investments. 

The Interstate Commerce Commission annually makes a 
report of the income and profit-and-loss account of all the rail- 
roads of the United States, considered as one system. For 
example, the statement for the year 1912 includes the following 
items. Operating revenues from rail operations $2,842,695,382; 
operating expenses due to rail operations $1,972,415,776, which 
is 69.4%. Interest on funded debt used up 13.9% of the rev- 
enues, and taxes 4.2%. There were other miscellaneous incomes 
and expenditures which caused a net loss of another 2.0% 

536 



§481. 



OPERATING EXPENSES. 



537 



of revenue, leaving 10.5% or $299,361;208 which were issued 
as dividends. These dividends are about 3.4% of the outstand- 
ing stock. The percentage to the amount of money actually- 
paid for the stock is unknown and unknowable. 

481. Operating expenses per train-mile. The uniformity in 
the average operating expenses per train mile for light-traffic 
and heavy-traffic roads and for long and short roads is very 
remarkable. This is illustrated by a comparison of figures for 
ten heavy traffic roads and ten small roads selected at random, 
except that each had a mileage of less than 100 miles, 

' OPERATING EXPENSES PER TRAIN-MILE ON LARGE AND SMALL 
ROADS (1904 AND 1910). 



Mileage. 



1904. 



Whole United States. 



Canadian Pacific 

C, B. & Q 

Chicago <fe Northwestern 

Southern Railway 

C, R. I. & P 

Northern Pacific 

A., T. & S. F 

Great Northern 

Illinois Central 

Atlantic Coast Line 



Average of ten 



Montpelier & Wells River. . . 

Somerset Railway Co.* 

Huntingdon & Broadtop 
Mountain 

Lehigh & New England 

Ligonier Valley 

Newburgh, Dutchess & Con- 
necticut t 

Susquehanna & New York. . 

Detroit & Charlevoix 

Harriman & Northeastern * 

Galveston, Houston & Hen- 
derson 



1910. 



220,112 



Average of ten (or nine) 



8,332 
8.326 
7,412 
7,197 
6,761 
5,619 
5,031 
4,489 
4,374 
4,229 



44 
42 

66 
96 
11 

59 
55 
51 
20 

50 



240,43^ 



Operating 

expensesper 

train-mile. 



1904. 



1.314 



10,271 
9,040 
7,629' 
7,050! 
7,396 
6,189 
7,460| 
7,147i 
4,551| 
4,4911 



1 . 320 
1.313 
1.136 
1.048 
1.199 
1.392 
1 . 305 
1.464 
1.107 
0.984 



1910. 



1.489 



50 
94 

70 

170 

16 



80 
51 
20 

50 



1.227 



1.169 
0.802 

. 950 
0.793 
1.427 

0.922 
1.368 
1.424 
2.162 

1.556 



1.257 



1.504 
1.710 
1.306 
1.234 
1.344 
1.824 
1.626 
1.808 
1.409 
1.213 



1.498 



1.430 
1.314 

2.052 
2.045 
1.480 



1.028 
1.010 
1.733 

1.759 



1.539 



Ratio expenses 

to earnings 

per cent. 



1904. 



67.79 



68.72 
64.35 
66.61 
70.30 
72.90 
52.26 
60.05 
49.72 
70.02 
58.95 



1910. 



66.29 



65, 
71. 



63.39 



80.73 
59.37 

52.10 
69.80 
69.33 

85.09 

78.47 
67.52 
79.26 

47.27 



68.89 



70.31 
67.43 
73.07 
61.71 
64 . 33 
60.53 
74.84 
62.44 



67.18 



75.08 
76.65 

96.40 
62.84 
49.15 



77.81 
99.53 
63. 7e 

70.37 



74.61 



b 



* Subsidiary road since 1904. 

t Merged since 1904; separate figures not available. 



538 



BAILEOAD CONSTRUCTION. 



§482. 



The fluctuations of the average cost per train-mile for several 
years past may be noted from the following tabular form: 



AVERAGE COST PER TRAIN-MILE (FOR 


WHOLE 


U. S.) IN CENTS. 


Year. 


Cents. 1 


Year. 


Cents. 


Year. 


Cents. 


Year. 


Cents. 


1890 
1891 
1892 
1893 
1894 
1895 


96.006 

95.707 

96.580 

97.272 i 

93.478 

91.829 


1896 
1897 
1898 
1899 
1900 
1901 


93.838 
92.918 
95.635 
98.390 
107.288 
112.292 


1902 
1903 
1904 
1905 
1906 
1907 


117.960 
126.604 
131.375 
132.140 
137.060 
146.993 


1908 
1909 
1910 
1911 
1912 


147.340 
143.370 
148.865 
154.338 
159.077 



The enforced economies after the panic of 1893 are well 
shown. The reduction generally took the form of a lowering 
of the standards of maintenance of way and of maintenance of 
equipment. The marked advance since 1895 is partly due to 
the necessity for restoring the roads to proper conditions, replen- 
ishing worn-out equipment and providing additional equip- 
ment to handle the greatly increased volume of business. The 
recent advance is chiefly due to the increase in wages and the 
generally increased cost of supplies. 

It may be noted from the I. C. C. reports that the cases where 
the operating expenses per train-mile and the ratio of expenses 
to earnings vary very greatly from the average are almost 
invariably those of the very small roads or of ^'junction roads'' 
where the operating conditions are abnormal. For example, one 
little road, with a total length of 13 miles and total annual opera- 
ting expenses of $5342, spent but 22ic. per train-mile, which pre- 
cisely exhausted its earnings. This precise equality of earnings 
and expenses suggests jugglery in the bookkeeping. As another 
abnormal case, a road 44 miles long spent S3. 81 per train-mile, 
which was nearly fourteen times its earnings. In another case a 
road 13 miles long earned $7.76 per train-mile and spent $6.03 
(78%) on operating expenses, but the fixed charges were abnor- 
mal and the earnings were less than half the sum of the operating 
expenses and fixed charges. The normal case, even for the 
small road, is that the cost 'per train-mile and the ratio of operat- 
ing expenses to earnings will agree fairly well with the average, 
and when there is a marked difference it is generally due to 
some abnormal conditions of expenses or of earning capacity. 

482. Reasons for uniformity in expenses per train-mile. 
The chief reason is that, although on the heavy -traffic road 
everything is kept up on a finer scale, better roadbed, heavier 
rails, better rolling stock, more employees, better buildings, 



§ 483. OPERATING EXPENSES. 539 

rails, better rolling stock, more employees, better buildings, 
stations, and terminals, etc., yet the number of trains is so much 
greater that the divisor is just enough larger to make the average 
cost about constant. This is but a general statement of a fact 
which will be discussed in detail under the different items of 
expense. 

483. Detailed classification of expenses with ratios to the 
total expense. The Interstate Commerce Commission now 
publishes each year a classification with detailed summation 
for the cost of each item. These summations are made up 
from reports furnished by railroads which have (in the reports 
recently made) represented over 99% of the total traffic han- 
dled. In the annexed tabular form (Table XLI) are shown the 
percentages which each item bears to the total. The railroads 
have been divided into two classes, ''large" and ''small," as 
indicated below. Large roads report on 116 items which are 
combined and condensed with 44 items for small roads. 

"Large roads" are those with mileage greater than 250 miles, 
or those with operating revenues greater than $1,000,000. 
Roads subsidiary to "large roads" are also included in this 
class. 

"Small roads" are those with mileage less than 250 miles 
and also with operating revenues less than $1,000,000. 

484. Amounts and percentages of the various items. The 
I. C. C. report for the year ending June 30, 1909, was the first 
to include the distribution of expenses according to the present 
classification. The items as given are reliable and may be utiHzed, 
as far as any such computations are to be depended on, in 
estimating future expenses. The chief purpose of this dis- 
cussion is to point out those elements of the cost of operating 
trains which may be affected by such changes of location as an 
engineer is able to make. There are some items of expense with 
which the engineer has not the shghtest concern, nor wiU they 
be altered by any change in alinement or constructive detail 
which he may make. In the following discussion such items 
wall be passed over with a brief discussion of the sub-items 
included. 

MAINTENANCE OF WAY AND STRUCTURES. 

485. Items 2 to 5. Track material. The relative cost of 
ballast, ties, rails and other track material, as shown by com- 



540 



EAILROAD CONSTRUCTION. 



§485, 



TABLE XLI. — ANALYSIS OF OPERATING EXPENSES OF ALL ''lARGE"* 
RAILROADS IN THE UNITED STATES FOR YEAR ENDING JUNE 30, 
1912, SHOWING PERCENTAGE OP EACH ITEM TO TOTAL AND COST 
IN CENTS PER TRAIN-MILE. 



Item. 
No. 



Account. 



Total 

Amount 

(thousands) 



Per cent 

of total 

Expenses 



Cents per 
Train- 
Mile. 



1 

2 
3 
4 
5 
6 
7 
8 
9 
10-12 

13-15 

16,17 

18 

19 
20,21 

22,23 



Maintenance of Way and Structures. 

Superintendence 

Ballast 

Ties 

Rails 

Other track material 

Roadway and track 

Removal of snow, sand, and ice . 

Tunnels 

Bridges, trestles, and culverts. . . 

Crossings, all; fences; snow struc- 
tures 

Signals, telegraph, electrical power 
transmission 

Buildings, grounds, docks, wharves 

Roadway tools and supplies 

Injuries to persons 

Stationery, printing and other ex- 
penses 

Joint tracks, etc. (net balance) .... 

Maintenance of Equipment. 

Superintendence 

Repairs, renewals and depreciation: 
Locomotives, steam and electric. 

Cars, passenger 

Cars, freight 

Equipment, electrical, car 

Equipment, floating 

Equipment, work 

Equipment, shop (machinery and 

tools) 

Equipment, power plant 

Injuries to persons 

Stationery, printing and other ex- 
penses 

Joint equipment, at terminals (net 
balance) 

Traffic Expenses. 
Agencies; advertising; fast freight 
lines; etc 



$18,789 

7,157 

55,463 

16,438 

17,346 

129,397 

6,920 

1,141 

27,712 

8,066 

13,681 

35,389 

4,480 

1,989 

1,038 
3,463 



0.990 

0.377 

2.921 

.866 

.914 

6.815 

.364 

.060 

1.460 

.425 

.720 

1.864 

.236 

.105 

.054 

.182 



$348,471 



18.353 



1.58 

.60 

4.65 

1.38 

1.45 

10.84 

.58 

.10 

2.32 

.68 

1.14 

2.96 

.38 

.17 

.09 
.29 



29.20 



24 

25-30 
31-33 
34-36 
37-39 
40-42 
43-45 
46 

47 

48 

49,50 

51,52 



$13,175 

175,889 

38,968 

183,968 

318 

1,333 

6,128 

10,418 

268 

1,818 

4,036 

676 



.694 

9.263 

2.052 

9.690 

.017 

.071 

.322 

.548 
.014 
.096 

.213 

.036 



1.10 

14.74 
3.26 

15.41 
.03 
.11 
.51 

.87 
.02 
.15 

.34 

.06 



$436,995 



23.016 



36.61 



53-60 



$59,047 



3.110 



4.95 



* The " large " roads here reported represent 88% of the total mileage. 



paring either the gross amounts or the percentages in Table XLI, 
is suggestive and instructive. The fact that ties cost con- 
siderably more than all other track material combined shows 



485. 



OPERATING EXPENSES. 



541 



TABLE XLi. {Continued). — analysis of operating expenses 

OF ALL '' LARGE " RAILROADS IN THE UNITED STATES FOR 
YEAR ENDING JUNE 30, 1912, SHOWING PERCENTAGE OF 
EACH ITEM TO TOTAL AND COST IN CENTS PER TRAIN-MILE. 



Item 
No. 



61, 62 

63 

64-66 

67-70 
71-76 



77,78 
104, 105 

79, 80 
81 

82 

83 

84, 85 

86, 87 

88 

89 

90-92 

93 
94-98 

99-103 



Account. 



Transportation Expenses. 

Superintendence and train dis- 
patching 

Station employees . 

Weighing; car service associa- 
tion; coal and ore docks 

Yards (wages, expenses, sup- 
plies) 

Yard locomotives (enginemen, 
fuel, water, lubricants, sup- 
plies) 

f Operating joint tracks, ter- 
\ minals, yards, and facilities 
[ (net balance) 

Motormen and road enginemen. 

Road locomotives, engine-house 
expenses 

Road locomotives, fuel 

Road locomotives, water 

Road locomotives, lubricants 
and other supplies 

Operating power plants, pur- 
chased power 

Road trainmen 

Train supplies and expenses . . . 

Interlockers, signals, flagmen, 
draw-bridges 

Clearing wrecks 

Telegraph, floating equipment, 
stationery, miscellaneous. . . . 

Loss and damage to property, 
personal injuries 



Total 

Amount 

(thousands). 



$40,743 
133,877 

15,949 

76,069 

74,370 



10,430 
120,966 

33,951 
194,142 

12,482 

7,430 

1,797 

128,339 

34,462 

17,831 
5,167 

20,009 

56,838 



$984,852 



Per cent 

of total 

expenses. 



2.146 
7.051 

.839 

4.007 

3.917 



.550 
6.371 

1.788 

10.225 

.657 

.392 

.095 
6.759 
1.815 

.939 
.272 

1.054 

2.994 



51.871 



Cents per 
train- 
mile. 



3.41 
11.22 

1.33 

6.37 

6.23 



.88 
10.14 

2.84 

16.27 

1.04 

.62 

.15 

10.75 

2.89 

1.49 
.43 

1.68 

4.76 



82.51 



106-116 



General Expenses. 
Salaries of general officers, 
clerks, etc.; law, insurance, 
pensions, miscellaneous 



69,297 



3.650 



5.81 



Total operating expenses. . . 



$1,898,662 



100.000 



159.08 



the importance of any possible saving in tie renewals. It is 
also significant that the relative importance of ties has increased 
in the last few years, and that the relative increase has not been 
due to a reduction in the cost of other track material. Appar- 
ently the lengthening of the average life of ties, due to pre- 
servative processes, the use of tie-plates, and greater care to 
avoid the premature withdrawal from the track of ties which 



542 



RAILROAD CONSTRUCTION. 



§486. / 



are still serviceable, has not kept pace with the increase in the/ 
average cost per tie. The cost of rails has advanced because 
of (a) the very general adoption of heavier rails; (6) the almost 
universal substitution of more expensive open-hearth steel for 
Bessemer, on account of greater reliability and durability, and 
(c) the increase in cost of all steel products. 

486. Item 6. Roadway and track. This item is three-eighths 
of the total cost of maintenance of way and structures. It 
consists chiefly of the wages of trackmen. There has been an 
almost steady increase in the daily wages of section foremen 
and other trackmen since 1900, as shown below: 





1900 


1901 


1902 


1903 


1904 


1906 


1906 


Section foremen 

Other trackmen 

No. of trackmen per 
100 miles 


1.68 
1.22 

118 


1.71 
1.23 

122 


1.72 
1.25 

140 


1.78 
1.31 

147 


1^78 
1.33 

136 


1.79 
1.32 

143 


1.80 
1.36 

155 







Section foremen 

Other trackmen. . , . . . 

No. of trackmen per 

100 miles 



1907 



1.90 
1.46 

162 



1908 



1.95 
1.45 

130 



1909 



1.96 
1.38 

136 



1910 



1.99 
1.47 

157 



1911 



2.07 
1.50 

147 



1912 



2.09 
1.50 

143 



The average number of section foremen per 100 miles of line 
has remained almost constant at 18. Although there have been 
fluctuations in the number of ^' other trackmen ^' required per 
100 miles of line, there has been in general a very substantial 
increase. These two causes combined (increased number and 
increased wages) have had a great influence in producing the 
regular and steady increase in the average cost of a train-mile, 
as shown in § 481. 

487. Items 8 to 15. Maintenance of track structures. As a 
matter of economics, the locating engineer has little or no concern 
with the cost of maintaining track structures. If he is com- 
paring two proposed routes it would be seldom that they would 
be so different that he would be justified in attempting to compute 
a train-mile difference in cost of operation, based on differences 
in these items. Of course, one proposed line might call for one 
or more tunnels which the alternate line might not have, and 
the annual cost of maintaining the tunnels would increase the 
cost of operation. Such a case would justify special considera- 



§488. OPERATING EXPENSES. 543 

tion. So far as the maintenance of small bridges and culverts 
are concerned it would usually be sufficiently accurate to consider 
that a proposed change of line, involving perhaps several miles 
of road, would require substantially the same number of bridges 
and culverts, and therefore that the cost of maintaining them 
would be the same by either line. The error involved in such 
an assumption would usually be insignificant, unless there was 
a very large and material difference in the two lines in this 
respect. Under such conditions special computations should be 
made. The items total less than 3% for small roads and still 
less for large roads. 

MAINTENANCE OF EQUIPMENT. 

488. Items 25 to 27. Repairs, renewals and depreciation of 
steam and electric locomotives. The item is of interest to the 
locating engineer because he must appreciate the effect on 
locomotive repairs and renewals of an addition to distance. 
A large part of the repairs of locomotives are due to the wear 
of wheels, which is largely caused by curvature. Therefore the 
value of any reduction of curvature is a matter of importance, 
and this will be considered in Chapter XXII. A considerable 
portion of the deterioration of a locomotive is due to grade, and 
the economic advantages of reductions of grade will be con- 
sidered in Chapter XXIII. 

This item includes the expenses of work whose effect is sup- 
posed to last for an indefinite period. It does not include the 
expense of cleaning out boilers, packing cylinders, etc., which 
occurs regularly and which is charged to items 72 or 81. It 
does include all current repairs, general overhauhng, and even 
the replacement of old and worn-out locomotives by new ones 
to the extent of keeping up the original standard and number. 
Of course additions beyond this should be considered as so 
much increase in the original capital investment. As a loco- 
motive becomes older the annual repair charge becomes a larger 
percentage on the first cost, and it may become as much as one- 
fourth and even one-third of the first cost. When a locomotive 
is in this condition it is usually consigned to the scrap-pile; the 
annual cost for maintenance becomes too large an item for its 
annual mileage. The effect on expenses of increasing the weight of 
engines is too compHcated a problem to be solved accurately, but 



544 RAILROAD CONSTRUCTION. § 48^. 

certain elements of it may be readily computed. While the cost 
of repairs is greater for the heavier engines, the increase is only 
about one-half as fast as the increase in weight — some of the 
subitems not being increased at all. 



, TRANSPORTATION, 

489. Items 71 to 76. Yard-engine expenses. By com- 
paring these items with the corresponding items (80 to 85) for 
road engines, it may be seen that the total expenses assignable to 
yard engines are about 20% of those of road engines; the relative 
fuel charge for 1912 was 15.6%. The number of switching 
locomotives in the United States in 1912 was 9529 or 15.3% of 
the total number, 62,262. The relative charge for wages of engine- 
men was 26.2%. This higher proportionate charge is probably 
due to the fact that the wages for yard enginemen must neces- 
sarily be on a per diem basis, but the wages of road enginemen 
are generally on a mileage basis, as explained later. On the 
other hand the mileage of a yard engine is usually comparatively 
low, and the coal consumed will be correspondingly, although 
not proportionately, low. It must also be remembered that 
these figures are exclusive of the work and equipment of switching 
and terminal companies. 

490. Item 80. Road enginemen. This item requires 6% of 
the total operating expenses. The enginemen are usually paid 
on a mileage basis, or by the trip, except on very small railroads. 
On very short roads, where a train crew may make two, three, 
or even four complete round trips per day, they may readily be 
paid by the day, so many round trips being considered as a 
day's work, but on roads of great length, where all trains, and 
especially freight-trains, are run day and night, weekday and 
Sunday, all trainmen are necessarily paid by the trip. The pay 
for a trip is figured on a mileage basis except that a trip is usually 
considered to have a minimum length of 100 miles or 10 hours of 
time. Eight hours was fixed as standard by the '^ Adamson " 
law, in 1916. All extra time is called " overtime '' and is paid 
for at an extra rate. The basis of train wages is too complicated 
for any brief discussion. Even the basis is constantly changing, 
the only uniform feature being a steady increase. 

The increase in the average wages paid to enginemen and 
firemen since 1900 is plainly shown by the following figures : 



§491, 



OPERATING EXPENSES. 



545 



INCREASE IN DAILY WAGES, FROM 1900 TO 1912. 





1900 


1901 


1902 


1903 


1904 


1905 


1906 


Enginemen 


$ 
3.75 
2.14 


$ 
3.78 
2.16 


$ 
3.84 
2.20 


$ 
4.01 
2.28 


$ 
4.10 
2.35 


$ 
4.12 
2.38 


4 12 


Firemen 


2.42 



Enginemen 
Firemen. . . 



1907 



$ 
4.30 
2.54 



1908 



$ 
4.45 
2.64 



1909 



$ 
4.44 
2.67 



1910 



$ 
4.55 
2.74 



1911 



4.79 
2.94 



1912 



$ 
5.00 
3.02 



491. Item 82. Fuel for road locomotives. This item in- 
cludfsS every subitem of the entire cost of the fuel until it is 
placed in the engine-tender. The cost therefore includes not only 
the first cost at the point of delivery to the road, but also the 
expense of hauling it over the road from the point of delivery 
to the various coaling-stations and the cost of operating the 
coal-pockets from which it is loaded on to the tenders. Even 
though the cost may be fairly regular for any one road, the cost 
for different roads is exceedingly variable. There has been an 
almost steady increase in the percentage of the cost of this item 
per train-mile since 1897. Items 73 and 82 amounted to nearly 
12% of the total operating expenses in 1912, and required an 
actual expenditure of nearly $225,000,000. It is the largest 
item in the whole cost of railroad operation. Although some 
roads, which traverse coal-regions and perhaps actually own the 
coal-mines, are able to obtain their coal for a cost which may be 
charged up as $1 per ton or less, there are many roads which 
are far removed from coal-fields which have to pay $3 or $4 
per ton, on account of the excessive distance over which the coal 
must be hauled. Unfortimately the figures published by tlffe 
Interstate Commerce Commission do not show the variations 
in the percentage of this item in different localities. A sur- 
prisingly large percentage of the fuel consumed is not utiHzed in 
drawing a train along the road. A portion of this percentage is 
used in firing-up. A portion is wasted when the engine is stand- 
ing still, which is a considerable proportion of the whole time. 
The pohcy of banking fires instead of drawing them reduces the 
injury resulting from great fluctuations in temperature, but in a 
general way we may say that there is but little, if any, saving in 
fuel by banking the fires, and therefore we may consider that 



546 RAILROAD CONSTRUCTION. §491. 

almost a fire-box full of coal is wasted whether the fires are 
banked or drawn. As given in § 464, the fuel used by a loco- 
motive in firing-up may be estimated as 510 lbs. per 1000 square 
feet of heating surface, based on using 12000 B.t.u. coaL But 
even the amount of coal required to produce the required steam- 
pressure in the boiler from cold water does not represent the 
total loss. The train-dispatcher, in his anxiety that engines 
shall be ready when needed, will sometimes order out the loco- 
motives which remain somewhere in the yard, perhaps exposed 
to cold weather, and blow off steam for several hours before they 
make an actual start. This loss has been estimated as 120 lbs. 
per hour per 1000 square feet of heating surface, but it would 
evidently be far greater on a windy winter day than on a calm 
summer day. A freight-train, especially on a single-track road, 
will usually spend several hours during the day on sidings, and 
when a single-track road is being run to the limit of its capacity, 
or when the management is not good, the time will be still greater. 
It is estimated that the amount lost through a 2|-inch safety- 
valve in one minute would represent the consumption of 15 
pounds of coal, which would be sufficient to haul 100 tons on a 
mile of track with easy grades. Again we see that the amount 
thus lost is exceedingly variable and almost non-computable, 
although as a rough estimate the amount has been placed 
at from 3 to 6% of the total. Another very large subitem 
of loss of useful energy is that occasioned by stopping and 
starting. A train running 30 miles per hour has enough kinetic 
energy to move it on a straight level track for more than two 
miles. Therefore, every time a train running at 30 miles per 
hour is stopped, enough energy is consumed by the brakes to 
run it about two miles. There is a double loss^ not only due 
to the fact of the loss of energy, but also because the power of 
the locomotive has been consumed in operating the brakes. 
When the train is again started, this kinetic energy must be 
restored to the train in addition to the ordinary resistances which 
are even greater, on account of the greater resistance at very 
low velocities. Of course, the proportion of fuel thus con- 
sumed depends on the frequency of the stops. It was demon- 
strated by some tests on the Manhattan Elevated Road in New 
York City, where the stops average one in every three-eighths 
of a mile, that this cause alone would account for the consump- 
tion of nearly three-fourths of the fuel. On ordinary railroads 



§492. 



OPERATING EXPENSES. 



547 



the proportion, of course, will not be nearly so great, but there 
is reason to believe that 10 to 20% is not excessive as an 
average figure. 

492. Item 88. Road trainmen. This item includes the wages 
of conductors and " other trainmen." As in the case of all 
other employees, the average daily wages have advanced since 
1900 as shown below: 



AVERAGE DAILY WAGES OF CONDUCTORS AND OTHER TRAINMEN, 

1900 TO 1912. 





1900 


1901 


1902 


1903 


1904 


1905 


1906 


Conductors 

Other trainmen 


$ 
3.17 
1.96 


$ 
3.17 
2.00 


S 
3.21 
2.04 


$ 
3.38 
2.17 


$ 
3.50 

2.27 


$ 
3.50 
2.31 


$ 
3.51 
2.35 





1907 


1908 


1909 


1910 


1911 


1912 


Conductors 

Other trainmen 


.$ 

3.69 

2.54 


$ 
3.81 
2.60 


$ 
3.81 
2.59 


$ 
3.91 
2.69 


$ 
4.16 

2.88 


$ 
4.29 
2.96 



These figures are of vital importance from an economic stand- 
point, since they show a constant tendency to increase and thereby 
raise the average cost of a train-mile. And as there is no present 
indication of any limit to this increase, all economic calculations 
which attempt to predict future expenses, even for a few years 
in advance, must allow for these and other increased expenses. 

493. Item 89, Train supplies and expenses. These items, 
which average about 1.8%, include the large list of consumable 
supplies such as lubricating oil, illuminating-oil or gas, ice, fuel 
for heating, cleaning materials, etc., which are used on the cars 
and not on the locomotives. The consumption of some of these 
articles is chiefly a matter of time. In other cases it is a function 
of mileage. The effect of changes which an engineer may make 
on this item will be considered when estimating the effect of the 
changes. 

494. Items 93, 99 to 103. Clearing wrecks, loss, damage and 
injuries to persons and property. These expenses are fortuitous 
and bear no absolute relation either to the number of miles of 
road or the number of train-miles. While they depend largely 
on the standards of discipline on the road, even the best of roads 
have to pay some small proportion of their earnings to these 



L 



548 RAILROAD CONSTRUCTION. § 495. 

items. While we might expect that a road with heavy traffic 
would have a larger proportion of train accidents than a road of 
light traffic, it is usually true that on the heavy-traffic roads the 
precautions taken are such that they are usually freer from acci- 
dents than the light-traffic roads. During recent years there 
has been a very perceptible increase in the percentages of these 
items, particularly in the compensations paid for ^ injuries to 
persons." The increase in this item coincides with the increase 
already noted in the number of passengers killed during recent 
years. The possible relation between curvature and accidents 
has already been discussed, but otherwise the locating engineer 
has no concern with these items. 

495. Items 104, 105. Operating joint tracks and facilities, 
Dr. and Cr. A large part of these debit and credit charges 
are those for car per diem and mileage charges. This is a charge 
paid by one road to another for the use of cars, which are chiefly 
freight-cars. To save the rehandling of • freight at junctions, 
the policy of running freight-cars from one road to another is 
very extensively adopted. Since the foreign road receives its 
mileage proportion of the freight charge, it justly pays to the road 
owning the car at a rate which is supposed to represent the 
value of the use of the freight-car for the number of miles 
traveled. The foreign road then loads up the freight-car with 
freight consigned to some point on the home road and sends it 
back, paying mileage for the distance traveled on the foreign 
road, a proportional freight charge having been received for that 
service. All of these movements of freight-cars are reported 
to a car association, which, by a clearing-house arrangement, 
settles the debit and credit accounts of the various roads with 
each other. Such is the simple theory. In practice the cars are 
not sent back to the home road at once, but wander off according 
to the local demand. As long as a strict account is kept of the 
movements of every car, and as long as the home road is paid 
the charge which really covers the value of lost service, no harm 
is done to the home road, except that sometimes, when business 
has suddenly increased, the home road cannot get enough cars 
to handle its own business. The value of the car is then abnor- 
mally above its ordinary value, and the home road suffers for 
lack of the rolHng stock which belongs to it. Formerly such 
charges were paid strictly according to the mileage. This 
developed the intolerable condition that loaded cars would be, 



§ 495. OPERATING EXPENSES. 549 

run onto a siding and left there for several days, simply because 
it was not convenient to the consignee to unload the car imme- 
diately. On the mileage basis the car would be earning nothing, 
and, since the road on which the car then was had no particular 
interest in the car, the car was allowed to stand to suit the con- 
venience of the consignee. To correct this evil a system of per 
diem charges has been developed, so that a railroad has to pay a 
per diem charge for every foreign car on its lines. To reduce this 
charge as much as possible the railroads compel consignees, 
under penalty of heavy demurrage charges, to unload cars 
promptly. The running of freight-cars on foreign lines is now 
settled almost exclusively on the per diem basis, but the running 
of passenger-cars over other lines, as is done on account of the 
advantages of through-car service, as well as the running of 
Pullmans and other special cars, is still paid for on the mileage 
basis. To the extent to which this charge is settled on the mile- 
age basis, any change in distance which the engineer may be able 
to effect in the length of the road will have its influence on this 
item, but when the freight-car business, which comprises by far 
the larger part of the running of cars over foreign Hues, is settled 
on the per diem basis no changes in alinement which the engineer 
may make will affect the item appreciably. 

Switching Charges. Where two or more railroads intersect 
there will be a considerable amount of shifting of cars, chieflj^ 
freight-cars, from one road to the other. This shifting at any 
one junction may be done entirely by the engines of one road 
or perhaps by those of both roads. A portion of the expense 
of this work is charged up against the other road by the road 
which does the work. The total amount of this work is care- 
fully accoimted for by a clearing-house arrangement, and the 
balance is charged up against the road which has done the least 
work. The item is very small, is fairly uniform year by year, 
and is seldom, if ever, affected by changes of alinement. 

Other Items. All of the remaining items, as stated in Table 
XLI, are of no concern to the locating engineer. They are either 
general expenses, such as the salaries of general officers, insurance 
or law expenses, or are special items, such as advertising or the 
operation of marine equipment which will not be changed by 
any variations in distance, curvature, or grades which a locating 
engineer may make. There is therefore no need for their further 
discussion here. 



CHAPTER XXL 

DISTANCE. 

. 496. Relation of distance to rates and expenses. Rates 
are usually based * on distance traveled, on the apparent 
hypotheses that each additional mile of distance adds its pro- 
portional amount not only to the service rendered but also to 
the expense of rendering it. Neither hypothesis is true. The 
value of the service of transporting a passenger or a ton of 
freight from A to 5 is a more or less uncertain gross amount 
depending on the necessities of the case and independent of 
the exact distance. Except for that very small part of passen- 
ger traffic which is undertaken for the mere pleasure of traveling, 
the general object to be attained in either passenger or freight 
traffic is the transportation from A to B, however it is attained. 
A mile greater distance does not improve the service rendered ; 
in fact, it consumes valuable time of the passengers and perhaps 
deteriorates the freight. From the standpoint of service ren- 
dered, the railroad which adopts a more costly construction and 
thereby saves a mile or more in the route between two places 
is thereby fairly entitled to additional compensation rather 
than have it cut down as it would be by a strict mileage rate. 
The actual value of the service rendered may therefore vary 
from an insignificant amount which is less than any reasonable 
charge (which therefore discourages such traffic) and its value 
in cases of necessity — a value which can hardly be measured in 
money. If the passenger charge between New York and Phila- 
delphia were raised to $5, $10, or even $20, there would still be 
some passengers who would pay it and go, because to them 
it would be worth $5, $10, or $20, or even more. Therefore, 
when they pay $2.25 they are not paying what the service is 
worth to them. The service rendered cannot therefore be 
made a measure of the charge, nor is the service rendered pro- 
portional to the miles of distance. 

The idea that the cost of transportation is proportional to 

550 



§ 497. DISTANCE. 551 

the distance is much aiore prevalent and is in some respects 
more justifiable, but it is still far from true. This is especially 
true of passenger service. The extra cost of transporting a single 
passenger is but little more than the cost of printing his ticket. 
Once aboard the train, it makes but little difference to the rail- 
road whether he travels one mile or a hundred. Of course there 
are certain very large expenses due to the passenger traffic 
which must be paid for by a tariff which is rightfully demanded, 
but such expenses have but little relation to the cost of an 
additional mile or so of distance inserted between stations. 
The same is true to a slightly less degree of the freight traffic. 
As shown later, the items of expense in the total cost of a train- 
mile, which are directly affected by a small increase in distance, 
are but a small proportion of the total cost. 

497. The conditions other than distance that affect the cost; 
reasons why rates are usually based on distance. Curvature 
and minor grades have a considerable influence on the cost of 
transportation, as will be shown in detail in succeeding chap- 
ters, but they are never considered in making rates. Ruling 
grades have a very large influence on the cost, but they are hke- 
wise disregarded in making rates. An accurate measure of 
the effect of these elements is difficult and complicated and 
would not be appreciated by the general public. Mere dis- 
tance is easily calculated; the public is satisfied vdth. such 
a method of calculation; and the railroads therefore adopt a 
tariff which pays expenses and profits even though the charges 
are not in accordance with the expenses or the ser^dce rendered. 

EFFECT OF DISTANCE ON RECEIPTS. 

498. Classification of traffic. There are various methods 
of classifying traffic, according to the use it is intended to make 
of the classification. The method here adopted will have ref- 
erence to its competitive or non-competitive character and also 
to the method of division of the receipts on through traffic. 
Traffic may be classified first as '' through '^ and '' local " — 
through traffic being that traveling over two (or more) lines, 
no matter how short or non-competitive it may be; '^ local '^ 
traffic is that confined entirely to one road. A fivefold classifica- 
tion is however necessary — ^which is: 

A. NoH'Competitive local-^on one road with no choice of route* 



552 RAILROAD CONSTRUCTION. § 499. 

B. Non-competitive through — on two (or more) roads, but 
with no choice. 

C. Competitive local — a choice of two (or more) routes, but 
the entire haul may be made on the home road. 

D. Competitive through — direct competition between two 
or more routes each passing over two or more lines. 

E. Semi-competitive through — a non-competitive haul on the 
home road and a competitive haul on foreign roads. 

There are other possible combinations, but they all reduce to 
one of the above forms so far as their essential effect is concerned. 

499. Method of division of through rates between the 
roads run over. Through rates are divided between the 
roads run over in proportion to the mileage. There may 
be terminal charges and possibly other more or less arbitrary 
deductions to be taken from the total amount received, but 
when the final division is made the remainder is divided accord- 
ing to the nlileage. On account of this method of division and 
also because non-competitive rates are always fixed according 
to the distance, there results the unusual feature that, unlike 
curvature and grade, there is a compensating advantage in 
increased distance, which applies to all the above kinds of 
traffic except one (competitive local), and that the compensation 
is sometimes sufficient to make the added distance an actual 
source of profit. It has been estimated that the cost of hauling 
a train an additional mile is only 33 to 49% of the average cost. 
Therefore in all non-competitive business (local or through) 
where the rate is according to the distance, there is an actual 
profit in all such added distance. In competitive local busi- 
ness, in which the rate is fixed by competition and has practically 
no relation to distance, any additional distance is dead loss. In 
competitive through business the profit or loss depends on the 
distances involved. This may best be demonstrated by exam- 
ples. 

500. Effect of a change in the length of the home road on 
its receipts from through competitive traffic. Suppose the 
home road is 100 miles long and the foreign road is 150 miles 

long. Then the home road will receive = 40% of the 

100 -j- 150 

through rate. 

Suppose the home road is lengthened 5 miles; then it will 



§ 501. DISTANCE. . 553 

105 
receive = 41.176% of the through rate. The traffic 

being competitive, the rate will be a j&xed quantity regardless 
of this change of distance. By the first plan the rate received 
is 0.4% per mile; adding 5 miles, the rate for the original 100 
miles may be considered the same as before; and that the addi- 
tional 5 miles receive 1.176%, or 0.235% per mile. This is 59% 
of the original rate per mile, and since this is more than the 
cost per mile for the additional distance, the added distance is 
evidently in this case a source of distinct profit. On the other 
hand, if the line is shortened 5 miles, it may be similarly shown 
that not only are the receipts lessened, but that the saving in 
operating expenses by the shorter distance is less than the 
reduction in receipts. 

A second example mil be considered to illustrate another 
phase. Suppose the home road is 200 miles long and the foreign 
road is 50 miles long. In this case the home road will receive 

^ =80% of the through rate. Suppose the home road is 

205 

lengthened 5 miles ; then it will receive ^^rrz — ~ = 80 , 392% 

of the through rate. By the first plan the rate received is 
0.400% per mile; adding 5 miles, there is a surplus of 0.392, 
or 0.0784 per mile, which is but 19.6% of the original rate. 
At this rate the extra distance evidently is not profitable, al- 
though it is not a dead loss — there is some compensation. 

501. The most advantageous conditions for roads forming 
part of a through competitive route. From the above it may 
be seen that when a road is but a short link in a long com- 
petitive through route, an addition to its length will increase 
its receipts and increase them more than the addition to the 
operating expenses. 

As the proportionate length of the home road increases the 
less will this advantage become, until at some proportion an 
increase in distance will just pay for itself. As the proportionate 
length grows greater the advantage becomes a disadvantage 
until, when the competitive haul is entirely on the home road, 
any increase in distance becomes a net loss without any com- 
pensation. It is therefore advantageous for a road to be a 
short link in a long competitive route; an increase in that link 



554 RAILROAD CONSTRUCTION. § 502. 

will be financially advantageous; if the total length is less than 
that of the competing line, the advantage is still greater, for 
then the rate received per mile will be greater. 

502. Effect of the variations in the length of haul and the 
classes of the business actually done. The above distances 
refer to particular lengths of haul and are not necessarily the 
total lengths of the road. Each station on the road has 
traffic relations with an indefinite number of traffic points 
all over the country. The traffic between each station on 
the road and any other station in the country between which 
traffic may pass therefore furnishes a new combination, the 
effect of which will be an element in the total effect of a 
change of distance. In consequence of this, any exact solution 
of such a problem becomes impracticable, but a sufficiently 
accurate solution for all practical purposes is frequently ob- 
tainable. For it frequently happens that the great bulk of a 
road's business is non-competitive, or, on the other hand, it 
may be competitive-through, and that the proportion of one 
or more definite kinds of traffic is so large as to overshadow 
the other miscellaneous traffic. In such cases an approximate 
but sufficiently accurate solution is possible. 

503. General conclusions regarding a change in distance. 
(a) In all non-competitive business (local and through) the 
added distance is actually profitable. Sometimes practically- 
all of the business of the road is non-competitive ; a considerable 
proportion of it is always non-competitive. 

(b) When the competitive local business is very large and the 
competitive through business has a very large average home 
haul compared with the foreign haul, the added distance is 
a source of loss. Such situations are unusual and are generally 
confined to trunk lines. 

(c) The above may be still further condensed to the general 
conclusion that there is always some compensation for the added 
cost of operating an added length of line and that it frequently 
is a source of actual profit. 

(d) There is, however, a limitation which should not be lost 
sight of. The above argument may be carried to the logical 
conclusion that, if added distance is profitable, the engineer 
should purposely lengthen the line. But added distance means 
added operating expenses. A sufficient tariff to meet these is a 



§ 504. DISTANCE. 555 

traffic. It is contrary to public policy to burden a community 
with, an avoidable expense. But, on the other hand, a railroad 
is not a charitable organization, but a money-making enter- 
prise, and cannot be expected to unduly load up its first cost 
in order that subsequent operating expenses may be unduly 
cheapened and the tariff unduly lowered. A common reason 
for increased distance is the saving of the first cost of a very 
expensive although shorter line. 

(e) Finally, although there is a considerable and uncom- 
pensated loss resulting from curvature and grade which will 
justify a considerable expenditure to avoid them, there is by 
no means as much justification to incur additional expenditure 
to avoid distance. Of course needless lengthening should be 
avoided. A moderate expenditure to shorten the line may be 
justifiable, but large expenditures to decrease distance are 
never justifiable except when the great bulk of the traffic is 
exceedingly hea\^ and is competitive. 

504. Justification of decreasing distance to save time. It 
should be recalled that the changes which an engineer may 
make which are physically or financially possible will ordi- 
narily have but little effect on the time required for a trip^ 
The time which can thus be saved will have practically no value 
for the freight business — at least any value which would justify 
changing the route. When there is a large directly competitive 
passenger traffic between two cities (e.g. New York to Phila- 
delphia) a difference of even 10 minutes in the tirne required 
for a run might have considerable financial importance, but 
such cases are comparatively rare. It may therefore be con- 
cluded that the value of the time saved by shortening distance 
will not ordinarily be a justification for increased expense to 
accomplish it. 

505. Effect of change of distance on the business done. 
The above discussion is based on the assumption that the busi- 
ness done is unaffected by any proposed change in distance. 
If a proposed reduction in distance involves a loss of business 
obtained, it is almost certainly unwise. But if by increasing 
the distance the original cost of the road is decreased (because 
the construction is of less expensive character), and if the receipts 
are greater, and are increased still more by an increase in busi- 
ness done, then the change is probably wise. While it is almost 
impossible in a subject of such complexity to give a general 



556 RAILROAD CONSTRUCTION. § 505. 

rule, the following is generally safe : Adopt a route of such length 
that the annual traffic per mile of line is a maximum. This 
statement raay be improved by allowing the element of original 
cost to enter and say, adopt a route of such length that the annual 
traffic per mile of line divided by the average cost per mile is 
a maximum. Even in the above the operating cost per mile, 
as affected by the curvature and grades on the various routes, 
does not enter, but any attempt to formulate a general rule 
which would allow for variable operating expenses would evi- 
dently be too complicated for practical application. 



CHAPTER XXII. 

CURVATURE. 

506. General objections to curvature. In the popular mind 
curvature is one of the most objectionable features of railroad 
alinement. The cause of this is plain. The objectionable 
qualities are on the surface, and are apparent to the non-tech- 
nical mind. They may be itemized as follows: 

1. Curvature increases operating expenses by increasing (a) 
the required tractive force, (b) the wear and tear of roadbed 
and track, (c) the w^ear and tear of equipment, and (d) the 
required number of track- walkers and watchmen. 

2. It may affect the operation of trains (a) by limiting the 
length of trains, and (b) by preventing the use of the heaviest 
tj^pes of engines. 

3. It may affect travel (a) bj'' the difficulty of making time, 
(b) on account of rough riding, and (c) on account of the appre- 
hension of danger. 

4. There is actually an increased danger of collision, derail- 
ment, or other form of accident. 

Some of these objections are quite definite and their true 
value may be computed. Others are more general and vague 
and are usually exaggerated. These objections will be dis- 
cussed in inverse order. 

507. Financial value of the danger of accident due to curva- 
ture. At the outset it should be realized that in general the 
problem is not one of curvature vs. no curvature, but simply 
sharp curvature vs. easier curvature (the central angle remain- 
ing the same), or a greater or less percentage of elimination 
of the degrees of central angle. A straight road between ter- 
mini is in general a financial (if not a physical) impossibility. 
The practical question is then, how much is the financial value 
of such diminution of danger that may result from such elimi- 
nations of curvature as an engineer is able to make? 

557 



558 RAILROAD^CONSTRUCTION. § 508. 

In the year 1898 there were 2228 railroad accidents reported 
by the Railroad Gazette, whose Ksts of all accidents worth re- 
porting are very complete. Of these a very large proportion 
clearly had no relation whatever to curvature. But suppose 
we assume that 50% (or 1114 accidents) were directly caused 
by curvature. Since there are approximately 200,000 curves 
on the railroads of the country, there was on the average an 
accident for every 179 curves during the year. Therefore* we 
may say, according to the theory of probabilities, that the 
chances are even that an accident may happen on any particular 
curve in 179 years. This assumes all curves to be equally danger- 
ous, which is not true, but we may temporarily consider it to be 
true. If, at the time of the construction of the road, $1.00 were 
placed at compound interest at 5% for 179 years, it would pro- 
duce in that time $620.89 for each dollar saved, wherewitti to pay 
all damages, while the amount necessary to eliminate that cur- 
vature, even if it were possible, would probably be several thou- 
sand dollars. The number of passengers carried one mile for 
one killed in 1898-99 was 61,051,580. If a passenger were to 
ride continuously at the rate of sixty miles per hour, day and 
night, year after year, he would need to ride for more than 116 
years before he had covered such a mileage, and even then the 
probabilities of his death being due to curvature or to such a 
reduction of curvature as an engineer might accomplish are 
very small. Of course particular curves are often, for special 
reasons, a source of danger and justify the employment of 
special watchmen. They would also justify very large expen- 
ditures for their elimination if possible. But as a general 
proposition it is^evidently impossible to assign a definite money 
value to the danger of a serious accident happening on a par- 
ticular curve which has no special elements of danger. 

Another element of safety on curved track is that trait of 
human nature to exercise greater care where the danger is more 
apparent. Many accidents are on record which have been 
caused by a carelessness, of locomotive engineers on a straight 
track when the extra watchfulness usually observed on a curved 
track would hav^e avoided them. 

508. Effect of curvature on travel, (a) Difficulty in making 
time. The growing use of transition curves has largely elimi- 
nated the necessity for reducing speed on curves, and even when 
the speed is reduced it is done so easily and quickly by means 



§509. ' CURVATURE. 559 

of air-brakes that but little time is lost, If two parallel lines 
were competing sharply for passenger traffic, the handicap of 
sharp curvature on one road and easy curvature on the other 
might have a considerable financial value, but ordinarily the 
mere reduction of time due to sharp curvature will not have any 
computable financial value. 

(b) On account of rough riding. Again, this is much reduced 
by the use of transition curves. Some roads suffer from a gen- 
eral reputation for crookedness, but in such cases the excessive 
curvature is practically unavoidable. This cause probably 
does have some effect in influencing competitive passenger 
traffic. 

(c) On account of the apprehension of danger. This doubtless 
has its influence in deterring travel. The amount of its influence 
is hardly computable. When the track is in good condition 
and transition curves are used so that the riding is smooth, 
even the apprehension of danger will largely disappear. 

Travel is doubtless more or less affected by curvature, but 
it is impossible to say how much. Nevertheless the engineer 
should not ordinarily give this item any financial weight what- 
soever. Freight traffic (two thirds of the total) is unaffected 
by it. It chiefly affects that limited class of sharply competi- 
tive passenger traffic — a traffic of which most roads have not a 
trace. 

509. Effect on operation of trains, (a) Limiting the length 
of trains. When curvature actually limits the length of trains, 
as is sometimes true, the objection is valid and serious. But 
this can generally be avoided. If a curve occurs on a ruling 
grade without a reduction of the grade sufficient to compensate 
for the curvature, then the resistance on that curve will be a 
maximum and that curve will limit the trains to even a less 
weight than that which may be hauled on the ruling grade. 
In such cases the unquestionably correct policy is to "com- 
pensate for curvature, '^ as explained later (see §§510, 511), and 
not allow such an objection to exist. It is possible for curvature 
to limit the length of trains even without the effect of grade. 
On the Hudson River R. R. the total net fall from Albany to 
New York is so small that it has practically no influence in 
determining grade. On the other hand, a considerable portion 
of the route follows a steep rock}^ river bank which is so crooked 
that much curvature is unavoidable and very sharp curvature 



560 RAILROAD CONSTRUCTION. § 509. 

can only be avoided by very large expenditure. As a consequence 
sharp curvature has been used and the resistance on the curves 
is far greater than that of any fluctuations of grade which it 
was necessary to use. Or, at least, a comparatively small 
expenditure would safiice to cut down any grade so that its 
resistance would be less than that of some curve which could 
not be avoided except at an enormous cost. And as a resultj 
since the length of trains is really limited by curvature, minor 
grades of 0.3 to 0.5% have been freely introduced which 
might be removed at comparatively small expense The above 
case is verj?- unusual. Low grades are usually associated with 
generally level country where curvature is easily avoided — 
as in the Camden and Atlantic R. R. Even in the extreme 
case of the Hudson River road the maximum curvature is 
only equivalent to a comparatively low ruling grade. 

(b) Preventing the use of the heaviest types of engines. The 
validity of this objection depends somewhat on the degree of 
curvature and the detailed construction of the engine. While 
some types of engines might have difficulty on curves of ex- 
tremely short radius, yet the objection is ordinarily invalid. 
This will best be appreciated when it is recalled that the " Con- 
solidation" type was originally designed for use on the sharp 
curvature of the mountain divisions of the Lehigh Valley R. R., 
and that the type has been found so satisfactory that it has 
been extensively employed elsewhere. It should also be re- 
membered that during the Civil War an immense traffic daily 
passed over a hastily constructed trestle near Petersburg, Va., 
the track having a radius of 50 feet. As a result of a test made 
at Renovo on the Philadelphia and Erie R. R. by Mr. Isaac 
Dripps, Gen. Mast. Mech., in 1875,* it was claimed that a 
Consolidation engine encountered less resistance per ton than 
one of the "American" type. Whether the test was strictly 
reliable or not, it certainly demonstrated that there was no 
trouble in using these heavy engines on very sharp curvature, 
and w^e may therefore consider that, except in the most extreme 
cases, this objection has no force whatsoever. 

* * Seventh An. Rep. Am. Mast. Mech. Assn. 



§ 510. CURVATURE. 561 



COMPENSATION FOR CURVATURE. 

510. Reasons for compensation. The effect of curvature on 
a grade is to increase the resistance by an amount which is equiv- 
alent to a material addition to that grade. On n^nor grades 
the addition is of little importance, but when the grade is nearly 
or quite the ruling grade of the road, then the additional resist- 
ance induced by a curve will make that curve a place of maxi- 
mum resistance and the real maximum will be a 'Sdrtual grade" 
somewhat higher than the nominal maximum. If, in Fig. 211, 






^^^C Tang. D 
.--Tan'Sl-...:^^!^^ A^tJIiTgrade 



Fig. 211. 

Alsl represents an actual uniform grade consisting of tangents 
and curves, the ^'virtual grade" on curves at BC and DE may 
be represented by BC and DE. If BC and DE are very long, 
or if a stop becomes necessary on the curve, then the full dis- 
advantage of the curve becomes developed. If the whole grade 
may be operated without stoppage, then, as elaborated further 
in the next chapter, the whole grade may be operated as if equal 
to the average grade, AF, which is better than BCj although 
much worse than AN. The process of ''compensation" con- 
sists in reducing the grade on every curve by such an amount 
that the actual resistance on each curve, due to both curvature 
and grade, shall precisely equal the resistance on the tangent. 
The practical effect of such reduction is that the "virtual" grade 
is kept constant, while the nominal grade fluctuates. 

One effect of this is that (see Fig. 212) instead of accomplish- 
ing the vertical rise from A to G (i.e., HG) in the horizontal 
distance AH, it requires the horizontal distance AK, Such an 
addition to the horizontal distance can usually be obtained by 
proper development, and it should always be done on a ruling 



562 



RAILROAD CONSTRUCTION. 



§ 511 



grade. Of course it is possible that it will cost more to accom- 
plish this than it is worth, but the engineer should be sure of 
this before allowing this virtual increase of the grade. 




Fig. 212. 



European engineers early realized the significance of unre- 
duced curvature and the folly of laying out a uniform ruling 
grade regardless of the curvature encountered. Curve compen- 
sation is now quite generally allowed for in this country, but 
thousands of miles have been laid out without any compensa- 
tion. A very common limitation of curvature and grade has 
been the alliterative figures 6° curvature and 60 feet per mile 
of grade, either singly or in combination. Assuming that the 
resistance on a 6° curve is equivalent to a 0.3% grade (15.84 feet 
per mile), then a 6° curve occurring on a 60-foot grade would 
develop more resistance than a 75-foot grade on a tangent. 
The *' mountain cut-off" of the Lehigh Valley Railroad near 
Wilkesbarre is a fine example of a heavy grade compensated 
for curvature, and yet so laid out that the virtual grade is uni- 
form from bottom to top, a distance of several miles. 

511. The proper rate of compensation. This evidently is the 
rate of grade of which the resistance just equals the resistance 
due to the curve. But such resistance is variable. It is greater 
as the velocity is lower ; it is generally about 2 lbs. per ton 
(equivalent to a 0.1% grade) per degree of curve when starting 
a train. On this account, the compensation for a curve which 
occurs at a known stopping-place for the heaviest trains should 
be 0.1% per degree of curve. The resistance is not even strictly 
proportional to the degree of curvature, although it is usually 
considered to be so. In fact most formula) for curve resistance 
are based on such a relation. But if the experimentally deter- 
mined resistances for low curvatures are applied to the excessive 
curvature of the New York Elevated road, for example, the 



§ 512. CURVATURE. 563 

rules become ridiculous. On this account the compensation 
per degree of curve may be made less on a sharp curve than on 
an easy curve. The compensation actually required for very 
fast trains is less than for slow trains, say 0.02 or 0.03% per 
degree of curve; but since the comparatively slow and heavy 
freight trains are the trains which are chiefly limited by ruling 
grade, the compensation must be made with respect to those 
trains. From 0.04 to 0.05% per degree is the rate of compen- 
sation most usually employed for average conditions. Curves 
which occur below a known stopping-place for all trains need 
not be compensated, for the extra resistance of the curve will 
be simply utilized in place of brakes to stop the train. If a curve 
occurs just above a stopping-place, it is very serious and should 
be amply compensated. Of course the down-grade traffic need 
not be considered. 

It sometimes happens that the ordinary rate of compensa- 
tion will consume so much of the vertical height (especially if 
the curvature is excessive) that a steeper through grade must 
be adopted than was first computed, and then the trains might 
stall on the tangents rather than on the curves. In such cases 
a slight reduction in the rate of compensation might be justi- 
fiable. 

The following rules have been approved by the Amer. Rwy. 
Eng. Assoc. 

1. Compensate .03% per degree (a) when the length of curve 
is less than haK the length of the longest train; (6) when a curve 
occurs within the first 20 feet of rise of a grade; (c) when cur- 
vature is in no sense limiting. 

2. Compensate .035% per degree (a) when curves are be- 
tween one-haK and three-quarters as long as the longest train; 
(6) when the curve occurs between 20 feet and 40 feet of rise 
from the bottom of the grade. 

3. Compensate .04% per degree (a) where the curve is habit- 
ually operated at low speed; (6) where the length of the curve 
is longer than three-quarters of the length of the longest train; 
(c) where elevation is excessive for freight trains; {d) at all places 
where curvature is likely to be limited. 

4. Compensate .05% per degree wherever the loss of elevation 
can be spared. 

512. The limitations of maximum curvature. What is the 
maximum degree of curvature which should be allowed on any 



564 RAILROAD CONSTRUCTION. § 512, 

road? It has been sliown that sharp curvature does not prevent 
the use of the heaviest types of engines, and although a sharp 
curve unquestionably increases operating expenses, the increase 
is but one of degree with hardly any definite limit. The general, 
character of the country and the gross capital available (or 
the probable earnings) are generally the true criterions. 

A portion of the road from Denver to Leadville, Col., is an 
example of the necessity of considering sharp curvature. The 
traffic that might be expected on the line was so meagre and 
yet the general character of the country was so forbidding 
that a road built according to the usual standards would have 
cost very much more than the traffic could possibly pay for. 
The line as adopted cost about $20,000 per mile, and yet in a 
stretch of 11.2 miles there are about 127 curves. One is a 25° 
20' curve, twenty-four are 24° curves, twenty-five are 20° curves, 
and seventy-two are sharper than 10°. If 10° had been made 
the limit (a rather high limit according to usual ideas), it is 
probable that the line would have been found impracticable 
(except with prohibitive grades) unless four or five times as 
much per mile had been spent on it, and this would have ruined 
the project financially. 

For many years the main-line traffic of the Baltimore and 
Ohio R. R. has passed over a 300-foot curve (19° 10') and a 
400-foot curve (14° 22') at Harper's Ferry. A few years ago 
some reduction was made in this by means of a tunnel, but 
the fact that such a road thought it wise to construct and operate 
such curves (and such illustrations on the heaviest-traffic roads 
are quite common) shows how foolish it is for an engineer to 
sacrifice money or (which is much more common) sacrifice 
gradients in order to reduce the rate of curvature on a road 
which at its best is but a second- or third-class road. 

Of course such belittling of the effects of curvature may 
be (and sometimes is) carried to an extreme and cause an engi- 
neer to fail to give to curvature its due consideration. Degrees 
of central angle should always be reduced by all the ingenuity 
of the engineer, and should only be limited by the general rela- 
tion between the financial and topographical conditions of the 
problem. Easy curvature is in general better than sharp curva- 
ture and should be adopted when it may be done at a small 
financial sacrifice, especially since it reduces distance generally 
and may even cut down the initial cost of that section of the 



§ 512. CURVATURE. 565 

road. But large financial expenditures are rarely, if ever, jus- 
tifiable where the net result is a mere increase in radius without 
a reduction in central angle. An analysis of the changes which 
have been so extensively made during late years on the Penn. 
R. R. and the N. Y., N. H. & H. R. R. will show invariably a 
reduction of distance, or of central angle, or both, and perhaps 
incidentally an increase in radius of curvature. There are but 
few, if any, cases where the sole object to be attained by the 
improvement is a mere increase in radius. 

The requirements of standard M. C. B. car-couplers have 
virtually placed a limitation on the radius on account of the 
corners of adjacent cars striking each other on very sharp 
curves. This limitation has been crystallized into a rule on 
the P. R. R. that no curve, even that of a siding, can have a 
less radius than 175 feet, which is nearly the radius of a 33° 
curve. Of course only the most peremptory requirements of 
yard work would justify the employment of such a radius. 



CHAPTER XXIII. 

GRADE. 

513. Two distinct effects of grade. The effects of grade on 
train expenses are of two distinct kinds; one possible effect is 
very costly and should be limited even at considerable expen- 
diture; the other is of comparatively little importance, its cost 
being slight. As long as the length of the train is not limited, 
the occurrence of a grade on a road simply means that the engine 
is required to develop so many foot-pounds of work in raising 
the train so many feet of vertical height. For example, if a 
freight train weighing 600 tons (1,200,000 lbs.) climbs a hill 
50 feet high, the engine performs an additional work of creating 
60,000,000 foot-pounds of potential energy. If this height is 
surmounted in 2 miles and in 6 minutes of actual time (20 
miles per hour), the extra work is 10,000,000 foot-pounds per 
minute, or about 303 horse-power. But the disadvantages of 
such a rise are always largely compensated. Except for the fact 
that one terminus of. a road is generally higher than the other, 
every up grade is followed, more or less directly, by a down grade 
which is operated partly by the potential energy acquired during 
the previous climb. But when we consider the trains running 
in both directions even the difference of elevation of the termini 
is largely neutralized. If we could eliminate frictional resist- 
ances and particularly the use of brakes, the net effect of minor 
grades on the operation of minor grades in both directions would 
be zero. Whatever was lost on any up grade would be regained 
on a succeeding down grade, or at any rate on the return trip. 
On the very lowest grades (the limits of which are defined later) 
we may consider this to be literally true, viz., that nothing is 
lost by their presence ; whatever is temporarily lost in climbing 
them is either immediately regained on a subsequent light down 
grade or is regained on the return trip. If a stop is required 
at the bottom of a sag, there is a net and uncompensated loss 

of energy. 

566 



§ 514. GRADE. 567 

On the other hand, if the length of trains is limited by the 
grade, it will require more trains to handle a given traffic. The 
receipts from the traffic are a definite sum. The cost of hand- 
ling it will be nearly in proportion to the number of trains. 
Assume that by lowering the rate of ruling grade it becomes pos- 
sible to handle such an increased number of cars with one engine 
that four engines can haul as many cars on the reduced grade as 
five engines could haul on the higher grade and at a cost but 
slightly more than four-fifths as much. The effect of this on 
dividends may readily be imagined. 

514. Application to the movement of trains of the laws of 
accelerated motion. When a train starts from rest and acquires 
its normal velocity, it overcomes not only the usual tangent 
resistances (and perhaps curve and grade resistances), but it 
also performs work in storing into the train a vast fund of kinetic 
energy. This work is not lost, for every foot-pound of such 
energy may later be utilized in overcoming resistances, pro- 
vided it is not wasted by the action of train-brakes. If for a 
moment we consider that a train runs without any friction, 
then, when running at a velocity of v feet per second, it possesses 
a kinetic energy which would raise it to a height h feet, when 

h = — , in which g is the acceleration of gravity =32.16. Assum- 

ing that the engine is exerting just enough energy to overcome 
the frictional resistances, the train would climb a grade until the 
train was raised h feet above the point where its velocity was n 
When it had climbed a height h^ (less than h) it would have a 
velocity Vi=\/2g{h — h'), As a numerical illustration, assume 

v = 30 miles per hour = 44 feet per second. Then /i = — = 30. 1 feet, 

and assuming that the engine was exerting just enough force 
to overcome the rolling resistances on a level, the kinetic 
energy in the train would carry it for two miles up a grade of 
15 feet per mile, or half a mile up a grade of 60 feet per mile. 
When the train had climbed 20 feet, there would still be 10.1 

feet left and its velocity would be t^i = \/2^(10.1) =25.49 feet 
per second = 17.4 miles per hour. These figures, however, must 
be slightly modified on account of the weight and the revolving 
action of the wheels, which form a considerable percentage 
of the total weight of the train. When train velocity is being 



568 RAILEOAD CONSTRUCTION. § 515. 

acquired, part of the work done Is spent in imparting the energy 
of rotation to the driving-wheels and various truck-wheels of 
the train. Since these wheels run on the rails and must turn 
as the train moves, their rotative kinetic energy is just as effect- 
ive — as far as it goes — in becoming transformed back into 
useful work. The proportion of this energy to the total kinetic 
energy has already been demonstrated (see Chapter XVI, 
§ 435). The value of this correction is variable, but an average 
value of 5% has been adopted for use in the accompanying 
tabular form (Table XLII), in which is given the corrected 
*' velocity head " corresponding to various velocities in miles 
per hour. The table is computed from the following formula : 

XT , ., , 1 !;2 in ft. per sec. 2.151F2in m.per h. ^^^^,,,,, 

Velocity head = -^^ = = 0.0334472 

64.32 64.32 

adding 5%for the rotative kinetic energy of the wheels, 0.00167F2 
The corrected velocity head therefore equals 0.0351 17^ 

Part of the figures of Table XLII were obtained by inter- 
polation and the final hundredth may be in error by one unit, 
but it may readily be shown that the final hundredth is of no 
practical importance. It is also true that the chief use made 
of this table is with velocities much less than 45 miles per hour. 
Corresponding figures may be obtained for higher velocities, if 
desired, by multiplying the figure for half the velocity by four. 

515. Construction of a virtual profile. The following simple 
demonstration will be made on the basis that the ordinary 
tractive resistances and also the tractive force of the locomo- 
tive are independent of velocity. For a considerable range of 
velocity which includes the most common freight-train velocities 
the first assumption is practically true; the second assumption 
is so nearly true under certain possible operative conditions that it 
may serve as a prehminary to the more accm-ate solution. It may 
best be illustrated by considering a simple numerical example. 

Assume that Fig. 213 shows the profile of a section of road and 
that the grade of AE is 0.40%, which is 21.12 feet per mile. 
Assume also that a freight engine is climbing up the grade at a 
uniform velocity of 20 miles per hour. But since the train is 
moving at 20 miles per hour it has a kinetic energy corresponding 
to a velocity of 14.05 feet (see Table XLII). At A it encounters a 
down-grade of 0,20 per cent, which is 1500 feet long. Although 



§415. 



GRADE. 



569 



AB has a down-grade of only 0.20%, its grade with respect to 

the up-grade of AE (0.40%) is 

0.60%. Therefore B is 9.00 feet 

below B\ Since the work done 

by the engine would have carried 

the train up to the point B' with 

a velocity of 20 miles per hour, 

the virtual drop of 9 feet will 

increase the velocity head from 

14.05 feet to 23.05 feet, which 
corresponds to the velocity of 

25.6 miles per hour, and this 
will actually be the velocity of 
the train at the point B. At B 
the grade changes to a 1.0% up- 
grade for a distance of 2300 feet. 

The approach of the grade BC 
to the grade B'C is at the rate 
of 1.0-0.4 = 0.6% and therefore, 
the point C will be reached in 
1500 feet. In the remaining 800 
feet the hne will chmb to D, 
which is 4.8 feet above D\ Al- 
though at B the train is moving 
at the rate of 25.6 miles per 
hour and the engine is working 
at such a rate that it will carry 
the train up a 0.4% grade, yet 
when cHmbing up a 1.0% grade 
it consumes its kinetic energy in 
overcoming the additional grade. 
When it reaches C, it has lost 
the additional kinetic energy 
which it gained from A to B, and 
as it continues it loses even more. 
When it reaches D, it has lost 4.8 
feet more and its velocity head 
is reduced to 14.05 -4.8 = 9.25 ft., 
which corresponds to a velocity 
of 16.2 miles per hour. At D 
the grade changes to +0.1%. 




570 



RAILEOAD CONSTRUCTION. 



§515. 



TABLE XLII — VELOCITY HEAD (REPRESENTING THE KINETIC 
energy) of trains MOVING AT VARIOUS VELOCITIES. 



Vel. 






















mi. 
hr. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


5 


0.88 


0.91 


0.95 


0.99 


1.02 


1.06 


1.10 


1.14 


1.18 


1.22 


6 


1.26 


1.31 


1.35 


1.40 


1.44 


1.48 


1.53 


1.58 


1.62 


1.67 


7 


1.72 


1.77 


1.82 


1.87 


1.92 


1.97 


2.03 


2.08 


2.14 


2.19 


8 


2.25 


2.30 


2.36 


2.42 


2.48 


2.54 


2.60 


2.66 


2.72 


2.78 


9 


2.85 


2.91 


2.97 


3.04 


3.10 


3.17 


3.24 


3.30 


3.37 


3.44 


10 


3.51 


3.58 


3.65 


3.72 


3.79 


3.87 


3.95 


4.02 


4.10 


4.17 


11 


4.25 


4.33 


4.41 


4.49 


4.57 


4.65 


4.73 


4.81 


4.89 


4.97 


12 


5.06 


5.15 


5.23 


5.32 


5.41 


5.50 


5.58 


5.67 


5.75 


5.84 


13 


5.93 


6.02 


6.12 


6.21 


6.31 


6.40 


6.50 


6.59 


6.69 


6.78 


14 


6.88 


6.98 


7.08 


7.19 


7.29 


7.39 


7.49 


7.60 


7.70 


7.80 


15 


7.90 


8.00 


8.11 


8.22 


8.33 


8.44 


8.55 


8.66 


8.77 


8.88 


16 


8.99 


9.10 


9.21 


9.32 


9.43 


9.55 


9.67 


9.79 


9.91 


10.03 


17 


10.15 


10.27 


10.39 


10.51 


10.63 


10.75 


10.87 


10.99 


11.12 


11.25 


18 


11.38 


11.50 


11.63 


11.76 


11.89 


12.02 


12.15 


12.28 


12.41 


12.55 


19 


12.68 


12.81 


12.95 


13.08 


13.22 


13.35 


13.49 


13.63 


13.77 


13.91 


20 


14.05 


14.19 


14.33 


14.47 


14.61 


14.75 


14.89 


15.04 


15.19 


15.34 


21 


15.49 


15.64 


15.79 


15.94 


16.09 


16.24 


16.39 


16.54 


16.69 


16.84 


22 


17.00 


17.15 


17.30 


17.46 


17.62 


17.78 


17.94 


18.10 


18.26 


18.42 


23 


18.58 


18.74 


18.90 


19.06 


19.22 


19.38 


19.55 


19.72 


19.89 


20.06 


24 


20.23 


20.40 


20.57 


20.74 


20.91 


21.08 


21.25 


21.42 


21.59 


21.77 


25 


21.95 


22.12 


22.30 


22.48 


22.66 


22.84 


23.02 


23.20 


23.38 


23.56 


26 


23.74 


23.92 


24.10 


24.28 


24.46 


24.65 


24.84 


25.03 


25.22 


25.41 


27 


25.60 


25.79 


25.98 


26.17 


26.36 


26.55 


26.74 


26.93 


27.13 


27.33 


28 


27.53 


27.73 


27.93 


28.13 


28.33 


28.53 


28.73 


28.93 


29.13 


29.33 


29 


29.53 


29.73 


29.93 


30.13 


30.34 


30.55 


30.76 


30.97 


31.18 


31.39 


30 


31.60 


31.81 


32.02 


32.23 


32.44 


32.65 


32.86 


33.08 


33.30 


33.52 


31 


33.74 


33.96 


34.18 


34.40 


34.62 


34.84 


35.06 


35.28 


35.50 


35.72 


32 


35.95 


36.17 


36.39 


36.62 


36.85 


37.08 


37.31 


37.54 


37.77 


38.00 


33 


38.23 


38.46 


38.69 


38.92 


39.15 


39.38 


39.62 


39.86 


40.10 


40.34 


34 


40.58 


40.82 


41.06 


41.30 


41.54 


41.78 


42.02 


42.26 


42.51 


42.76 


35 


43.01 


43.26 


43.51 


43.76 


44.01 


44.26 


44.51 


44.76 


45.01 


45.26 


36 


45.51 


45.76 


46.01 


46.26 


46.52 


46.78 


47.04 


47.30 


47.56 


47.82 


37 


48.08 


48.34 


48.60 


48.86 


49.12 


49.38 


49.64 


49.91 


50.18 


50.45 


38 


50.72 


50.99 


51.26 


51.53 


51.80 


52.07 


52.34 


52.61 


52.88 


53.15 


39 


53.42 


53.69 


53.96 


54.23 


54.51 


54.79 


55.07 


55.35 


55.63 


55.91 


40 


56.19 


56.47 


56.75 


57.03 


57.31 


57.59 


57.87 


58.16 


58.45 


58.74 


41 


59.03 


59.32 


59.61 


59.90 


60.19 


60.48 


60.77 


61.06 


61.35 


61.64 


42 


61.94 


62.23 


62.52 


62.82 


63.12 


63.42 


63.72 


64.02 


64.32 


64.62 


43 


64.92 


65.22 


65.52 


65.82 


66.12 


66.43 


66.74 


67.05 


67.36 


67.67 


44 


67.98 


68.29 


68.60 


68.91 


69.22 


69.53 


69.84 


70.15 


70.46 


70.78 



Here we have the rather surprising condition that, although 
the grade is actually rising, it is virtually a down-grade under the 
given conditions, for the engine is working harder than is re- 
quired to run up merely a 0.1% grade and hence will gain in 
velocity. At E, a distance of 1600 feet from D, it reaches what 



§ 515. GRADE. 571 

would have been a uniform 0.4% grade from A to E and the 
grade continues at that rate. Although the train has actually 
cUmbed 1.6 feet from D to E^ it has virtually fallen the 4.8 feet 
between D and D\ and the velocity head has increased from its 
value of 9.25 feet at D to 14.05 feet, and its velocity is again 20 
miles per hour. The upper hhe represents the " virtual profile/' 
which may always be drawn by measuring off to the proper scale 
at every point an ordinate which is the velocity head at that 
point. Since the engine is working uniformly, the virtual profile 
is in this case a straight Hne. 

As another case, assume that a train is climbing the grade AE 
and exerting a pull just sufficient to maintain a constant velocity 
up that grade. Then A'B' (par- ^ ^f 

allel to AB) is the virtual profile, ^ ^r ^^ p 

AA' representing the velocity 
head. A stop being required at 
(7, steam is shut off and brakes 
are appHed at B, and the velocity ^^^ 214. 

head BB' reduces to zero at C. 

The train starts from C, and at D attains a velocity correspond- 
ing to the ordinate DD', At D the throttle may be shghtly 
closed so that the velocity will be uniform and the virtual grade 
is D'J^', paraUel to DE, 

From the above it may be seen that a virtual profile has the 
following properties: 

(a) When the velocity is uniform, the virtual profile is parallel 

with the actual. 

(6) When the velocity is increasing the profiles are separating; 
when decreasing the profiles are approaching. 

(c) When the velocity is zero the profiles coincide. 

\d) The virtual grade at any place is a measure of the work 
required of the engine beyond that required to overcome merely 
the tractive resistances. If it is horizontal it shows that the 
engine is doing nothing besides overcoming the tractive re- 
sistances. If it is upward and is uniform, as in Fig. 213, it 
shows that it is working uniformly and is storing in the train 
" potential " energy which may be utilized on the return trip 
if it is not utiHzed to overcome tractive resistance in moving 
down a succeeding down-grade. If it is downward, as from B' 
to C, Fig. 214, it shows that the train is giving up kinetic energy, 
probably consuming most of it in brakes, but utihzing some of it 



572 RAILROAD CONSTRUCTION. § 516. 

to furnish the tractive power to run from J5 to C and also to 
overcome the grade from B to C, 

516. Variation in draw-bar pull. The above demonstration 
has been made on the basis that the draw-bar pull is constant 
throughout. It is shown in Chapter XVIII that, when the 
engine is working at its full capacity the draw-bar pull decreases 
as the velocity increases^ which is chiefly due to the fact that if 
we attempt to use full stroke at 2 M or 3 M velocity the steam 
will be so rapidly exhausted from the boiler that the pressure will 
fall. Therefore the valves are set to cut off so as to use the 
steam expansively but as this reduces the average pressure in 
the cylinder, then (see Eq. 103), the tractive power must be less. 
The reduction of tractive power for several multiples of M is 
shown in Table XXXIX. For example, in the numerical prob- 
lem given above, and assuming the use of the Mikado engine 
whose characteristics have already been computed, the velocity 
at A =20 -i- 6. 167 = 3.25 M and the tractive power at this velocity 
is 49.23% of its power at M velocity. From the tabular form in 
§ 460 the draw-bar pull at 3.25 ikf -velocity may be found by 
interpolation to be 16587 lbs. Similarly at B the velocity is 
expected to be 25.6 m.p.h. =4.15 ikf, and then the tractive power 
is 38.48% and the draw-bar pull only 12484 lbs., about 75% 
of the pull at A. But since the draw-bar pull is so much reduced 
the velocity evidently would not be increased the theoretical 
amount due to the virtual drop BB\ On the other hand, when 
the train reaches D, where the velocity is supposed to be 16.2 
m.p.h. =2.62 M, the draw-bar pull would be 20144, which is over 
121% of the normal pull at 3.25 M velocity. The average pull 
between B and D is 16314 or within 2 % of the normal 16587. 
The average bdtween A and Ey assuming that the theoretical 
velocities at B and D were actually realized, would be about 2% 
below the assumed pull at A. The 3000-foot sag ABC will be 
passed in 90 seconds and no very great reduction in boiler power 
could take place in that time , especially if the fireman used extra 
care to maintain the pressure. Investigators have declared 
that tests of trains, with a dynamometer car between the tender 
and cars, have shown a practically uniform draw-bar pull, with 
an unchanged throttle and with velocities varying substantially 
on the principles indicated above. If the sag ABC is excessively 
long or deep the reduction of tractive force with increased 
velocity would be so great that the error of the method would be 



§517 GRADE. 573 

too great for practical use. But experience has proven that 
for ordinary cases the method can be used with substantial 
accuracy. 

517. Use, value, and possible misuse. The essential feature 
respecting grades is the demand on the locomotive. From the 
foregoing it may readily be seen that the ruling grade of a road 
is not necessarily the steepest nominal grade. When a grade 
may be operated by momentum, i.e., when every train has an 
opportunity to take " a run at the hill,'' it may become a very 
harmless grade and not Hmit the length of trains, while another 
grade, actually much less, which occurs at a stopping-place 
for the heaviest trains, will require such extra exertion to get 
trains started that it may be the worst place on the road. 
Therefore the true way to consider the value of the grade at 
any critical place on the road is to construct a virtual profile 
for that section of the road. The required length of such a 
profile is variable, but in general may be said to be Umited by 
points on each side of the critical section at which the velocity 
is definite, as at a stopping-place (velocity zero), or a long heavy 
grade where it is the minimum permissible, say M miles per 
hour. 

Since the velocities of different trains vary, each train will 
have its own virtual profile at any particular place. Fast 
passenger trains are less affected than slow freight trains. The 
requirement of high average speed necessitates the use of power- 
ful engines, and grades which would stall a heavy freight will 
only cause a momentary and harmless reduction of speed of 
the fast passenger train. 

A possible misuse of virtual profiles lies in the chance that a 
station or railroad grade crossing may be subsequently located 
on a heavy grade that was designed to be operated by momen- 
tum. But this should not be used as an argument against the 
employment of a virtual profile. The virtual profile shows the 
actual state of the case and only points out the necessity, if an 
unexpected requirement for a full stoppage of trains at a critical 
point has developed, of changing the location (if a station), or 
of changing the grade by regrading or by using an overhead 
crossing. 

518. Undulatory grades. Advantages. Money can generally 
be saved by adopting an actual profile which is not strictly 
imiform — the matter of compensation for curvature being here 



574 KAILROAD CONSTRUCTION. § 518. 

ignored. Its effect on the operation of trains is harmless pro- 
vided the sag or hump is not too great. In Fig. 215 the undu- 
latory grade may actually be operated as a uniform grade AG. 
The sag at C must be considered as a sag, even though BC is actu- 
ally an up grade. But the engine is supposed to be working 
hard enough to carry a train at uniform velocity up a grade AG. 
Therefore it gains in velocity from B to C, and from C to D loses 
an equal amount. It may even be proven that the time re- 




Fig. 215. 



quired to pass the sag will be slightly less than the time required 
to run the uniform grade. 

Disadvantages. The hump at F is dangerous in that, if the 
velocity at E is not equal to that corresponding to the extra 
velocity-head ordinate at F, the train will be stalled before 
reaching F. In practice there should be considerable margin. 
Any train should have a velocity of at least M (see § 455) 
in passing any summit. An extra heavy head wind, slippery 
rails, etc., may use up any smaller margin and stall the train. 
If the grade AG is a ruling grade, then no bump should be allowed 
under any circums tances. For the heaviest trains are supposed 
to be so made up that the engine will just haul them up the 
ruling grades — of course with some margin for safety. Any 
increase of this grade, however short, would probably stall the 
train. 

Safe limits. Since over 99.4% of all freight cars are now 
equipped with train brakes and automatic couplers, there is 
not now the limitation which formerly existed about operating 
freight trains at high speeds, but it may frequently happen 
that it would be undesirable to run a freight train through a 
deep sag at such a velocity as would result from a free run and 
it would therefore become necessary to use brakes, which will 
add a distinct element of cost. 

The term " safe limits'' as used here, refers to the Umits within 



§ 519. GRADE. ' 575 

which a freight train may be safely operated without the appK- 
cation of brakes or varying the work of the engine. Of course 
much greater undulations are frequently necessary and are 
safely operated, but it should be remembered that they add a 
distinct element to the cost of operating trains and that they 
must not be considered as harmless or that they should be 
introduced unless really necessary. 



RULING GRADES. 

519. Definition. Ruling grades are those which limit the 
weight of the train of cars which may be hauled by one engine. 
The subject of '' pusher grades " will be considered later. For 
the present it will suffice to say that on all well-designed roads 
the large majority of the grades on any one division are kept 
below some hmit which is considered the ruhng grade. If a 
heavier grade is absolutely necessary no special expense will 
be made to keep it below a rate where the resistance is twice 
(or possibly three times) the resistance on the ruhng grade, and 
then the trains can be hauled unbroken up these few special 
grades with the help of one (or two) pusher engines. So far 
as limitation of train length is concerned, these pusher grades 
are no worse than the regular ruling grades and, except for the 
expense of operating the pusher engines (which is a separate 
matter), they are not appreciably more expensive than any 
ruhng grade. As before stated, the engineer cannot alter very 
greatly the ruling grade of the road when the general route 
has been decided on. He may remove sags or humps, or 
he may lower the natural grade of the route by development 
in order to bring the grade within the adopted hmit of ruhng 
grade. 

520. Choice of ruling grade. It is of course impracticable for 
an engine to drop off or pick up cars according to the grades 
which may be encountered along the hne. A train load is made 
up at one terminus of a division and must rim to the other 
terminus. Excluding from consideration any short but steep 
grades which may always be operated by momentum, and also 
all pusher grades, the maximum grade on that division is the 
ruhng grade. 

It will evidently be economy to reduce the few grades which 
naturally would be a Httle higher than the great majority of 



576 RAILROAD CONSTRUCTION. § 521. 

others until such a large amount of grade is at some uniform 
limit that a reduction at all these places would cost more than 
it is worth. The precise determination of this limit is prac- 
tically impossible, but an approximate value may be at once 
determined from a general survey of the route. The distance 
apart of consecutive control points (see § 18) into their differ- 
ence of elevation is a first trial figure for the rate of the grade. If 
a grade even approximately uniform is impossible owing to the 
elevation of intervening ground, the worst place may be selected 
and the natural grade of that part of the route determined. 
If this grade is much steeper than the general run of the natural 
grades, it may be policy to reduce it by development or to boldly 
plan to operate that place as a pusher grade. The choice of 
possible grades thus has large limitations, and it justifies very 
close study to determine the best combination of grades and 
pusher grades. When the choice has narrowed down to two 
limits, the lower of which may be obtained by the expenditure 
of a definite extra sum, the choice may be readily computed, as 
will be developed. 

521. Maximum train load on any grade. The Mikado loco- 
motive, whose characteristics were analyzed in Chapter XVIII^ 
has a net pulling power at the rim of the drivers, at M velocity, , 
of 35758 lbs. which is 23.3% of 153,200, the weight on the drivers. 
This percentage is slightly over -^. Increasing the percentage 
6% on account of increased power at starting we have 24.7% 
or nearly -|-. On the other hand, wet, slippery rails may render 
the adhesion as low as 3- and thus limit the actual drawing power. 
Although the real power of a locomotive depends on the velocity 
at which it seems desirable to run, the maximum tractive power 
at ** ikf " velocity can always be approximately estimated as 
^ of the weight on the drivers. In Table XLIII are given the 
weights of several types of locomotives together with their 
tractive powers at three ratios of adhesion. These values are 
useful when the more elaborate method detailed in Chapter 
XVIII is not considered necessary. 

The maximum train load on any grade depends on the character 
and number of the cars, as well as on their gross weight. The 
approximate resistance of cars is given by Eq. 121 as -R = 2.2 t 
+122 n. Applying this to a steel box-car weighing 40 tons net 
and loaded with 100,000 lbs., the resistance would be 310 lbs. or 
3.55 lbs. per ton. Empty, the resistance would be 5.25 lbs. per 



§521. 



GRADE. 



577 



TABLE XLIII. — TRACTIVE POWER OF VARIOUS TYPES OF STANDARD- 
GAUGE LOCOMOTIVES AT VARIOUS RATES OF ADHESION. 



Type of locomotive. 


Total weight 

of engine 
and tender. 


Weight 

of 
engine 
only. 


Weight 
on the 
drivers. 


Tractive power when 

ratio of adhesion 

is 




Lbs. 


Tons. 


I 


4% 


i 


Atlantic, 4-4-2 

Atlantic, 4-4-2, four 
cylinder compound 

Pacific, 4-6-2 

Pacific, 4-6-2 

Ten-wheel, 4-6-0. . . 

Prairie, 2-6-2 

Consolidation, 2-8-0 
Consolidation, 2-8-0 
Mikado, 2-8-2 


340,000 

368,800 
343,600 
403,780 
321,000 
366,500 
214,000 
366,700 
405,500 


170.0 

184.4 
171.8 
201.9 
160.5 
183.2 
107.0 
183.3 
202.7 


199,400 

206,000 
218,000 
226,700 
201,000 
212,500 
120,000 
221,500 
259,000 


105,540 

115,000 
142,000 
151,900 
154,000 
154,000 
106,000 
197,500 
196,000 


26,385 

28,750 
35,500 
37,975 
38,500 
38,500 
26,500 
49,375 
40,000 


23,740 

25,875 
31,950 
34,180 
34,650 
34,650 
23,850 
44,440 
44,100 


21,100 

23,000 
28,400 
30,380 
30,800 
30,800 
21,200 
39,500 
39,200 



ton. Applying the formula to a wooden box-car weighing 15 
tons net and carrying 60000 lbs., the resistances for the car full 
and empty would be 4.9 and 10.3 lbs. per ton, respectively. 
Three and 10 pounds per ton are the ordinary extremes. Al- 
though resistances of less than 3 lbs. per ton have been 
measured for whole trains of heavy-loaded coal cars, there 
are usually enough light-weight cars and empties in a train to 
increase the average per ton resistance to perhaps 6 lbs. per ton. 
The Mikado locomotive, referred to above, had a draw-bar 
pull on a level at M velocity (6.167 m.p.h.) of 35419 lbs. How 
much of a load could it draw up a 1.2% grade at M velocity? 
Assume that the cars have a weight and character such that 
the average resistance would be 6 lbs. per ton. The grade 
resistance of the locomotive is 315,000 X. 012 = 3780, which sub- 
tracted from 35419 leaves 31639, the pull available for the cars. 
Then, calling T the tons weight of cars 



and 



31639 = 6r+(20Xl.2Xr)=30r, 
r = 1054. 



This allows only 6% margin for extra starting resistance if it 
should be necessary to stop and start on the grade, and makes 
no allowance for acceleration. It represents a* limit, for the 
given condition, which would probably not be used. 



578 RAILROAD CONSTRUCTION. § 522. 

522. Proportion of the traffic affected by the ruling grade. 

Some very light traffic roads are not so fortunate as to have 
a traffic which will be largely affected by the rate of the ruHng 
grade. When passenger traffic is light, and when, for the sake 
of encouraging traffic, more frequent trains are run than are 
required from the standpoint of engine capacity, it may happen 
that no passenger trains are really limited by any grade on the 
road — i.e., an extra passenger car could be added if needed. 
The maximum grade then has no worse effect (for passenger 
trains) than to cause a harmless reduction of speed at a few points. 
The local freight business is frequently affected in practically 
the same way. All coal, mineral, or timber roads are affected 
by the rate of ruling grade as far as such traffic is concerned. 
Likewise the through business in general merchandise, especially 
of the heavy traffic roads, will generally be affected by the rate 
of ruling grade. Therefore in computing the effect of ruHng 
grade, the total number of trains on the road should not ordi- 
narily be considered, but only the trains to which cars are added, 
until the limit of the hauling power of the engine on the ruling 
grades is reached. 

PUSHER GRADES. , 

523. General principles underlying the use of pusher engines. 

On nearly all roads there are some grades which are greatly 
in excess of the general average rate of grade, and these heavy 
grades cannot usually be materially reduced without an expend- 
iture which is excessive and beyond the financial capacity 
of the road. If no pusher engines are used, the length of all 
heavy trains is limited by these grades. The financial value 
of the reduction of such ruling grades has already been shown. 
But in the operation of pusher grades there is incurred the 
additional cost of pusher-engine service, for a pusher engine 
must run twice over the grade for each train which is assisted. 
It is possible for this additional expense to equal or even exceed 
the advantage to be gained. In any case it means the adoption 
of the lesser of two evils, or the adoption of the more economical 
method. The work of overcoming the normal resistances of so 
many loaded cars over so many miles of track and of lifting so 
many tons up the gross differences of elevation of predetermined 
points of the line is approximately the same whatever the exact 



§ 524. GRADE. 579 

route, and if the grades are so made that fewer engines working 
more constantly can accomplish the work as well as more engines 
which are not hard worked for a considerable proportion of the 
time, the economy is very apparent and unquestionable. Wel- 
lington expresses it concisely : ^* It is a truth of the first importance 
that the objection to high gradients is not the work which the 
engines have to do on them, but it is the work which they do 
not do when they thunder over the track with a Hght train be- 
hind them, from end to end of a division, in order that the 
needed power may be at hand at a few scattered points where 
alone it is needed. '* 

524. Balance of grades for pusher service. Assume that both 
pusher and through engines are the Mikado engine with dimen- 
sions already given (§ 453), and that they will be operated at 
their most effective velocity, ikf = 6.167 m.p.h., and that the 
effective draw-bar pull of each is 37190-1771=35419 lbs., 
less the locomotive grade resistance, which on a 1.9% grade 
is 20X1.9X157.5 = 5985 lbs. The net draw-bar pull on this 
grade for each engine is, therefore, 29434 lbs. Assume that 
the train considered is made up of coal cars weighing 40000 lbs. 
net and carrying 100,000 lbs. each; also a caboose weighing 12 
tons. Utilizing Eq. 121, the tractive resistance of a loaded 
coal car will be 2.2X70+122 = 276, and the grade resistance 
20X1.9X70=2660, making a total of 2936. The total for the 
caboose is 148+456 = 604. The two engines have a net draw- 
bar pull of 2X29434 = 58868 lbs. Subtracting 604 for the 
caboose, there is left 58264 for coal cars. 58264 -^2936 = 19.84, 
the number of cars. Although the number of cars must, of 
course, be a whole number, the computation of the relative 
through and pusher grades requires that we use the fractional 
number. The tractive resistance of the 19.84 cars and caboose 
is 2.2 [(19.84X70) +12] + (122X20.84) =5624. The force avail- 
able for grade is 35419-5624 = 29795. The tonnage on the 
single engine grade is 157.5 (engine) plus 19.84X70 = 1388.8 
(coal cars), plus 12 (caboose), or 1558.3 tons. 29795-^1558.3 
= 19.12 lbs. per ton, which is the grade resistance for a 0.956% 
grade. This means that the through grade can be made 0.956% 
and the corresponding pusher grade may be 1.9%. If the same 
problem is worked out on the basis of some other type of engine, 
which, perhaps, weighs considerably less, very nearly the same 
through grade to correspond with the pusher grade will be 



580 RAILKOAD CONSTRUCTION. § 525. 

obtained. The above combination of unit car weights must be 
worked as 19 coal cars and a caboose and have a considerable 
margin of unused power. A different combination of car 
weights would use up the power with less or no margin, but in 
any case the computation of the corresponding lower grade, or 
the computation of an allowable pusher grade on the basis of 
a given through grade, should be made by using a fractional 
number of cars. 

Since the pusher engine service is intermittent, and since it is 
working at full power for much less than half the time, it is 
jSracticable for the fireman to feed coal faster than the standard 
of 4000 lbs. of coal per hour while going up the pusher grade. 
The above computation was made on the basis of power pro- 
duction at the 4000-lb. rate. In § 457, it is shown that increas- 
ing the rate of coal consumption increases the value of M, and 
conversely when the locomotive is run at a velocity less than M 
the tractive power is increased, although the increase is dis- 
proportionately small. Increasing the tractive power of the 
pusher engine will increase the number of cars, although probably 
not as much as one car. Then the increase in car number will 
increase the computed resistance and decrease the amount avail- 
able for grade. This decreased amount is divided by an in- 
creased number of tons and the amount of available for grade 
per ton is less and the computed through grade is less. Con- 
sidering the very shght and disproportionate difference made by 
increasing the rate of coal consumption beyond the 4000-lb. 
standard, it is, perhaps, wisest to make the ratio of the grades 
oft the basis of engines of equal power. 

525. Two-pusher grades. It may happen, although rarely, 
that three systems of ruhng grades may be necessary on one 
division, which may be so balanced that one unbroken train is 
handled with equal facility on through grades with one engine, 
on one-pusher grades with two engines and on two-pusher 
grades with three engines. The relation of these three grades 
may be computed on the same principles as are used above. 

526. Operation of pusher engines. The maximum efficiency 
in operating pusher engines is obtained when the pusher engine 
is kept constantly at work, and this is facilitated when the pusher 
grade is as long as possible, i.e., when the heavy grades and the 
great bulk of the difference of elevation to be surmounted is 
at one place. For example, a pusher grade of three miles fol- 



§526. GRADE. 581 

lowed by a comparatively level stretch of three miles and then 
by another pusher grade of two miles cannot all be operated as 
cheaply as a continuous pusher grade of five miles. Either 
the two grades must be operated as a continuous grade of eight 
miles (sixteen pusher miles per trip) or else as two short pusher 
grades, in which case there would be a very great loss of time 
and a difficulty in so arranging the schedules that a train need 
not wait for a pusher or the pushers need not waste too much 
time in idleness waiting for trains. If the level stretch were 
imperative, the two grades would probably be operated as one, 
but an effort should be made to bring the grades together. It 
is not necessary to bring the trains to a stop to uncouple the 
pusher engine, but a stop is generally made for coupHng on, and 
the actual cost in loss of energy and in wear and tear of stopping 
and starting a heavy train is as great as the cost of running 
an engine light for several miles. 

There are two ways in which it is possible to economize in 
\he use of pusher engines, (a) When the traffic of a road is 
so very light that a pusher engine will not be kept reasonably 
busy on the pusher grade it may be worth while to place a 
siding long enough for the longest trains both at top and bottom 
of the pusher grade and then take up the train in sections. 
Perhaps the worst objection to this method is the time lost 
while the engine runs the iextra mileage, but with such very 
light traffic roads a little time more or less is of smaU consequence. 
On light traffic roads this method of surmounting a heavy grade 
will be occasionally adopted even if pushers are never used. 
If the traffic is fluctuating, the method has the advantage 
of only requiring such operation when it is needed and avoiding 
the purchase and operation of a pusher engine which has but 
little to do and which might be idle for a considerable proportion 
of the year, (h) The second possible method of economizing 
is only practicable when a pusher grade begins or ends at or 
near a station yard where switching-engines are required. In 
such cases there is a possible economy in utilizing the switching- 
engines as pushers, especially when the work in each class is 
small, and thus obtain a greater useful mileage. But such cases 
are special and generally imply small traffic. 

A telegraph-station at top and bottom of a pusher grade is 
generally indispensable to effective and safe operation. 

527. Length of a pusher grade. The virtual length of the 



582 RAILROAD CONSTRUCTION. § 528. 

pusher grade, as indicated by the mileage of the pusher engine, 
is always somewhat in excess of the true length of the grade 
as shown on the profile, and sometimes the excess length is 
very great. If a station is located on a lower grade within a 
mile or so of the top or bottom of a pusher grade, it w^ill ordina- 
rily be advisable to couple or uncouple at or near the station, 
since the telegraph-station, switching, and signaHng may be 
more economically operated at a regular station. If the extra 
engine is coupled on ahead of the through engine (as is some- 
times required by law for passenger trains) the uncoupling at 
the top of the grade may be accomplished by running the assist- 
ant engine ahead at greater speed after it is uncoupled, and, 
after running it on a siding, clearing the track for the train. 
But this requires considerable extra track at the top of the grade. 
Therefore, when estimating the length of the pusher grade, 
the most desirable position for the terminal sidings must be 
studied and the length determined accordingly rather than 
by measuring the mere length of the grade on the profile. Of 
course these odd distances are always excess; the coupling or 
uncoupling should not be done while on the grade. 

528. The cost of pusher-engine service. When we analyze 
the elements of cost, we will find that many of them are dependent 
only on time, while others are dependent upon mileage. Still 
others are dependent on both. Very much wiU depend on the 
constancy of the service, and this in turn depends on the train 
schedule and on a variety of local conditions which must be 
considered for each particular case. The effect of a pusher- 
engine on maintenance of way may be considered on the basis 
that an engine is responsible for one-half of the deterioration of 
maintenance of way and structiu-es, and, therefore, one-half of 
the percentage of the first 19 items in Table XLI or 9.06% of 
the average cost of a train-mile will be considered as chargeable 
for each mile of pusher engine service. Although the cost of 
repairs and renewals of engines is evidently a function of the 
mileage, and would therefore be somewhat less for a pusher- 
engine which did httle work than for an engine which was 
worked to the limit of its capacity, yet it is only safe to make 
the same allowance as for other engines. Other items of main- 
tenance of equipment are evidently to be ignored. The item of 
wages of enginemen will evidently depend upon the system 
employed on the particular road. Whatever the precise system 



§528. GRADE. 583 

TABLE XLIV. — COST FOR EACH MILE OF PUSHER-ENGINE SERVICE. 



Item , 
number. 


Item (abbreviated). 


Normal 
average. 


Per cent 
affected. 


Cost per 

engine 

mile, 

per cent. 


1-19 
25-27 
80,81 


Track material, labor, bridges. . . . 

Steam locomotives 

Road enginemen and engine-house 


18.12% 
9.24 

8.12 

11.27 

1.21 


50 
100 

100 
100 
100 


9.06 
9.24 

8.12 


82-85 
90,91,94 


Fuel and other engine supplies .... 
Signaling, flagmen, and telegraph. . 


11.27 
1.21 










38.90 











the general result is to pay the enginemen as much in wages 
as the average payment for regular service, and therefore the 
full allowance for Item 80 will be made. Similarly we must 
allow the full cost of the items for engine supphes. While the 
engine is doing its heavy work in cHmbing up the grade, the 
consumption of fuel and water is certainly greater than the 
average; but, on the other hand, on the return trip, when the 
engine is running Hght, it probably runs for a considerable por- 
tion of the distance actually without steam, and therefore the 
consumption of fuel and water will nearly, if not quite, average 
the consumption for an engine running up and down grade 
along the whole hne. That portion of fuel consumption which 
is due to radiation, blowing-off steam, and the many other 
causes previously enumerated, will be the same regardless of 
the work done. We therefore allow 100% for all of these items 
of engine supplies. In general we must add 100% for Items 90, 
91, and 94, the cost of switchmen and telegraphic service. While 
there might be cases where there would be no actual addition 
to the pay-rolls or the operating expenses on account of these 
items, we are not justified in general in neglecting to add the 
full quota for such service. Collecting these items we will have 
38.90% of the average cost of a train-mile for the cost of each 
mile run by the pusher engine. On the basis that the average 
cost of a train mile is SI. 60, the cost of one mile of pusher engine 
service would be .3890 X$1.60 = 62.24 cents. Assume that the 
pusher engine grade is five miles long but that the engine actually 
runs 11 miles on a round trip and that it makes 5 round trips or 
55 miles per day. Then the daily cost would be .6224X55 = 
$34.23 per day. Probably $25 to $30 per day should be charged 



584 RAILROAD CONSTRUCTION. § 529. 

up even if the mileage did not amount to as much, since many of 
the items in the cost of service are largely independent of mileage. 
On the other hand the pusher engine service renders imnecessary 
the extra trains which would have been required to handle the 
traffic with one engine over the steeper grades. Tbe cost of 
these must be computed for each particular case. 

BALANCE OF GRADES FOR UNEQUAL TRAFFIC. 

529. Nature of the subject. It sometimes happens, as when 
a road runs into a mountainous country for the purpose of 
hauling therefrom the natural products of lumber or minerals, 
that the heavy grades are all in one direction — that the whole 
line consists of a more or less unbroken climb having perhaps 
a few comparatively level stretches, but no do\\Ti grade (except 
possibly a slight sag) in the direction of the general up grade. 
With such lines this present topic has no concern. But the 
majority of railroads have termini at nearly the same level 
(500 feet in 500 miles has no practical effect on grade) and 
consist of up and down grades in nearly equal amounts and 
rates. The general rate of ruling grade is determined by the 
character of the country and the character and financial backing 
of the road to be built. It is always possible to reduce the grade 
at some point by '^ development '\ or in general by the expen- 
diture of more money. It has been tacitly assumed in the 
previous discussions that when the ruling grade has been de- 
termined all grades in either direction are cut down to that 
limit. If the traffic in both directions were the same this would 
be the proper policy and sometimes is so. But it has developed, 
especially on the great east and west trunk lines, that the weight 
of the eastbound freight traffic is enormously greater than that 
of the westbound — that westbound trains consist very largely of 
*' empties" and that an engine which could haul twenty loaded 
cars up a given grade in eastbound traffic could haul the same 
cars empty up a much higher grade when running west. As 
an illustration of the large disproportion which may exist, the 
eastbound ton-mileage on the P. R. R. between the years 1851 
and 1885 was 3.7 times the westbound ton-mileage. Between 
the years 1876 and 1880 the ratio rose to more than 4.5 to 1. 
On such a basis it is as important and necessary to obtain, say, 
a 0.6% ruling grade against the eastbound traffic as to have, 



§ 530. GRADE. 585 

say, a 1.0% grade against the westbound traffic. This is the 
basis of the following discussion. It now remains to estimate 
the probable ratio of the traffic in the two directions and from 
that to determine the proper "balance" of the opposite ruling 
grades. 

530. Computation of the theoretical balance. Assume first, 
for simplicity, that the exact business in either direction is 
accurately known. A little thought will show the truth of the 
following statements. 

1. The locomotive and passenger-car traffic in both directions 
is equal. 

2. Except as a road may carry emigrants, the passenger 
traffic in both directions is equal. Of course there are in^aumer- 
able individual instances in which the return trip is made by 
another route, but it is seldom if ever that there is any marked 
tendency to uniformity in this. Considering that a car load 
of, say, 50 passengers at 150 pounds apiece weigh but 7500 
pounds, which is -^ of the 75000 pounds which the car may 
weigh, even a considerable variation in the number of passengers 
will not appreciably affect the hauling of cars on grades. On 
parlor-cars and sleepers the ratio of live load to dead load (say 
20 passengers, 3000 pounds, and the car, 125000 pounds) is 
even more insignificant. The effect of passenger traffic on 
balance of grades may therefore be disregarded. 

3. Empty cars have a greater resistance per ton than loaded 
cars. Therefore in computing the hauling capacity of a loco- 
motive hauHng so many tons of '' empties," a larger figure must 
be used for the ordinary tractive resistances — say four pounds 
per ton greater. 

4. Owing to greater or less imperfections of management a 
small percentage of cars will run empty or but partly full in 
the direction of greatest traffic. 

5. Freight having great bulk and weight (such as grain, 
lumber, coal, etc.) is run from the rural districts toward the 
cities and manufacturing districts. 

6. The return traffic — manufactured products — although worth 
as much or more, do not weigh as much. 

As a simple numerical illustration assume that the weight 
of the cars is J and the live load f of the total load when 
the cars are *'fuir' — although not loaded to their absolute 
limit of capacity. Assume that the relative weight of live load 



586 RAILROAD CONSTRUCTION. § 530. 

to be hauled in the other direction is but J; assume that the 
grade against the heaviest traffic is 0.9%. Since the tractive 
resistance per ton is considerably greater in the case of unloaded 
cars than it is in the case of loaded cars, allowance must be 
made for this in calculating the train resistance. Mr. A. C. 
Dennis, of the Canadian Pacific Railway Company, has made 
some elaborate tests of train resistance for trains which were 
alternately loaded and empty, and found that the tractive resist- 
ance of loaded cars was very uniform at 4.7 pounds per ton- 
when the weight of the empty cars was J of the total weight. 
He also found that the tractive resistance of empty cars was 
very uniform at 8.9 pounds per ton. Although the live load 
capacity of a box-car is usually considerably more than twice 
the weight of the empty car, it will probably coincide more 
nearly with actual running conditions to consider that the live 
load is just twice the dead load. Assume that these loads are 
being hauled by a consolidation engine with a total weight, 
including engine and tender, of 107 tons, of which 106000 pounds 
is on the drivers. We will assume that the tractive resistance 
of the locomotive is likewise 4.7 pounds per ton. On the 0.9% 
grade, the grade resistance will be 18 pounds per ton, and there- 
fore the total resistance is 22.7 pounds per ton. Assume that 
this engine is working with a tractive adhesion of }; the trac- 
tive power at the circumference of the drivers will be J of 
106000 pounds, or 26500 pounds. Dividing this by 22.7, we 
obtain 1167 as the gross load of the train in tons. Subtracting 
the weight of the locomotive, 107 tons, we have 1060 t^ns as 
the weight of the loaded cars which could be hauled by this 
locomotive up a 0.9% grade, assuming an adhesion of }. Since 
the traffic in the other direction is but J, we will assume that 
f of the return cars are empty. We then have 353 tons of 
loaded cars with a locomotive weighing 107 tons, and 236 tons 
of empty cars in the return. train. The loaded cars with the 
locomotive will weigh 460 tons, and their tractive resistance 
will be 4.7 pounds per ton, or 2162 pounds. The 236 tons of 
empty cars will have a resistance of 8.9 pounds per ton, or a 
total tractive resistance of 2100 pounds. This makes a total 
of 4262 pounds of tractive resistance. Subtracting this from 
the 26500 of total adhesion of the drivers, we have left 22238 
as the amount of pull available for grade. But the return train 
weighs 696 tons. Dividing this into 22238, we find that 32 



§ 531. GRADE. 587 

pounds per ton is available for grade, which is the resistance on 
a 1.60% grade. Therefore, under the above conditions^ a 0.9% 
grade against the heaviest traffic will correspond with a 1.60% 
grade against the lighter traffic. 

Of course these figures will be slightly modified by variations 
in the assumptions as to the tractive resistance of loaded and 
unloaded cars, and more especially by variations in the ratio 
of live load to dead load in the two directions. Therefore no 
great accuracy can be claimed for the ratio of these two grades 
in opposite directions, nevertheless the above calculation shows 
unmistakably that under the given conditions, a very consider- 
£ible variation in the rate of grade in opposite directions is not 
on^y justifiable, but a neglect to allow for it would be a gi^eat 
economic error. 

531. Computation of relative traffic. Some of the principal 
elements have already been referred to, but in addition the 
following facts should be considered. 

(a) The greatest disparity in traffic occurs through the hand- 
ling of large amounts of coal, lumber, iron ore, grain, etc. On 
roads which handle but little of these articles or on which for 
local reasons coal is hauled one way and large shipments of 
grain the other way the disparity will be less and will perhaps 
be insignificant. 

(&) A marked change in the development of the country iray, 
and often does, cause a marked difference in the disparity of 
traffic. The heaviest traffic (in mere weight) is always toward 
manufacturing regions and away from agricultural regions. But 
when a region, from being purely agricultural or mineral, be- 
comes largely manufacturing, or when a manufacturing region 
develops an industry which will cause a growth of heavy freight 
traffic from it, a marked change in the relative freight movemer t 
will be the result. 

(c) Very great fluctuations in the relative traffic may be 
expected for prolonged intervals. 

(d) An estimate of the relative traffic may be formed by 
the same general method used in computing the total traffic 
of the road (see § 473, Chap. XIX) or by noting the relative 
traffic on existing roads which may be assumed to have practically 
the same traffic as the proposed road will obtain. 



CHAPTER XXIV. 
THE IMPROVEMENT OF OLD LINES. 

532. Classification of improvements. The improvements here 
considered are only those of alignment — horizontal and vertical. 
Strictly there is no definite limit, either in kind or magnitude, 
to the improvements which may be made. But since a railroad 
cannot ordinarily obtain money, even for improvements^ to 
an amount greater than some small proportion of the pre- 
viously invested capital, it becomes doubly necessary to expend 
such money to the greatest possible advantage. It has been 
previously shown that securing additional business and increas- 
ing the train load are the two most important factors in increas- 
ing dividends. After these, and of far less importance, come 
reductions of curvature, reductions of distance (frequently of 
doubtful policy, see Chap. XXI, § 503), and elimination of sags 
and humps. These various improvements will be briefly dis- 
cussed. 

(a) Securing additional business. It is not often possible 
by any small modification of alignment to materially increase 
the business of a road. The cases which do occur are usually 
those in which a gross error of judgment was committed during 
the original construction. For instance, in the early history 
of railroad construction many roads were largely aided by the 
towns through which the road passed, part of the money neces- 
sary for construction being raised by the sale of bonds, which 
were assumed or guaranteed and subsequently paid by the 
towns. Such aid was often demanded and exacted by the 
promoters. Instances are not unknown where a failure to 
come to an agreement has caused the promoters to deliberately 
pass by the town at a distance of some miles, to the mutual 
disadvantage of the road and the town. If the town subsequent- 
ly grew in spite of this disadvantage, the annual loss of business 
might readily amount to more than the original sum in dispute. 

588 



§ 533. IMPROVEMENT OF OLD LINES. 589 

Such an instance would be a legitimate opportunity for study 
of the advisability of re-location. 

As another instance (the original location being justifia- 
ble) a railroad might have been located along the bank of a 
considerable river too wide to be crossed except at consider- 
able expense. When originally constructed the enterprise would 
not justify the two extra bridges needed to reach the town. 
A growth in prosperity and in the business obtainable 
might subsequently make such extra expense a profitable invest- 
ment. 

(b) Increasing the train load. On account of its importance 
this will be separately considered in § 535 et seq. 

(c) Reduction in curvature and distance and the elimination 
of sags and humps. Such improvements are constantly being 
made by all progressive roads. The need for such changes 
occurs in some cases because the original location was very 
faulty, the revised location being no more expensive than the 
original, and in other cases because the original location was the 
best that was then financially possible and because the present 
expanded business will justify a change. 

(d) Changing the location of stations or of passing sidings. 
The station may sometimes be re-located so as to bring it nearer 
to the business center and thus increase the business done. 
But the principal reasons for re-locating stations or passing 
sidings is that starting trains may have an easier grade on which 
to overcome the additional resistances of starting. Such changes 
will be discussed in detail in § 537. 

533. Advantages of re-locations. There are certain undoubted 
advantages possessed by the engineer who is endeavoring to 
improve an old line. 

(a) The gross traffic to be handled is definitely known. 

(b) The actual cost per train-mile for that road (which may 
differ very greatly from the average) is also known, and therefore 
the value of the proposed improvement can be more accurately 
determined. 

(c) The actual performance of such locomotives as are used 
on the road may be studied at leisure and more rehable data 
may be obtained for the computations. 

534. Disadvantages of re-locations. The disadvantages are 
generally more apparent and frequently appear practically 
insuperable — more so than they prove to be on closer inspection. 



590 RAILROAD CONSTRUCTION. i § 534. 

(a) It frequently means the abandonment of a greater or less 
length of old line and the construction of new line. At first 
thought it might seem as if a change of line such as would permit 
an increase of train-load of 50 or perhaps 100% could never 
be obtained, or at least that it could not be done except at an 
impracticable expense. On the contrary a change of 10% 
of the old line is frequently all that is necessary to reduce the 
grades so that the train-loads hauled by one engine may be 
nearly if not quite doubled. And when it is considered that 
the cost of a road to sub-grade is generally not more than one- 
third of the total cost of construction and equipment per mile, 
it becomes plain that an expenditure of but a small percentage 
of the original outlay, expended where it will do the most good, 
will often suffice to increase enormously the earning capacity. 

(6) One of the most difficult matters is to convince the finan- 
cial backers of the road that the proposed improvement will 
be justifiable. The cause is simple. The disadvantages of the 
original construction He in the large increase of certain items 
of expense which are necessary to handle a given traffic. And 
yet the fact that the expenditures are larger than they need 
be are only apparent to the expert, and the fact that a saving 
may be made is considered to be largely a matter of opinion 
until it is demonstrated by actual trial. On the other Rand 
the cost of the proposed changes is definite, and the very fact 
that the road has been uneconomically worked and is in a poor 
financial condition makes it difficult to obtain money for im- 
provements. 

(c) The legal right to abandon a section of operated line 
and thus reduce the value of some adjoining property has 
sometimes been successfully attacked. A common instance 
would be that of a factory which was located adjoining the right 
of way for convenience of transportation facilities. The abandon- 
ment of that section of the right of way would probably be fatal 
to the successful operation of the factory. The objection may 
be largely ehminated by the maintenance of the old right of 
way as a long siding (although the business of the factory might 
not be worth it), but it is not always so easy of solution, and 
this phase of the question must always be considered. 



§ 535. IMPROVEMENT OF OLD LINES. 591 



^iEDUCTION OF VIRTUAL GRADE. 

535. Obtaining data for computations. As developed in the 
last chapter (§§ 515-517) the real object to be attained is the 
reduction of the virtual grade. The method of comparing grades 
under various assumed conditions was there discussed. When 
the road is still "on paper ^^ some such method is all that is 
possible; but when the road is in actual operation the virtual 
grade of the road at various critical points, with the rolling 
stock actually in use, may be determined by a simple test and 
the effect of a proposed change may be reliably computed. 
Bearing in mind the general principle that the virtual grade 
line is the locus of points determined by adding to the actual 
grade profile ordinates equal to the velocity head of the train, 
it only becomes necessary to measure the velocity at various 
points. Since the velocity is not usually uniform, its precise 
determination at any instant is almost impossible, but it will 
generally be found to be sufficiently precise to assume the velocity 
to be uniform for a short distance, and then observe the time 
required to pass that short space. Suppose that an ordinary 
watch is used and the time taken to the nearest second. At 
30 railes per hour, the velocity is 44 feet per second. To obtain 
the time to within 1%, the time would need to be 100 seconds 
and the space 4400 feet. But with variable velocity there 
would be too great error in assuming the velocity as uniform 
for 4400 feet or for the time of 100 seconds. Using a stop- 
watch registering fifths of a second, a 1% accuracy would 
require but 20* seconds and a space of 880 feet, at 30 miles per 
hour. Wellington suggests that the space be made 293 feet 
4 inches, or -^^ of a mile; then the speed in miles per hour 
equals 200 -^ s, in which s is the time in seconds required to 
traverse the 293' 4''. For instance, suppose the time required 
to pass the interval is 12.5 seconds. -^-^ mile in 12.5 seconds = 
one mile in 225 seconds, or 16 miles per hour. But likemse 
200^12.5=16, the required velocity. The following features 
should be noted when obtaining data for the computations: 

(a) All critical grades on the road should be located and 
their profiles obtained — by a survey if necessary. 

(6) At the bottom and top of all long grades (and perhaps at 
intermediate points if the grades are very long) spaces of known 



592 RAILROAD CONSTRUCTION. § 536. 

length (preferably 293J feet) should be measured off and marked 
by flags, painted boards, or any other serviceable targets. 

(c) Provided with a stop-watch marking fifths of seconds 
the observer should ride on the trains affected by these grades 
and note the exact interval of time required to pass these spaces. 
If the space is 293 J feet, the velocity in miles per hour =200 -f- 
interval in seconds. In general, 

T- distance in feet X 3600 



time in seconds X 5280* 



(d) Since these critical grades are those which require the 
greatest tax on the power of the locomotive, the conditions 
under which the locomotive is working must be known — i.e., 
the steam pressure, point of cut-off, and position of the throttle. 
Economy of coal consumption as well as efficient w^orking at 
high speeds requires that steam be used expansively (using an 
early cut-off), and even that the throttle be partly closed; but 
when an engine is slowly climbing up a maximum grade mth a 
full load it is not exerting its maximum tractive power unless 
it has its maximum steam pressure, wide-open throttle, and is 
cutting off nearly at full stroke. These data must therefore 
be obtained so as to know whether the engine is developing 
at a critical place all the tractive force of which it is capable. 
The condition of the track (wet and slippery or dry) and the 
approximate direction and force of the wind should be noted 
with sufficient accuracy to judge whether the test has been made 
under ordinary conditions rather than under conditions which 
are exceptionally favorable or unfavorable. • 

(e) The train-loading should be obtained as closely as possible. 
Of course the dead weight of the cars is easily found, and the 
records of the freight department will usually give the live 
load with all sufficient accuracy. 

536. Use of the data obtained. A very brief inspection 
of the results, freed from refined calculations or imcertainties, 
will demonstrate the following truths: 

(a) If, on a uniform grade, the velocity increases, it shows 
that, under those conditions of engine working, the load is less 
than the engine can handle on that grade 

(h) If the velocity decreases, it shows that the load is greater 
than the engine can handle on an indefinite length of such 



§ 536. IMPROVEMENT OF OLD LINES. 593 

grade. It shows that such a grade is being operated by momen- 
tum. Frcm the rate of decrease of velocity the maximum 
practicable length of such a grade (starting with a given velocity) 
may be easily computed. 

(c) By combining results under different conditions of grade 
but with practically the same engine working, the tractive 
power of the engine may be determined (according to the prin- 
ciples previously demonstrated) for any grade and velocity. 
For example: On an examination of the profile of a division 
of a road the maximum grade was found to be 1.62% (85.54 
feet per mile). At the bottom and near the top of this grade 
two lengths of 293' 4" are laid off. The distance between the 
centers of these lengths is 6000 feet. A freight train moving 
up the grade is timed at 9f seconds on the lower stretch and 7f 

seconds on the upper. These times correspond to ^— - and =—x 

or 21.3 and 26.3 miles per hour respectively. It is at once 

observed that the velocity has increased and that the engine 

could draw even a heavier load up such a grade for an indefinite 

distance. How much heavier might the load be? 

For simpHcity we will assume that the conditions were 

normal, neither exceptionally favorable nor unfavorable, and 

that the engine was worked to its maximum capacity. The 

engine is a ^'consoHdation'' weighing 128700 pounds, with 

112600 pounds on the drivers. The train-load behind the 

engine consists of ten loaded cars weighing 465 tons and eleven 

empties weighing 183 tons, thus making a total train- weight of 

712 tons. Applying Eq. 106, we find that the additional force 

which the engine has actually exerted per ton in increasing the 

velocity from 21.3 to 26.3 miles per hour in a distance of 6000 

feet is 

70 224 
P - -^^ (26 . 32 - 21 . 32) =2 . 78 pounds per ton 

The grade resistance on a 1.62% grade is 32.4 pounds per 
ton. The average train resistance may be computed similarly 



» 



to the method adopted in § 439: 



\ tons at 4.7 pounds per ton = 2486 pounds 



465 

64 

183 

, tt 

712 4115 



594 RAILROAD CONSTRUCTION. § 537. 

The average tractive resistance is therefore 41 15 -^ 712 = 5.78 
pounds per ton. Adding the grade resistance (32.4) we have 
a total train resistance of 38.18 pounds per ton. But, com- 
puting from the increase in velocity, the locomotive is evidently 
exerting a pull of 2.78 pounds per ton in excess of the computed 
required pull on that grade, or a total pull of 40.96 pounds 
per ton. Therefore the train load might have been increased 
proportionately and might have been made 



712X?^^U||^=764 tons. 



This shows that 52 tons additional might have been loaded 
on to the train, or say, three more empties or one additional 
loaded car. 

A pull of 40.96 pounds per ton means a total adhesion at the 
drivers of 29164 pounds, which is about 26% of the weight on 
the drivers — 112600 pounds. This indicates average condi- 
tions as to traction, although better conditions than can be 
depended on for regular service. 

The above calculation should of course be considered simply 
as a ''single observation." The performance of the same engine 
on the same grade (as well as on many other grades) on succeed- 
ing days should also be noted. It may readily happen that 
variations in the condition of the track or of the handling of the 
engine may make considerable variation in the results of the 
several calculations, but when the work is properly done it is 
always possible to draw definite and very positive deductions. 

537. Reducing the starting grade at stations. The resistance 
to starting a train is augmented from two causes : (a) the trac- 
tive resistances are usually about 20 pounds per ton instead 
of, say, 6 pounds, and (b) the inertia resistance must be overcome. 
The inertia resistance of a freight train (see § 435) which is 
expected to attain a velocity of 15 miles per hour in a distance 
of 1000 feet is (see Eq. 140) 

70 224 
p = ir_f^(152_o) =15.8 pounds per ton, which is the equiva- 
lent of a 0.79% grade. Adding this to a grade which nearly or 
quite equals the ruling grade, it virtually creates a new and 
higher ruling grade. Of course that additional force can be 
greatly reduced at the expense of slower acceleration, but even 



§537, 



IMPROVEMENT OF OLD LINES, 



595 



this cannot be done indefinitely, and an acceleration to only 
15 miles per hour in 1000 feet is as slow as should be allowed 
for. With perhaps 14 pounds per ton additional tractive 
resistance, we have about 30 pounds per ton additional — equiva- 



S~*,il2> 




Fig. 216. 



lent to a 1.5% grade. Instances are known where it has proven 
wise to create a hump (in what was otherwise a uniform grade) 
at a station. The effect of this on high-speed passenger trains 
moving up the grade would be merely to reduce their speed 
very slighth^ No harm is done to trains moving down the 
grade. Freight trains moving up the grade and intending to 
stop at the station will merely have their velocity reduced as 
they approach the station and will actually save part of the 
wear and tear otherwise resulting from applying brakes. When 
the trains start they are assisted by the short down grade, 
just where they need assistance most. Even if the grade CD 
is still an up grade, the pull required at starting is less than that 
required on the uniform grade by an amount equal to 20 times 
the difference of the grade in per cent. 



APPENDIX. 

THE ADJUSTMENTS OF INSTEUMENTS. 

The accuracy of instrumental work may be vitiated by any 
one of a large number of inaccuracies in the geometrical relations 
of the parts of the instruments. Some of these relations are so 
apt to b^ pltered by ordinary usage of the instrument that the 
makers have provided adjusting-screws so that the inaccuracies 
may be readily corrected. There are other possible defects, 
which, however, will seldom be found to exist, provided the 
instrument was properly made and has never been subjected to 
treatment sufficiently rough to distort it. Such defects, when 
found, can only be corrected by a competent instrument-maker 
or repairer. 

A WARNING is necessary to those who would test the accuracy 
of instruments, and especially to those whose experience in such 
work is small. Lack of skill in handling an instrument will 
often indicate an apparent error of adjustment when the real 
error is very different or perhaps non-existent. It is always a 
safe plan when testing an adjustment to note the amount of the 
apparent error ; then, beginning anew, make another independent 
determination of the amount of the error. When two or more 
perfectly independent determinations of such an error are made 
it will generally be found that they differ by an appreciable 
amount. The differences may be due in variable measure to 
careless inaccurate manipulation and to instrumental defects 
which are wholly independent of the particular test being made. 
Such careful determinations of the amounts of the errors are 
generally advisable in view of the next paragraph. 

Do NOT DISTURB THE ADJUSTING-SCREWS ANY MORE THAN 

NECESSARY. Although metals are apparently rigid, they are 
really elastic and yielding. If some parts of a complicated 
mechanism, which is held together largely by friction, are sub- 
jected to greater internal stresses than other parts of the mech- 

596 



APPENDIX. 597 

anism, the jarring resulting from handling will frequently cause 
a slight readjustment in the parts which will tend to more nearly 
equaHze the internal stresses. Such action frequently occurs 
with the adjusting mechanism of instruments. One screw may 
be strained more than others. The friction of parts may pre- 
vent the opposing screw from immediately taking up an equal 
stress. Perhaps the adjustment appears perfect under these 
conditions Jarring diminishes the friction between the parts, 
and the unequal stresses tend to equalize. A motion takes place 
which, although microscopically minute, is sufficient to indicate 
an error of adjustment. A readjustment made by unskillful 
hands may not make the final adjustment any more perfect. 
The frequent shifting of adjusting-screws wears them badly, 
and when the screws are worn it is still more difficult to keep 
them from moving enough to vitiate the adjustments. It is 
therefore preferable in many cases to refrain from disturbing the 
adjusting-screws, especially as the accuracy of the work done is 
not necessarily affected by errors of adjustment, as may be 
illustrated : 

(a) Certain operations are absolutely unaffected by certain 
errors of adjustment. 

Q)) Certain operations are so slightly affected by certain small 
errors of adjustment that their effect may properly be neglected. 

(c) Certain errors of adjustment may be readily allowed for 
and neutralized so that no error results from the use of the 
unadjusted instrument. Illustrations of all these cases will be 
given under their proper heads. 

ADJUSTMENTS OF THE TRANSIT. 

1. To have the plate-hubhles in the center of the tubes when the 
axis is vertical. Clamp the upper plate and, with the lower 
clamp loose, swing the instrument so that the plate-bubbles are 
parallel to the lines of opposite leveling-screws. Level up until 
both bubbles are central. Swing the instrument 180°. If the 
bubbles again settle at the center, the adjustment is perfect. If 
either bubble does not settle in the center, move the leveling- 
screws until the bubble is half-way back to the center. Then, 
before touching the adjusting-screws, note carefully the position 
of the bubbles and observe whether the bubbles always settle at 
the same place in the tube, no matter to what position the in- 



598 RAILROAD CONSTRUCTION. 

stniment may be rotated. AVhen the instrument is so leveled, 
the axis is truly vertical and the discrepancies between this 
constant position of the bubbles and the centers of the tubes 
measure the errors of adjustment. By means of the adjusting- 
screws bring each bubble to the center of the tube. If this is 
done so skillfully that the true level of the instrument is not 
disturbed, the bubbles should settle in the center for all positions 
of the instrument. Under unskillful hands, two or more such 
trials may be necessary. 

When the plates are not horizontal, the measured angle is greater than 
the true horizontal angle by the difference between the measured angle 
and its projection on a horizontal plane. When this angle of inclination 
is small, the difference is insignificant. Therefore when the plate-bubbles 
are very nearly in adjustment, the error of measurement of horizontal 
ingles may be far within the lowest unit of measurement used. A small 
Irror of adjustment of the plate-bubble perpendicular to the telescope will 
Affect the horizontal angles by only a small proportion of the error, which 
w^ill be perhaps imperceptible. Vertical angles will be affected by the 
Bame insignificant amount. A small error of adjustment of the plate- 
bubble parallel to the telescope will affect horizontal angles very slightly, 
but will affect vertical angles by the full amount of the error. 

All error due to imadjusted plate-bubbles may be avoided by noting in 
what positions in the tubes the bubbles will remain fixed for all positions 
of azimuth and then keeping the bubbles adjusted to these positions, for 
\he axis is then truly vertical. It will often save time to work in this way 
temporarily rather than to stop to make the adjustments. This should 
especially be done when accurate vertical angles are required. 

When the bubbles are truly adjusted, they should remain stationary 
regardless of whether the telescope is revolved with the upper plate loose 
and the lower plate clamped or whether the whole instrument is revolved , 
the plates being clamped together. If there is any appreciable difference, 
it shows that the two vertical axes or *' centers" of the plates are not con- 
centric. This may be due to cheap and faulty construction or to the exces- 
sive wear that may be sometimes observed in an old instrument originally 
well made. In either case it can only be corrected by a maker. 

2. To make the revolving axis of the telescope perpendicular to 
the vertical axis of the instrument. This is best tested by using 
a long plumb-line, so placed that the telescope must be pointed 
upward at an angle of about 45° to sight at the top of the plumb- 
line and downward about the same amount, if possible, to 
sight at the lower end. The vertical axis of the transit must 
be made truly vertical. Sight at the upper part of the line> 
clamping the horizontal plates. Swing the telescope down 
and see if the cross-wire again bisects the cord. If so, the 
adjustment is probably perfect (a conceivable exception will be 



APPENDIX. 599 

noted later) ; if not, raise or lower one end of the axis by mi ana 
of the adjusting-screws, placed at the top of one of the standards, 
until the cross-wire will bisect the cord both at top and bottom. 
The plumb-bob may be steadied, if necessary, by hanging it 
in a pail of water. As many telescopes cannot be focused 
on an object nearer than 6 or 8 feet from the telescope, this 
method requires a long plumb-line swung from a high point, 
which may be inconvenient. 

Another method is to set up the instrument about 10 feet 
from a high wall. After leveling, sight at some convenient 
mark high up on the wall. Swing the telescope down and make 
a mark (when working alone some convenient natural mark may 
generally be found) low down on the wall. Plunge the telescope 
and revolve the instrument about its vertical axis and again sight 
at the upper mark. Swing down to the lower mark. If the 
wire again bisects it, the adjustment is perfect. If not, fix a 
point half-way between the two positions of the lower mark. 
The plane of this point, the upper point, and the center of the 
instrument is truly vertical. Adjust the axis to these upper and 
lower points as when using the plumb-line. 

3. To make the line of collimation perpendicular to the revolving 
axis of the telescope. With the instrument level and the telescope 
nearly horizontal point at some well-defined point at a distance 
of 200 feet or more. Plunge the telescope and establish a point 
in the opposite direction. Turn the whole instniment about the 
vertical axis until it again points at the first mark. Again 
plunge to '^direct position" {i,e., with the level-tube under 
the telescope). If the vertical cross- wire again points at the 
second mark, the adjustment is perfect. If not, the error is 
one-fourth of the distance between the two positions of the 
second mark. Loosen the capstan screw on one side of the 
telescope and tighten it on the other side until the vertical 
wire is set at the one-fourth mark. Turn the whole instrument 
by means of the tangent screw until the vertical wire is midway 
between the two positions 'of the second mark. Plunge the 
telescope. If the adjusting has been skillfully done, the cross- 
wire should come exactly to the first mark. As an "erecting 
eyepiece" reinverts an image already inverted, the ring carrying 
the cross-wires mufit be moved in the same direction as the 
apparent error in order to correct that error. 



600 KAILROAD CONSTRUCTION. 

/ 

The necessity for the third adjustment lies principally in the practice 
of producing a Une by plunging the telescope, but when this is required to 
be done with great accuracy it is always better to obtain the forward point 
by reversion (as described above for making the test) and take the mean 
of the two forward points. Horizontal and vertical angles are practically 
unaffected by small errors of this adjustment, unless, in the case of hori- 
zontal angles, the vertical angles to the points observed are very different. 

Unnecessary motion of the adjusting-screws may sometimes be avoided 
by carefully establishing the forward point on line by repeated reversions 
of the instrument, and thus determining by repeated trials the exact amount 
of the error. Differences in the amoimt of error determined would be 
evidence of inaccuracy in manipulating the instrument, and would show 
that an adjustment based on the first trial would probably prove unsatis- 
factory. 

The 2d and 3d adjustments are mutually dependent. If either adjust- 
ment is badly out, the other adjustment cannot be made except as follows 1 

(a) The second adjustment can be made regardless of the third when 
the lines to the high point and the low point make equal angles with the 
horizontal. 

(6) The third adjustment can be made regardless of the second whej> 
the front and rear points are on a level with the instrument. 

When both of these requirements are nearly fulfilled, and especially 
when the error of either adjustment is small, no trouble will be found in 
perfecting either adjustment on account of a small error in the other ad- 
justment. 

If the test for the second adjustment is made by means of the plumb- 
line and the vertical cross-wire intersects the line at all points as the tele- 
scope is raised or lowered, it not only demonstrates at once the accuracy 
of that adjustment, but also shows that the third adjustment is either 
perfect or has so small an error that it does not affect the second. 

4. To have the bubble of the telescope-level in the center of the 
tube when the Une of collimation is horizontal. The line of coUi- 
mation should coincide with the optical axis of the telescope. 
If the object-glass and eyepiece have been properly centered, 
the previous adjustment will have brought the vertical cross- 
wire to the center of the field of view. The horizontal cross- 
-wire should also be brought to the center of the field of view, 
and the bubble should be adjusted to it. 

a. Peg method. Set up the transit at one end of a nearly 
level stretch of about 300 feet. Clamp the telescope with its 
bubble in the center. Drive a stake vertically under the eye- 
piece of the transit, a,nd another about 300 feet away. Observe 
the height of the center of the ej^epiece (the telescope being 
level) above the stake (calling it a) ; observe the reading of the 
rod when held on the other stake (caUing it b) ; take the instru- 
ment to the other stake and set it up so that the eyepiece is 



APPENDIX. 601 

' 1 

vertically over the stake, observing the height, c ; take a reading 
on the first stake, calling it d. If this adjustment is perfect, 
then 

a — d = h — c, 
or (a— c?)-(6-c)=0. 
CaU (a-d)-(b-c)=2m. 
When m is positive, the line points downward] 
" m '* negative, " " " upward. 

To adjust: if the line points up, sight the horizontal cross- 
wire (by moving the vertical tangent screw) at a point which is 
m lower, then adjust the bubble so that it is in the center. 

By taking several independent values for a, 6, c, and d, a mean value 
for m is obtained, which is more reliable and which may save much im- 
necessary working of the adjusting-screws. 

b. Using an auxiliary level. When a carefully adjusted level 
is at hand, this adjustment may sometimes be more easily 
made by setting up the transit and level, so that their lines of 
coUimation are as nearly as possible at the same height. If a 
point may b^ found which is half a mile or more away and 
which is on the horizontal cross-wire of the level, the horizontal 
cross-wire of the transit may be pointed directly at it, and the 
bubble adjusted accordingly. Any slight difference in the 
heights of the lines of coUimation of the transit and level (say 
J'') may almost be disregarded at a distance of J mile or more, 
or, if the difference of level would have an appreciable effect, 
even this may be practically eliminated by making an estimated 
allowance when sighting at the distant point. Or, if a distant 
point is not available, a level-rod with target may be used at a 
distance of (say) 300 feet, making allowance for the carefully 
determined difference of elevation of the two lines of coUimation. 

5. Zero of vertical circle. When the Hne of coUimation is truly 
horizontal and the vertical axis is truly vertical, the reading 
of the vertical circle should be 0*^. If the arc is adjustable, 
it should be brought to 0°. If it is not adjustable, the index 
error should be observed, so that it may be applied to all readings 
of vertical angles. 

ADJUSTMENTS OF THE WYE LEVEL. 

1. To make the line of coUimation coincide with the center of 
the rings. Point the intersection of the cross-wires at some 



602 RAILROAD CONSTRUCTION. 

well-defined point which is at a considerable distance. The in- 
strument need not be level, which allows much greater liberty 
in choosing a convenient point. The vertical axis should be 
clamped, and the clips over the wyes should be loosened and 
raised. Rotate the telescope in the wyes. The intersection of 
the cross-wires should be continually on the point. If it is not^ 
it requires adjustment. Rotate the telescope 180° and adjust 
one-half of the error by means of the capstan-headed screws that 
move the cross-wire ring. It should be remembered that, with 
an erecting telescope, on account of the inversion of the image, 
the ring should be moved in the direction of the apparent error. 
A^djust the other half of the error with the leveling-screws. 
Then rotate the telescope 90° from its usual position, sight 
accurately at the point, and then rotate 180° from that position 
and adjust any error as before. It may require several trials, 
but it is necessary to adjust the ring until the intersection of 
the cross-wires will remain on the point for any position of 
rotation. 

If such a test is made on a very distant point and again on a point only 
10 or 15 feet from the instrument, the adjustment may be found correct 
for one point and incorrect for the other. This indicates that the object- 
slide is improperly centered. Usually this defect can only be corrected by 
an instrument-maker. If the difference is very small it may be ignored, 
but the adjustment should then be made on a point which is at about the 
mean distance for usual practice — say 150 feet. 

If the whole image appears to shift as the telescope is rotated, it indi- 
cates that the eyepiece is improperly adjusted. This defect is likewise 
usually corrected only by the maker. It does not interfere with instru- 
mental accuracy, but it usually causes the intersection of the cross-wires 
to be eccentric with the field of view. 

2. To make the axis of the level-tube parallel to the line of colli" 
mation. Raise the clips as far as possible. Swing the level 
so that it is parallel to a pair of opposite leveling-screws and 
clamp it. Bring the bubble to the middle of the tube by means 
of the leveling-screws. Take the telescope out of the wyes and 
replace it end for end, using extreme care that the wyes are iiot 
jarred by the action. If the bubble does not come to the center, 
correct one-half of the error by the vertical adjusting-screws at 
one end of the bubble. Correct the other half by the leveling- 
screws. Test the work by again changing the telescope end for 
end in the wyes. 

Care should be taken while making this adjustment to see 



^^^ 



APPENDIX. 603 

that the level-tube is vertically under the telescope. With the 
bubble in the center of the tube, rotate the telescope in the wyes 
for a considerable angle each side of the vertical. If the first 
half of the adjustment has been made and the bubble moves, it 
shows that the axis of the wyes and the axis of the level-tube 
are not in the same vertical plane although both have been made 
horizontal. By moving one end of the level-tube sidewise by 
means of the horizontal screws at one end of th% tube, the two 
axes may be brought into the same plane. As this adjustment 
is hable to disturb the other, both should be alternately tested 
until both requirements are complied with. 

By these methods the axis of the bubble is made parallel to 
the axis of the wyes; and as this has been made parallel to the 
lines of collimation by means of the previous adjustment, the 
axis of the bubble is therefore parallel to the line of collimation. 

3. To make the line of collimation perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveling carefully 
over one pair of screws, revolve the telescope 180°. If it is not 
level, adjust half of the error by means of the capstan-headed 
screw under one of the wyes, and the other half by the leveling- 
screws. Reverse again as a test. 

When the first two adjustments have been accurately made, good level- 
ing may always be done by bringing the bubble to the center by means of 
the leveling-screws, at every sight if necessary, even if the third adjust- 
ment is not made. Of course this third adjustment should be made as a 
matter of convenience, so that the line of coUimation may be always level 
no matter in what direction it may be pointed, but it is not necessary to 
stop work to make this adjustment every time it is found to be defective. 

ADJUSTMENTS OF THE DUMPY LEVEL. 

1. To make the axis of the level-tube perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveHng-screws. Then, after leveling care- 
fully over one pair of screws, revolve the telescope 180°. If 
it is not level, adjust one-half of the error by means of the adjust- 
ing-screws at one end of the bubble, and the other half by 
means of the leveling-screws. Reverse again as a test. 

2. To make the line of collimation perpendicular to the vertical 
axis. The method of adjustment is identical with that for 
the transit (No. 4, pi. 505) except that the cross-wire must be 



604 RAILROAD CONSTRUCTION. 

adjusted to agree with the level-bubble rather than vice versa, as 
is the case with the corresponding adjustment of the transit; 
i.e., with the level-bubble in the center, raise or lower the hori- 
zontal cross-wire until it points at the mark known to be on 
a level with the center of the instrument. 

If the instrument has been well made*and has not been dis- 
torted by rough usage, the cross-wires will intersect at the 
center of the field of view when adjusted as described. If they 
do not, it indicates an error which ordinarily can only be cor- 
rected by an instrument-maker. The error may be due to any 
one of several causes, which are 

(a) faulty centering of object-slide; 

(6) faulty centering of eyepiece; 

(c) distortion of instrument so that the geometric axis of 
the telescope is not perpendicular to the vertical axis. If the 
error is only just perceptible, it will not probably cause any 
error in the work. 



AZIMUTH. 

The azimuth of a line on the surface of the earth is its angle 
with a true meridian through a point on the line. It is the 
true bearing as distinguished from ^' magnetic bearing." Fed- 
eral law requires that all surveys of government lands shall be 
made by '' Solar Observations '' (rather than with the magnetic 
needle) so as to obtain true bearings. 

Solar Azimuth may be obtained in two general ways, (a) by 
direct observation on the sun with an ordinary " complete " 
transit, provided with a colored glass shade, and (6) by the use 
of a ^' solar attachment " or a solar compass. The first method 
only requires as special equipment a colored glass shade costing 
but a few dollars, but it requires the separate solution of a for- 
mula for each observation made. Even the colored glass shade 
is not always necessary — as when the disc of the sun is just seen 



^^ 



APPENDIX. 



605 



through thin clouds and is not too bright to be observed with the 
naked eye. The ^' colored glass shade " may be merely a piece 
of colored glass fitted over the eye-piece, or the glass may be 
set into a frame very similar to the object glass cover and readily 
taken off and put on. In the latter case the glass must be 
'' optically perfect/' i.e., with the sides perfectly plane and 
parallel, so that there shall be no refraction of the image, or 
such glass as is used for the sun shade of a sextant. 

The second method (6) does not require any calculation of a 
formula; the true meridian is given directly but it requires the 
use of a special instrument, whose adjustments must be made 
with great care or the resulting azimuth will often be in error by 
a much larger amoxmt than the error in the adjustment. A 
proper appreciation of either method requires an understand- 
ing of certain astronomical relations. 



/T) ' 


7 


X 






oo:^ 


K 


H 


^rv-- 






\ 


h \" 


yy 







Fig. 1. 



Fig. 1 represents the orthographic projection of the celestial 
sphere, projected on the plane of the meridian of the observer. 
H P Z E represents the meridian of the observer. 

Z = the zenith. 
CP = the polar axis of the earth. 
C^ = the plane of the equator. 

>S = the position of the sun. 
EZ = the latitude of the observer = <^. 

ZP =90° -ct> = GO cfy. 

/SG = the true altitude of the sun = /i. 

SZ = 90°-h = coh. 

ST = the decHnation of the sun, north or south of the equator 

= 5. 
^P = 90°-5 = co 5. 



606 RAILROAD CONSTRUCTIQN. 

The essential sign of 5 must be considered. If the sun is 
south of the equator (as it is from about September 21 to March 
21), 5 is negative and if the dechnation is (say) S 20^, 5= —20°. 
Then CO 5 = 90°- 5-90°- (-20°) =110°. 

Z = the angle from the position of the sun to the true north = 
the spherical angle &ZF, 

Then, from spherical trigonometry, we have, in the spherical 
triangle aSZP, 

q- 1 v— pi° (aS — CO h) sin (>S — co <^) 



4 



sin CO h sin cp <> 



in which ^5 = J[co /i+co <^-f-co 5]. 

The sun describes each day a path which is approximately 
parallel with the equator, the change in declination being very 
small during June and December and fastest when the sun is 
crossing the equator in March and September, the greatest rate 
of change being about 59 seconds of arc per hour. The declina- 
tion of the sun must be known for the time of observation. This 
is obtainable from the Nautical Almanac or Ephemeris. 

Example. — Declination for Philadelphia, Feb. 20, 1914, at 
8:10 A. M., standard time, 75th meridian. Since " standard 
time " is a definite time interval from Greenwich mean local 
time, we may use it here regardless of precise longitude or mean 
local time, 8:10 A. M. on the 75° meridian is 1:10 P. M. mean 
time, at Greenwich. 1.17/iX53''.64 = 62^.58 = 1' 2''.6 and 
-11°7'1'M+0°1'2''.6=-11°5'58^5 which is south dech- 
nation. 

Refraction. Refraction causes the sun to appear higher than 
it actually is. Therefore when the altitude of the sun is observed, 
the computed refraction should be subtracted from the apparent 
altitude to obtain the true altitude. The amount of the refrac- 
tion is a very complicated function of the temperature and of 
the barometric pressure. For refined astronomical work, large 
refraction tables should be used, making due allowance for 
temperature and pressure, but for such work as may be done 
with an ordinary transit the values given in the following table 
will suffice. 

Angular diameter of sun. The sun's angular diameter is 
about 0° 32^ With the comparatively high power telescopes 
now generally used on transits, this fills a large part of the field 
of view and it is impossible to accurately bisect such a large 



APPENDIX. 



607 



MEAN REFRACTIONS [bESSEL] TRUE FOR BAROMETER AT 29". 6, 

TEMP. 48° F. 



Alt. 



0° 0' 
10 
20 
30 
40 
50 

1° 
10 
20 



Refr. 



34' 54" 

32 49 

30 52 

29 03 

27 23 

25 50 

24 25 

23 07 

21 56 



Alt. 



30' 
40 
50 


30 


30 


30 



Refr. 



20' 51' 

19 52 

18 58 

18 09 

16 01 

14 15 

12 48 

11 39 

10 40 



Alt. 



0' 
30 


30 


30 


30 





Refr- 



9' 46" 
9 02 
8 23 
7 49 
7 20 



6 53 
6 30 
6 08 
5 49 



Alt. 


Refr. 


Alt. 


Refr. 


Alt. 


Refr. 


9° 30' 


5' 32" 


18° 


2' 56" 


j 30° 


1' 40" 


10 


5 16 


19 


2 46 


35 


1 22 


11 


4 48 


20 


2 37 


40 


1 09 


12 


4 25 


21 


2 29 


45 


58 


13 


4 05 


22 


2 22 


50 


48 


14 


3 47 


23 


2 15 


60 


33 


15 


3 32 


24 


2 09 


70 


21 


16 


3 19 


26 


1 58 


80 


10 


17 


3 07 


28 


1 48 


90 






P.M. 



A.M. 




angular width especially as the apparent motion of the sun across 
the field of view is very rapid. It therefore becomes advisable 
(when sighting directly at the sun with the transit telescope) to 
sight the cross wires on the edges of the 
sun, as shown in Fig. 2, and make due 
allowance for the semi-diameter of the 
sun. The effect of this is to obtain an 
altitude which differs from the true alti- 
tude by the angular value of the semi- 
diameter. The observed azimuth differs 
from the true azimuth by the semi- 
diameter ^ cos h. When the sun is at 
the horizon, cos /i = 1, and the allowance equals the semi-diameter 
both for altitude and azimuth. For higher altitudes the allow- 
ance for azimuth is much larger than the semi-diameter, since 
the divisor (cos h) is small. If several observations are taken 
within a short interval, the change in this allowance for azimuth 
during this short interval may be too small for notice and one 
value may be sufficiently accurate for all the observations. 

There is a slight variation in the semi-diameter as is shown in 
the accompanying tabular form, giving average values, which 



Fig. 2. 



608 



RAILROAD CONSTRUCTION. 



may be used by interpolation, if a closer value than the nearest 
minute is desired. 



Time. 


Semi-diam. of the Sun 
in minutes of arc. 


Jan. 1 . . . . 
April 1 . . . 
July 1.... 
Oct. 1 


16'.30 (max) 

16.03 

15 .76 (min) 

16.01 



Latitude. If the latitude of the place of observation is not 
known to the nearest minute, it may readily be obtained by 
observing the altitude of the sun at culmination at noon. The 
horizontal cross wire should be sighted at the upper (or the lower) 
edge of the disc of the sun. 

If cZ = angular diameter of sun r= refraction 

<}> = latitude 8 = declination 

A' = observed angle of elevation 

then <l> = 90°-W-r-d±id] 

in which | d is + for an observation on the lower edge, 
and I d is— for an observation on the upper edge. 

Set up the transit several minutes before noon, taking sufficient 
time to level up with the utmost care. Set the horizontal cross 
wire on the upper (or lower) edge of the sun and with the tangent 
screw follow the motion of the sun. As the required angle is 
found at culmination, the motion of the telescope should cease 
when the highest altitude is obtained and the sun begins to 
descend. 

Azimuth by an Observation with the transit telescope. Set 
up the transit at a convenient station from which an unobstructed 
view of the sun may be obtained at all times and from which a 
convenient permanent azimuth mark (e.g., a distant steeple or 
chimney) may be observed. Point at the azimuth mark with 
the horizontal plates reading zero. With the upper plate loose, 
point at the sun observing the time, altitude and the horizontal 
angle from the azimuth mark. Three or more such observations 
are generally advisable, especially as they are so easily and 
quickly taken and are such a valuable check on each other. A 
single observation may be vitiated by some inaccuracy or blunder 



APPENDIX. 



609 



in manipulation or reading which would not be discovered unless 
more than one observation is taken, in which case the error 
would hardly be precisely repeated both in nature and amount. 
Finally, point at the azimuth mark to test whether the lower 
plate has slipped. The reading on the azimuth mark should 
beO°. 

Reducing the Observations. Compute the declinations for 
the given times of observation. If several observations are 
taken, it is generally best to compute the declinations for the 
times of the first and last observations and interpolate for the 
others. The observations may most readily be reduced by using 
a regular form as given below. The six observations quoted 
were taken in 15 minutes by one of the author's students. 





Apparent 
Altitude 


a 


h 


d 


! 

Z 


Semi- 
diam. 


True Azi. 


Time 


cos.ap. 
alt. 


of Mark. 


4:50 
4:53 
4:55 
4:58 
5:00 
5:03 


22° 48'.5 
22 12 .5 
21 44 .5 
21 19 .0 
20 49 .5 
20 28.0 


237° 41' 
238 11 
238 34 

238 55 

239 19 .5 
239 38.0 


22° 30'.3 
21 54 .3 
21 26 .2 
21 0.7 
20 31 .1 
20 9.5 


14° 45'.6 
45.6 
45 .6 
45.7 
45 .7 

14 45.7 


^9° 16'.6 
^8 46.6 
88 23 .3 
§8 02.4 
87 38 .0 
p 19 .9 


17'.2 
17.2 
17 .1 
17.1 
17 .0 
17.0 


213° 19^6 
19.6 
19.8 
19.7 
19.5 

213 19.1 



Mean = 213°19^55. 

Observations taken Apr. 29, 1897: Semi-diam. of Sun 15'. 9. 
Sun observed in lower left-hand corner. 

a = horizontal angle to azimuth mark, the angle being measured 

to the right. 
/i = app. alt.— refraction— semi-diam. of sun; semi-diam. is 

-f when sun is above hor. cross wire, —when below. 
5 = dechnation, and Z = computed angle (as illustrated below). 



Semi-diam. 

True azimuth of mark = 540 zb — dzZ— a, m which 

cos. app. alt. 

. ,, , ^ T^ ,;r 11 Semi-diam. . 
Z 18+for A. M. and -for P. M. and the ;— is+when the 



cos. app. alt. 

sua is on the left of the middle wire (as above); 
is— when the sun is on the right of the middle wire. 



Semi-diam. 
COS. app. alt. 



610 



RAILROAD CONSTRUCTION. 



As a numerical specimen of the reduction: — App. decl. Green- 
wich mean noon Apr. 29, 1897, 14° 38'.0; hourly change-|-077; 
diff. of time between Greenwich and Philadelphia 5.0 hours; 
5 P. M. at Philadelphia = 10 P. M. at Greenwich; therefore 5 
for 5 P. M. at Philadelphia = 14° 38'.0+10X0'.77 = 14° 45'.7. 
Using the equation 



sm 



-^^ 



sin (s — CO h) sin (s — co 0) 



sin CO h sin co </> 



CO ;i = 67° 29'.7 
CO = 50° 02 .0 
CO 5= 75° 14.4 



s-co ;i=28° 53'.3, sin =9.684041 
s-co</)=46° 21.0, sin =9.859480 



192° 46M 
5 = 96° 23 .0 



sin CO ;i =9.965599 
sin CO 0=9.884466 



9.850065 



iZ =44° 38'.3: Z=89° 16'.6 



9.543521 



9.850065 

2.|9. 693456 

9.846728 



=sin 44° 38'.3 



Semi-diam. Sun 



15.9 



7 = 17'.2 



COS. app. alt. cos 22° 48 

540°+17'.2= 540° 17'.2 
-Z-a=-89° 16'.6-237° 41' = -326° 57'.6 



213° 19'.6 =true azimuth of mark. 

The instrument used had a vertical circle reading 30" directly 
and could be estimated to 15". 



EXPLANATORY NOTE ON THE USE OF THE TABLES. 

The logarithms here given are '^five-place/' but the last 
figure sometimes has a special mark over it {e.g., 6) Vr-hich indi- 
cates that one-half a unit in the last place should be added. 
Tor example 



the value 
.69586 
.69586 



includes all values between 
.6958575000 + and .6958624999 
.6958625000 + and .6958674999 



The maximum error in any one value therefore does not 
exceed one-quarter of a fifth-place unit. 

When adding or subtracting such logarithms allow a half-unit 
for such a sign. For example 



.69586 . 


.69586 


.69585 


.10841 


.10841 


.10841 


.12947 


.12947 


.12947 



.93374 .93375 .93375 

All other logarithmic operations are performed as usual and 
are supposed to be understood by the student. 

611 







«« 


TABLE I.— RADII OF CURVES. 








Deg 


0° 


1° 


3° 


3° 


Deg 


Min 


Radiu 


s. Log i2 


Radius. 


logM 


Radius. 


Log^B 


Racfius. 


Log 22 


Min 





00 


00 


5729.6 


3.75813 


2864-9 


3.45711 


1910-1 


3-28105 





1 


343775 


5-53627 


5635.7 


-75095 


2841.3 


-45351 


1899-5 


-27864 


1 


2 


171887 


5.23524 


5544.8 


-74389 


2818.0 


.44993 


1889-1 


.27625 


2 


3 


114592 


5.05915 


5456.8 


.73694 


2795-1 


.44639 


1878-8 


.27387 


3 


4 


85944 


4-93421 


5371.6 


.73010 


2772-5 


.44287 


1868-6 


.27151 


4 


5 


68755 


4.83730 


5288.9 


-72336 


2750-4 


-43939 


1858.5 


-26915 


5 


6 


57296 


4.75812 


5208-8 


3-71673 


2728-5 


3-43593 


1848.5 


3-26681 


6 


7 


49111 


-69117 


5131-0 


.71020 


2707-0 


.43249 


1838.6 


-26448 


7 


8 


42972 


-63318 


5055-6 


.70377 


2685.9 


.42909 


1828-8 


-26217 


8 


9 


38197 


.58203 


4982.3 


.69743 


2665.1 


.42571 


1819.1 


-25986 


9 


10 


34377 


.53627 


4911.2 


-69118 


2644-6 


-42235 


1809-6 


-25757 


10 


11 


31252 


4.49488 


4842.0 


3-68502 


2624.4 


3-41903 


1800-1 


3-25529 


11 


12 


28648 


.45709 


4774.7 


.67895 


2604.5 


.41572 


1790-7 


•25303 


12 


13 


26444 


-42233 


4709.3 


.67296 


2584.9 


.41245 


1781.5 


-25077 


13 


14 


24555 


-39014 


4645.7 


.66705 


2565.6 


.40919 


1772.3 


-24853 


14 


15 


22918 


.36018 


4583.8 


-66122 


2546.6 


.40597 
3.40276 


1763.2 


-24629 


15 


16 


21486 


4.33215 


4523.4 


3-65547 


2527.9 


1754.2 


3-24407 


16 


17 


20222 


-30582 


4464.7 


-64979 2509-5 


.39958 


1745.3 


-24186 


17 


18 


19099 


•28100 


4407-5 


.64419 2491.3 


.39642 


1736-5 


.23967 


18 


19 


18093 


.25752 


4351-7 


.63865 2473.4 


.39329 


1727-8 


.23748 


19 


20 


17189 


-23524 


4297.3 


-63319 


2455.7 


•39017 


1719-1 


-23530 


20 


21 


16370 


4.21405 


4244.2 


3-62780 


2438.3 


3.38708 


1710-6 


3-23314 


21 


22 


15626 


.19385 


4192.5 


-62247 


2421.1 


.38401 


1702.1 


-23098 


22 


23 


14947 


.17454 


4142.0 


.61720 


2404.2 


.38097 


1693.7 


.22884 


23 


24 


14324 


.15606 


4092.7 


-61200 


2387.5 


.37794 


1685.4 


.22670 


24 


25 


13751 


.13833 


4044.5 


-60686 


2371.0 


.37494 


1677.2 


-22458 


25 


26 


13222 


4.12130 


3997.5 


3-60178 


2354.8 


3.37195 


1669.1 


3-22247 


26 


27 


12732 


.10491 


3951.5 


-59676 


2338.8 


.36899 


1661-0 


.22037 


27 


28 


12278 


.08911 


3906.6 


-59180 


2323.0 


.36604 


1653-0 


.21827 


28 


29 


11854 


.07387 


3862.7 


.58689 


2307.4 


.36312 


1645-1 


.21619 


29 


30 


11459 


.05915 


3819.8 


-58204 


2292.0 


•36021 


1637-3 


-21412 


30 


31 


11090 


4.04491 


3777.9 


3.57724 


2276.8 


3.35733 


1629-5 


3-21206 


31 


32 


10743 


.03112 


3736.8 


-57250 


2261.9 


.35446 


1621-8 


.21000 


32 


33 


10417 


.01776 


3696.6 


.56780 


2247.1 


.35162 


1614-2 


.20796 


33 


34 


10111 


4.00479 


3657.3 


-56316 


2232.5 


.34879 


1606-7 


.20593 


34 


35 


9822 


.2 3.99221 


3618.8 


-55856 
3-55401 


2218.1 


.34598 


1599.2 


-20390 


35 
36 


36 


9549 


.3 3.97997 


3581-1 


2203.9 


3.34318 


1591.8 


3-20189 


37 


9291 


.3 -96807 


3544.2 


.54951 


2189.8 


.34041 


1584.5 


.19988 


37 


38 


9046 


-7 .95649 


3508.0 


.54506 


2176.0 


.33765 


1577.2 


.19789 


38 


39 


8814 


.8 -94521 


3472.6 


.54065 


2162.3 


.33491 


1570.0 


.19590 


39 


40 


8594 


-4 .93421 


3437-9 


.53629 


2148.8 


.33219 


1562.9 


-19392 


40 
41 


41 


8384 


.8 3.92349 


3403.8 


3-53197 


2135.4 


3.32949 


1555.8 


3-19195 


42 


8185 


.2 .91302 


3370.5 


.52769 


2122.3 


.32680 


1548.8 


.18999 


42 


43 


7994 


8 .90281 


3337.7 


.52345 


2109.2 


-32412 


1541.9 


.18804 


43 


44 


7813 


1 .89282 


3305.7 


.51925 


2096.4 


.32147 


1535.0 


.18610 


44 


45 


7639 


5 .88306 


3274.2 


.51510 


2083. 7 


-31883 


1528.2 


-18417 


45 
46 


46 


7473 


4 3.87352 


3243.3 


3.51098 


2071.1 


3-31621 


1521.4 


3-18224 


47 


7314 


4 .86418 


3213.0 


.50691 


2058.7 


.31360 


1514.7 


-18032 


47 


48 


7162 


.85503 


3183.2 


.50287 


2046.5 


.31101 


1508.1 


-17842 


48 


49 


7015 


9 .84608 


3154.0 


.49883 


2034.4 


.30843 


1501.5 


-17652 


49 


50 


6875 


6 83731 


3125.4 


-49490 


2022.4 


•30587 


1495-0 


-17462 


50 
51 


51 


6740 


7 3.82871 


3097.2 


3-49097 


2010-6 


3.30332 


1488-5 


3-17274 


52 


6611 


1 -82027 


3069.6 


.48707 


1998.9 


-30079 


1482.1 


.17087 


52 1 


53 


6486 


4 .81200 


3042.4 


.48321 


1987.3 


-29827 


1475.7 


.16900 


53 


54 


.6366 


3 .80388 


3015.7 


.47939 


1975.9 


-29577 


1469.4 


.16714 


54 


55 


6250 


5 .79591 


2989.5 


.47559 


1964-6 


-29328 


1463 . 2 


-16529 


55 
56 


56 


6138- 


9 3.78809 


2963.7 


3-47183 


1953.5 


3-29081 


1457.0 


3-16344 


57 


6031. 


2 .78040 


2938.4 


.46811 


1942.4 


-28835 


1450.8 


.16161 


57 


58 


5927. 


2 .77285 


2913.5 


.46441 


1931.5 


-28590 


1444-7 


.15978 


58 


59 


5826. 


8 .76542 


2889.0 


.46075 


1920.7 


-28347 


1438-7 


.15796 


59 


60 


5729. 


6 .75813 


2864.9 


.45711 


1910.1 


.28105 1432-7 


.15615 


60 



612 











tablp: 


I. 


-RADII OF CURVES. 










Deg 

Min 


4^ 


5° 


6° 


7° 


Deg 


Radius. 


Log^K 


Radius. 


logH 


Radius. 


logH 


Radius. 


logH 


Min 



1 
2 
3 
4 
5 

6 
7 
8 
9 

10 


1432.7 
1426.7 
1420.8 
1415.0 
1409.2 
1403.5 


3 


15615 
15434 
15255 
15076 
14897 
14720 


1146 
1142 
1138 
1134 
1131 
1127 


3 
5 
7 
9 
2 
5 


3 


.05929 
.05784 
.05640 
.05497 
.05354 
05211 


955 
952 
950 
947 
944 
942 


-37 
72 
09 
48 
88 
29 


2 


98017 
97896 
97776 
97657 
97537 
97418 


819.02 
817.08 
815.14 
813.22 
811.30 
809-40 


2 


91329 
91226 
91123 
91021 
90918 
90816 




1 
2 
3 
4 
5 


1397.8 
1392.1 
1386.5 
1380.9 
1375.4 


3 


14543 
.14367 
.14191 
.14017 

13843 


1123 
1120 
1116 
1112 
1109 


8 

2 
5 
9 
3 


3 


05069 
04928 
04787 
04646 
04506 


939 
937 
934 
932 
929 


72 
16 
62 
09 
57 


2 


97300 
97181 
97063 
96945 
96828 


807.50 
805.61 
803.73 
801.86 
800.00 


2 


90714 
90612 
90511 
90410 
90309 


6 
7 
8 
9 
10 


11 
12 
13 
14 
15 


1369.9 
1364.5 
1359.1 
1353.8 
1348.4 


3 


.13669 
.13497 
.13325 
.13154 
12983 


1105 
1102 
1098 
1095 
1091 


8 

2 
7 
2 
7 


3 


04366 
04227 
04088 
03949 
03811 


927 
924 
922 
919 
917 


07 
58 
10 
64 
19 


2 


96711 
96594 
96478 
96361 
96246 


798.14 
796.30 
794-46 
792-63 
790-81 


2 


90208 
90107 
90007 
89907 
89807 


11 
12 
13 
14 
15 


16 
17 
18 
19 
20 


1343-2 
1338.0 
1332.8 
1327.6 
1322-5 


3 


12813 

.12644 

.12475 

12307 

12140 


1088 
1084 
1081 
1078 
1074 


3 
8 

4 
1 
7 


3 


03674 
03537 
03400 
03264 
03128 


914 
912 
909 
907 
905 


75 
33 
92 
52 
13 


2 


96130 
96015 
95900 
95785 
95671 


789-00 
787-20 
785-41 
783.62 
781.84 


2 


89708 
89608 
89509 
89410 
89312 


16 
17 
18 
19 
20 


21 
22 
23 
24 
25 


1317.5 
1312.4 
1307.4 
1302.5 
1297.6 


3 


11974 
.11808 
11642 
11477 
11313 


1071 
1068 
1064 
1061 
1058 


3 

7 
4 
2 


3 


02992 
02857 
02723 
02589 
02455 


902 
900 
898 
895 
893 


76 
40 
05 
71 
39 


2 


95557 
95443 
95330 
95217 
95104 


780.07 
778-31 
776-55 
774-81 
773-07 


2 


89213 
89115 
89017 
88919 
88821 


21 

22 
23 
24 
25 


26 
27 
28 
29 
30 


1292.7 
1287.9 
1283.1 
1278.3 
1273.6 


3 


11150 
10987 
10825 
10663 
10502 


1054 
1051 
1048 
1045 
1042 


9 
7 
5 
3 

1 


3 


02322 
02189 
02056 
01924 
01792 


891 
888 
886 
884 
881 


08 
78 
49 
21 
95 


2 


94991 
94879 
94767 
94655 
94544 


771-34 
769-61 
767-90 
766-19 
764-49 


2 


88724 
88627 
88530 
88433 
88337 


26 
27 
28 
29 
30 


31 
32 
33 
34 
35 


1268.9 
1264.2 
1259.6 
1255.0 
1250.4 


3 


10341 
10182 
10022 
09864 
09705 


1039 
1035 
1032 
1029 
1026 



9 
8 
7 
6 


3 


01661 
01530 
01400 
01270 
01140 


879 
877 
875 
873 
870 


69 
45 
22 
00 
80 


2 


94433 
94322 
94212 
94101 
93991 


762-80 
761-11 
759-43 
757-76 
756-10 


2. 


88241 
88145 
88049 
87953 
87858 


31 
32 
33 
34 
35 


36 
37 
38 
39 
40 


1245.9 
1241.4 
1236.9 
1232.5 
1228.1 


3 


09548 
09391 
09234 
09079 
08923 


1023 
1020 
1017 
1014 
1011. 


5 
5 
5 
5 
5 


3 


01010 
00882 
00753 
00625 
00497 


868 
866 
864 
862 
859 


60 
41 
24 
07 
92 


2 


93882 
93772 
93663 
93554 
93446 


754-44 
752-80 
751-16 
749-52 
747-89 


2. 


87762 
87668 
87573 
87478 
87384 


36 
37 
38 
39 
40 


41 
42 
43 
44 
45 


1223.7 
1219.4 
1215.1 
1210.8 
1206.6 


3 


08769 
08614 
08461 
08308 
08155 


1008 

1005. 

1002. 

999. 

996. 


6 

6 

7 

76 

87 


3 

3 

2 


00370 
00242 
00116 
99989 
99863 


857 
855 
853 
851 
849 


78 

65 
53 

42 
32 


2- 


93337 
93229 
93122 
93014 
92907 


746-27 
744-66 
743-06 
741-46 
739- 86 


2. 


87290 
87196 
87102 
87008 
86915 


41 
42 
43 
44 
45 


46 
47 
48 
49 
50 


1202.4 
1198.2 
1194.0 
1189.9 
1185.8 


3 


08003 
07852 
07701 
07550 
07400 


993. 
991. 
988. 
985- 
982- 


99 

13 
28 
45 
64 


2 


99738 
99613 
99488 
99363 
99239 


847 

845 

843. 

841 

838 


23 
15 
08 
02 
97 


2 


92800 
92693 
92587 
92480 
92374 


738-28 
736- 70 
735-13 
733-56 
732-01 


2- 


86822 
86729 
86636 
86544 
86451 


46 
47 
48 
49 
50 


51 
52 
53 
54 
55 


1181.7 
1177.7 
1173-6 
1169.7 
1165-7 


3 


07251 
07102 
06954 
06806 
06658 


979- 
977- 
974- 
971. 
968 


84 
06 
29 
54 
81 


2 


99115 
98992 
98869 
98746 
98624 


836 

834 

832 

830- 

828 


93 
90 
89 
88 
88 


2 


92269 
92163 
92058 
91953 
91849 


730-45 
728-91 
727-37 
725.84 
724-31 


2 


86359 
86267 
86175 
86084 
85992 


51 
52 
53 
54 
55 


56 
57 
58 
59 
60 


1161-8 
1157-9 
1154-0 
1150-1 
1146.3 


3 


06511 
06365 
06219 
06074 
05929 


966 
963 
960 
958 
955 


09 
39 
70 
03 
37 


2 


98501 
98380 
98258 
98137 
98017 


826 
824 
822 
820 
819 


89 

91 
93 
9V 
02 


2 


91744 
91640 
91536 
91433 
91329 


722.79 
721.28 
719.77 
718.27 
716.78 


2 


85901 
.85810 
.85719 
.85629 
•85538 


56 
57 
58 
59 
60 



613 



TABLE I.— RADII OF CURVES. 



Deg. 


8° 
Radius. 1 


.ogn 


9° 


10° 


11° 


Deg 


Min. 


Radius. 


logJR 


Radius. 


logli 


Radius. L 


■Og-K 


Min 





716.78 2 


.85538 


637-27 


2.80432 


573. 69 


2.75867 


521.67 2 


71739 





1 


715.29 


.85448 


636 


.10 


.80352 


572 


73 


.75795 


520 


88 


71674 


1 


2 


713.81 


.85358 


634 


.93 


.80272 


571 


78 


.75723 


520 


10 


71608 


2 


3 


712.34 


.85268 


633 


.76 


.80192 


570 


84 


.75651 


519 


32 


71543 


3 


4 


710.87 


.85178 


632 


.60 


.80113 


569 


90 


.75579 


518 


54 


71478 


4 


5 


709.40 


.85089 


631 
630 


.44 
r29 


.80033 


568 


96 


.75508 


517 


76 


71413 


5 


6 


707.95 2 


85000 


2.79954 


568 


02 


2-75436 


516 


99 2 


71348 


6 


7 


706.49 


84911 


629 


14 


.79874 


567 


09 


-75365 


516 


21 


71283 


7 


8 


705.05 


84822 


627 


99 


.79795 


566 


16 


.75293 


515 


44 


71218 


8 


9 


703.61 


84733 


626 


85 


.79716 


565 


23 


.75222 


514 


68 


71153 


9 


10 


702.17 


84644 


625 


71 


.79637 


564 


31 


.75151 


513 


91 


71088 
71024 


10 


11 


700-75 2 


84556 


624 


58 


2.79558 


563 


38 


2-75080 


513 


15 2 


11 


12 


699. 33 


84468 


623 


45 


.79480 


562 


47 


-75009 


512 


38 


70959 


12 


13 


697.91 


84380 


622 


32 


.79401 


561 


55 


-74939 


511 


63 


70895 


13 


14 


696.50 


84292 


621 


20 


.79323 


560 


64 


.74868 


510 


87 


70831 


14 


15 


695.09 


84204 


620 


09 


•79245 


559 


73 


-74798 


510 


11 


70767 


15 


16 


693.70 2 


84117 


618 


97 


2.79167 


558 


82 


2-74727 


509 


36 2 


70702 


16 


17 


692.30 


84029 


617 


87 


.79089 


557 


92 


.74657 


508 


61 


70638 


17 


18 


690.91 


83942 


616 


76 


.79011 


557 


02 


.74587 


507 


86 


70575 


18 


19 


689.53 


83855 


615 


66 


.78934 


556 


12 


.74517 


507 


12 


70511 


19 


20 


688.16 


83768 


614 


56 


.78856 


555 


23 


.74447 


506 


38 


70447 


20 


21 


686.78 2 


83682 


613 


47 


2.78779 


554 


34 


2.74377 


505 


64 2 


70383 


21 


22 


685.42 


83595 


612 


38 


.78702 


553 


45 


.74307 


504 


90 


70320 


22 


23 


684.06 


83509 


611 


30 


.78625 


552 


56 


.74238 


504 


16 


70257 


23 


24 


682.70 


83423 


610 


21 


•78548 


551 


68 


•74168 


503 


42 


70193 


24 


25 


681.35 


83337 


609 


14 


.78471 


550 


80 


•74099 


502 


69 


70130 


25 


26 


680.01 2 


83251 


608 


06 


2.78395 


549 


92 


2-74030 


501 


96 2 


70067 


26 


27 


678.67 


83166 


606 


99 


•78318 


549 


05 


•73961 


501 


23 


70004 


27 


28 


677.34 


83080 


605 


93 


•78242 


548 


17 


•73892 


5C0 


51 


69941 


28 


29 


676.01 


82995 


604 


86 


.78165 


547 


30 


•73823 


499 


78 


.69878 


29 


30 


674-69 


82910 


603 


80 


.78089 


546 


44 


-73754 


499 


06 


68815 


30 


31 


673.37 2 


82825 


602 


75 


2.78013 


545 


57 


2-73685 


498 


34 2 


69752 


31 


32 


672.06 


82740 


601 


70 


.77938 


544 


71 


•73617 


497 


62 


.69690 


32 


38 


670.75 


82656 


600 


65 


.77862 


543 


86 


-73548 


496 


91 


•69627 


33 


34 


669.45 


82571 


599 


61 


•77786 


543 


00 


-73480 


496 


19 


•69565 


34 


35 


668.15 


82487 


598 


57 


•77711 


542 


15 


-73412 


495 


48 


.69503 


35 


36 


666.86 2 


82403 


597 


53 


2^77636 


541 


30 


2-73343 


494 


.77 2 


. 69440 


36 


37 


665.57 


82319 


596 


50 


•77561 


540 


45 


.73275 


494 


-07 


•69378 


37 


38 


664.29 


82235 


595 


47 


•77486 


539 


61 


•73207 


493 


-36 


•69316 


38 


39 


663.01 


82152 


594 


44 


•77411 


538 


76 


-73140 


492 


-66 


•69254 


39 


40 


661.74 


82068 


593 


42 


.77336 


537 


92 


•73072 


491 


-96 


.69192 


40 


41 


660.47 2 


81985 


592 


40 


2.77261 


537 


.09 


2.73004 


491 


-26 2 


.69131 


41 


4^ 


659.21 


81902 


591 


38 


•77187 


536 


25 


.72937 


490 


• 56 


•69069 


42 


43 


657.95 


81819 


590 


37 


•77112 


535 


42 


-72869 


489 


.86 


•69007 


43 


44 


656.69 


81736 


589 


36 


•77038 


534 


59 


-72802 


489 


.17 


•68946 


44 


45 


655.45 


81653 


588 


36 


•76964 


533 


77 


-72735 
2-72668 


488 


.48 


.68884 


45 
46 


46 


654.20 2 


81571 


587 


36 


2-76890 


532 


94 


487 


.79 2 


.68823 


47 


652.96 


81489 


586 


36 


•76816 


532 


12 


-72601 


487 


.10 


.68762 


47 


48 


651.73 


81406 


585 


36 


•76742 


531 


30 


•72534 


486 


.42 


.68701 


48 


49 


650-50 


81324 


584 


37 


•76669 


530 


49 


.72467 


485 


73 


.68640 


49 


50 


649.27 


81243 


583 


38 


.76595 


529 


67 


72401 


485 


05 


.68579 


50 
51 


51 


648.05 2 


81161 


582 


40 


2.76522 


528 


86 


2-72334 


484 


37 2 


68518 


52 


646.84 


81079 


581 


42 


. 76449 


528 


05 


-72267 


483 


69 


68457 


52 


53 


645.63 


80998 


580 


44 


•76376 


527 


25 


.72201 


483 


02 


.68396 


53 


54 


644.42 


80917 


579 


47 


.76303 


526 


44 


•72135 


482 


34 


68335 


54 


55 


643.22 


80836 


578 


49 


.76230 


525 


64 


.72069 


481 


67 


68275 


55 
56 


56 


642.02 2 


80755 


577 


53 


2.76157 


524 


84 


2.72003 


481 


00 2 


68214 


57 


640. 83 


80674 


576 


56 


-76084 


524 


05 


•71937 


480 


33 


68154 


57 


58 


639.64 


80593 


575 


60 


•76012 


523 


25 


•71871 


479 


67 


68094 


58 


59 


638 .'45 


80513 


574 


64 


•75939 


522 


46 


•71805 


479 


00 


68033 


59 


dO 


637.27 


8a432 


573.69 


•75867 


521.67 


•71739 


478.34 


67973 
.... 


60 



614 











TABLE 


I. 


—RADII OF CURVES. 








Deg. 


Radius. 


LogJJ 


Deg. 


Radius. 


Log-B 


Deg. 


Radius. 


log a 


Deg. 

31° 

10 
20 
30 
40 
50 

33° 
10 
20 
30 
40 
50 

33° 

10 
20 
30 
40 
50 

34° 
10 
20 
30 
40 
50 

35° 

30 

36° 

30 

37° 
30 

38° 
30 

39° 

30 
30° 

30 

31° 

33 

33 

34 

35 

36 
37 

38 
39 
40 

41 
43 
43 
44 
45 

46 
47 
48 
49 
50 

53 
54 
56 

58 
60 


Radius 


LogJB' 


12° 

2 

t 

8 

10 
12 
14 
16 
18 


478.34 
477-02 
475.71 
474.40 
473. 10 


2 


.67973 

.67853 

67734 

67614 

67495 


14° 

2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
32 
. 34 
36 
38 
40 
42 
44 
46 
48 

50 
52 
54 
56 
58 

15° 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 
22 
24 
26 
28 
30 
32 
34 
36 
38 

40 
42 
44 
46 
48 
50 
52 
54 
56 
58 

16° 


410 
409 
408 
407 
406 


.28 
31 
34 
38 
42 


2 


.61307 
.61205 
.61102 
.61000 
60898 


16° 

5 

: 10 

15 
20 
25 

30 
35 
40 
45 
50 
55 

17° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

18° 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

19° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

30° 

5 
10 
15 
20 
25 

30 
35 
40 
45 
50 
55 

21° 


359 
357 
355 
353 
351 
350 


-26 
-42 
-59 
• 77 
98 
21 


2 


.55541 
.553J7 
.55094 
.54872 
54652 
54432 


274.37 
272.23 
270-13 
268.06 
266.02 
264.02 


2.43833 
.43494 
.43157 
•42823 
•42492 


471.81 
470.53 
469.25 
467.98 
466-72 


2 


67376 
67258 
67140 
67022 
66905 


405 
404 
403 
402 
401 


47 
53 
58 
65 
71 


2 


60796 
60694 
60593 
60492 
60391 


•42163 


348 
346 
344 
343 
341 
339 


45 
71 
99 
29 
60 
93 


2 


54214 
53997 
53780 
53565 
53351 
53138 


262.04 
260.10 
258.18 
256.29 
254.43 
252.60 


2.41837 
•41513 
.41192 
.40873 


20 
22 


465.46 
464.21 
462-97 
461-73 
460-50 


2 


66788 
66671 
66555 
66439 
66323 


400 
399 
398 
398 
397 


78 
86 
94 
02 
11 


2 


60291 
60190 
60090 
59990 
59891 


.40557 
-40243 


24 
26 
28 

30 
32 
34 


338 
336 
335 
333 
331 
330 


27 
64 
01 
41 
82 
24 


2 


52927 
52716 
52506 
52297 
52090 
51883 


250.79 
249-01 
247-26 
245-53 
243. 82 
242-14 


2-39931 
-39622 
•39315 


459-28 
458-06 
456-85 
455.65 
454-45 

453.26 
452.07 
450.89 
449.72 
448.56 


2 


66207 
66092 
65977 
65863 
65748 


396 
395 
394 
393 
392 


20 
30 
40 
50 
61 


2 


59791 
59692 
59593 
59494 
59396 


•39010 

•38707 

38407 


36 
38 


328 

327 

325 

324 

322. 

321. 


68 
13 
60 
09 
59 
10 


2. 


51677 
51472 
51269 
51066 
50864 
50663 


240-49 
238-85 
237-24 
235-65 
234-08 
232-54 


2.38109 
•37813 


40 
42 
44 
46 
48 


2 


65634 
65521 
65407 
65294 
65181 


391. 

390. 

389. 

389 

388. 


72 
84 
96 
08 

21 


2. 


59298 
59199 
59102 
59004 
58907 


.37519 
•37227 
.36937 
•36649 


319 
318 
316 
315 
313 
312 

311. 

309. 

308 

306. 

305 

304 

302 
301 
300 
299 
297 
296 


62 
16 
71 
28 
86 
45 
06 
67 
30 
95 
60 
27 
'94 
63 
33 
04 
77 
50 


2. 


50464 
50265 
50067 
49869 
49673 
49478 


231.01 
226.55 
222-27 
218-15 


2.36363 


50 
52 


447-40 
446-24 
445-09 
443-95 
442-81 


2. 


65069 
64957 
64845 
64733 
64622 


387. 

386 

385 

384 

383 


34 
48 
62 
77 
91 


2 


58809 
58713 
58616 
58519 
58423 


.35517 
•34688 
.33875 


56 
68 

13° 

2 
4 
6 
8 

10 
12 
14 
16 
18 

20 
22 
24 
26 
28 

30 
32 
34 
36 
38 

40 
42 
44 
46 
48 

50 
52 
54 
56 
58 

14° 


214-18 
210. 36 
206-68 
203-13 


2.33078 
.32296 


441-68 
440-56 
439-44 
438.33 
437-22 


2. 


64511 
64400 
64290 
64180 
64070 


383- 
382. 
381. 
380. 
379- 


06 
22 
38 
54 
71 


2. 


58327 
58231 
58135 
58040 
57945 


2. 


49284 
49090 
48898 
48706 
48515 
48325 


.31529 
.30776 


199-70 
196-38 
193-19 
190.09 


2.30037 
.2931S 
.28597 


436.12 
435.02 
433.93 
432.84 
431.76 


2. 


63960 
63851 
63742 
63633 
63524 


378- 
378- 
377- 
376. 
375 


88 
05 

23 
41 
60 


2. 


57850 
57755 
57661 
57566 
57472 


•27896 


2- 


48136 
47948 
47760 
47573 
47388 
47203 


187-10 
181-40 
176-05 
171-02 
166-28 
161-80 
157-58 
153.58 
149-79 
146-19 


2.27207 
.25863 
.24563 
.23303 


430.69 
429.62 
428.56 
427.50 
426.44 


2. 


63416 
63308 
63201 
63093 
62986 


374. 
373. 
373 

372 
371 


79 
98 
17 
37 
57 


2 


57378 
57284 
57191 
57097 
57004 


.22083 


2.20899 
.19749 
.1863a 
.17547 


295 
294 
292 
291 
290 
289 


25 
00 
77 
55 
33 
13 


2- 


47018 
46835 
46652 
46471 
46289 
46109 


425.40 
424.35 
423.32 
422-28 
421-26 


2 


62879 
62773 
62666 
62560 
62454 


370 
369 
369 
368 
367 

366 
366 
365 
364 
363 

363 
362 
361 
360 
360 


78 
99 
20 
42 
64 

86 
09 
31 
55 
78 
02 
26 
51 
76 
01 


2 


56911 
56819 
56726 
56634 
56542 


.16492 


142-77 
139.52 
136.43 
133-47 
130-66 

127-97 
125-39 
122.93 
120.57 
118.31 


2. 1546? 
. 1446^ 
.13489 
.12539 


287 
286 
285 
284 
283 
282 


94 
76 
58 
42 
27 
12 


2- 


45930 
45751 
45573 
45396 
45219 
45044 


420.23 
419.22 
418.20 
417.19 
416.19 
415.19 
414.20 
413.21 
412.23 
411.25 


2 


62349 
62243 
62138 
62034 
61929 


2 


56450 
56358 
56266 
56175 
56084 


.11613 


2.10709 
.09827 
.08965 
.08124 


280 
279 
278 
277 
276 
275 


99 
86 
75 
64 
54 
45 


2 


44869 
44694 
44521 
44348 
44176 
44004 


2 


61825 
61721 
61617 
61514 
61410 


2 


55993 
55902 
55812 
55721 
55631 


.07302 


114.06 
110.13 
106.50 
103.13 
100.00 


2.05713 
.04192 
.02736 
.01340 










410.28 


2 


61307 


359 


26 


2 


55541 


Z/4. o / 


J .^ooooi 


2.00000 


















6J 


[5 

















TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

FOR A 1*> CURVE. 



Tang. 



50 
58 
66 
75 
83 
91 

100 
108 
116 
125 
133 
141 



150 
158 
166 
175 
183 
191 

200 
208 
216 
225 
233 
241 



250 
258 
266 
275 
283 
291 



300 
308 
316 
325 
333 
342 



350 
358 
367 
375 
383 
392 



400 
409 
417 
425 
434 
442 

450 
459 
467 
476 
484 
492 



501 
509 
518 
526 
534 
543 
551 



Ext. 
Dist. 

1^. 



218 
297 
388 
491 
606 
733 



873 
024 
188 
364 
552 
752 



1.964 
2.188 
2.425 
2.674 
2.934 
3.207 



3.492 
3.790 
4.099 
4.421 
4.755 
5.100 



.459 
.829 
.211 
.606 
.013 
.432 



.863 
.307 
.762 
.230 
.710 
.202 



10.707 
11.224 
11.753 
12.294 
12.847 
13.413 



991 
582 
184 
799 
426 
066 



0823 
48 24 
89 24 
29 25 



717 
381 
058 
746 
447 
161 

886 
624 
375 
138 
913 
700 



Lonff 
Chord 
iO. 



100 
116 
133 
150 
166 
183 



199 
216 
233 
249 
266 
283 



299 
316 
333 
349 
366 
383 



399 
416 
433 
449 
466 
483 



499 
516 
533 
549 
566 
583 



.00 
.67 
.33 
.00 
.66 
^ 

.99 

.66 

.32 

• 98 

.65 

ill 

■ 97 

63 

29 

95 

61 

l27 

92 

58 

24 

89 

54 

_2p 

85 

50 

15 

80 

44 

09 



599 
616 
633 
649 
666 
682 



699 
716 
732 
749 
766 
782 



799 
815 
832 
849 
865 
882 



899 
915 
932 
948 
965 
982 



998 
1015 
1031 
1048 
1065 
1081 



70126-500 1098.33 21 



11° 

10 
20 
30 
40 
50 

13° 

10 
20 
30 
40 
50 



13° 

10 
20 
30 
40 
50 



14" 

10 
20 
30 
40 
50 



15° 

10 
20 
30 
40 
50 



16^^ 

10 
20 
30 
40 
50 



17° 
10 
20 
30 
40 
50 



18° 
10 
20 
30 
40 
50 



19° 

10 
20 
30 
40 
50 



30° 

10 
20 
30 
40 
50 



Tang. 



551 
560 
568 
576 
585 
593 



602 

610 

619 

627 

635. 

644. 



652 
661 
669 
678 
686 
695 



703 
711 
720 
728 
737 
745 



754 
762 
771 
779 
788 
796 



805 
813 
822 
830 
839 
847 



856 
864 
873 
881 
890 
898 



907 
916 
924 
933 
941 
950 



30 
82 
35 
88 
41 
_95 
49 
03 
58 
13 
69 
25 



Ext. 
Dist. 



500 
313 
137 
974 
824 
686 



561 
447 
347 
259 
183 
120 



37 
38 
39 
39 
40 
42. 

43. 

44. 

45. 

46 

47. 

48 



069 
031 
006 
993 
992 
004 



029 
066 
116 
178 
253 
341 



441 
554 
679 
818 
969 
132 



Long 
Chord 



1098. 
1114. 
1131. 
1148. 
1164. 
1181. 



1197 
1214 
1231 
1247 
1264 
1280 



1297 
1313 
1330 
1346 
1363 
1380 



1396 
1413 
1429 
1446 
1462 
1479 



31° 

10 
20 
30 
40 



33° 
10 
20 
30 
40 
50 

33° 

10 
20 
30 
40 
50 



34^ 

10 
20 
30 
40 
50 






1061 
1070 
1079 
1087 
1096 
1105 



1113 
1122 
1131 
1139 
1148 
1157 



1165 
1174 
1183 
1191 
1200 
1209 



309 
498 
699 
914 
141 
381 



958 
967 
975 
984 
993. 
1001 



1010 
1018 
1027 
1036 
1044 
1053 



1061 



63 

64 

66 

67. 

68. 

TOj 

71. 

72. 

74. 

75. 

76. 

78. 



634 
900 
178 
470 
774 
091 



421 
764 
119 
488 
869 
264 

671 
092 
525 
972 
431 
904 



888 
399 
924 
462 
013 
577 



1495 
1512 
1528 
1545 
1561 
1578 



1594. 
1611. 
1627. 
1644. 
1660. 
1677. 



1693 
1710 
1726 
1743 
1759 
1776 



1792 

1809 

1825 

1842. 

1858 

1874 



1891 
1907 
1924 
1940 
1957 
1973 



1989 
2006 
2022 
2039 
2055 
2071 
2088 



35' 

10 
20 
30 
40 
50 



36° 

10 
20 
30 
40 
50 



37° 
10 
20 
30 
40 
50 



38" 
10 
20 
30 
40 
50 



39° 

10 
20 
30 
40 
50 



30° 

10 
20 
30 
40 
50 



3r 



1217 
1226 
1235 
1244 
1252 
1261 



1270 
1279 
1287 
1296 
1305 
1314 



1322 
1331 
1340 
1349 
1358 
1366 



1375 
1384 
1393 
1402 
1410 
1419 



1428 
1437 
1446 
1455 
1464 
1472 



1481 
1490 
1499 
1508 
1517 
1526 



1535 
1544 
1553 
1562 
1571 
1580 



1589 



Ext. 
Dist. 
JE. 



97 
99 
100 
102 
103 
105 



107 
108. 
110. 
112. 
113. 
115. 



117 
119 
120 
122 
124 
126 



Long 
Chorci 
XC. 



2088.3 
2104.7 
2121.1 
2137.4 
2153.8 
2170.2 



128 
129 
131 
133 
135 
137 



139 

141. 

142 

144. 

146 

148. 

150. 
152. 
154. 
156. 
158. 
160. 



162 
164 
166 
169 
171 
173 



175 
177 
179 
181 
184 
186 

188 
190 
192 
195 
197 
199 



202 
204 
206 
209 
211 
213 



2186.5 
2202.9 
2219.2 
2235.6 
2251.9 
2268.3 



2284.6 
2301.0 
2317.3 
2333.6 
2349.9 
2366.2 



2382.5 
2398.8 
2415.1 
2431.4 
2447.7 
2464.0 



2480.2 

2496.5 

2512.8 i 

2529.0 

2545.3 

2561.5 



2577.8 
2594.0 
2610.3 
2626.5 
2642.7 
2658.9 



2675.1 
2691.3 
2707.5 
2723-7 
2739.9 
2756-1 



2772-3 
2788.4 
2804.6 
2820.7 
2836-9 
2853.0 



216.25 3062.4 



2869.2 
2885.3 
2901.4 
2917.6 
2933.7 
2949.8 



2965-9 
2982-0 
2998-1 
3014-2 
3030-2 
3046.3 



616 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

FOR A 1° CURVE. 



A 


Tang- 


Ext. 
Dist. 
JE. 


Chord 


A 


T^-^g' 


Ext. 
Dist. 


Long 
Chord 


A 


T^ng. 


Ext. 
Dist. 
JE. 


cte 


31°, 

10' 
20 
30 
40 
50 


1589 
1598 
1606 
1615 
1624 
1633 


.0 
.0 
.9 
.9 
• 9 
9 


216 
218 
221 
223 
225 
228 


.25 

• 66 
.08 
.51 

• 96 
.42 


3062 
3078 
3094 
3110 
3126 
3142 


.4 
4 
5 
5 
6 
6 


41° 

10 
20 
30 
40 
50 

43° 

10 
20 
30 
40 
50 

43° 

10 
20 
30 
40 
50 

44° 

10 
20 
30 
40 
50 

45° 

10 

20, 

30 

40 

50 

46° 

10 
20 
30 
40 
50 

47° 
10 
20 
30 
40 
50 

48° 
10 
20 
30 
40 
50 

49° 

10 
20 
30 
40 
50 

50° 

10 
20 
30 
40 
50 

51° 


2142.2 
2151.7 
2161.2 
2170.8 
2180.3 
2189.9 


387 
390 
394 
397 
400 
404 


.38 4013 
.714028 
.06 4044 
.43 4059 
82 4075 
22 4091 


1 
7 
3 
9 
5 
1 


51° 

10 
20 
30 
40 
50 

53° 
10 
20 
30 
40 
50 

53° 

10 
20 
30 
40 
50 

54° 

10 
20 
30 
40 
50 

55° 

10 
20 
30 
40 
50 

56° 

10 
20 
30 
40 
50 


2732 
2743 
2753 
2763 
2773 
2784 


.9 
.1 
.4 
.7 
.9 
2 


618 
622 
627 
631 
636 
640 


39 
.81 

• 24 

• 69 
16 
66 


4933.4 
4948.4 
4963.4 
4978.4 
4993.4 
5008-4 


33° 

10 
20 
30 
40 
50 


1643 
1652 
1661 
1670 
1679 
1688 


.0 
.0 

• 

• 
.1 
•1 


230 
233 
235 
238 
240 
243 


• 90 
39 
90 
43 
96 
52 


3158 
3174 
3190 
3206 
3222 
3238 


6 
6 
6 
6 
6 
6 


2199.4 
2209.0 
2218.6 
2228.1 
2237.7 
2247.3 


407 
411 
414 
417 
421 
424 


64 4106 
074122 
52'4137 
994153 
48 4168 
98 4184 


6 
2 
7 
3 
8 
3 


2794 
2804 
2815 
2825 
2835 
2846 


.5 
.9 
.2 
.6 
.9 
.3 


645 
649 
654 
658 
663 
668 


17 
70 
25 
83 
42 
03 


5023.4 
5038.4 
5053.4 
5068.3 
5083.3 
5098.2 


33° 

10 
20 
30 
40 
50 


1697 
1706 
1715 
1724 
1733 
1742 


• 2 
3 
3 
4 
5 
6 


24G 
248 
251 
253 
256 
259 


08 
66 
26 
87 
50 
14 


3254 
3270 
3286 
3302 
3318 
3334 


6 
6 
6 
5 
5 
4 


2257.0 
2266.6 
2276.2 
2285.9 
2295.6 
2305-2 


428 
432 
435 
439 
442 
446 


504199 
044215 
594230 
164246 
754261 
35 4277 


8 
3 
8 
3 
8 
3 


2856 
2867 
2877 
2888 
2898 
2908 


.7 672 
.1 677 

5 681 
.0 686 
.4 691 

9 696 


66 
32 
99 
68 
.40 
.13 


5113.1 
5128.0 
5142.9 
5157-8 
5172.7 
5187.6 


34° 

10 
20 
30 
40 
50 


1751 
1760 
1770 
1779 
1788 
1797 


7 
8 


1 
2 
4 


261 
264 
267 
269 
272 
275 


80 
47 
16 
86 
58 
31 


3350 
3366 
3382 
3398 
3414 
3430 


4 
3 
2 
2 

1 



2314. 9 
2324.6 
2334.3 
2344.1 
2353. 8 
2363.5 


449 
453 
457 
460 
464 
468 


98 4292 
62 4308 
27,4323 
954339 
64 4354 
35 4369 


7 
2 
6 

5 
9 


2919 
2929 
2940 
2951 
2961 
2972 


4 
9 
4 

5 
1 


700 
705 
710 
715 
720 
724 


.89 
66 
46 
28 
11 
97 


5202.4 
5217-3 
5232.1 
5246.9 
5261.7 
5276.5 


35° 

10 
20 
30 
40 
50 


1806 
1815 
1824 
1834 
1843 
1852 


6 
7 
9 
1 
3 
5 


278 
280 
283 
286 
289 
292 


05 
82 
60 
39 
20 
02 


3445 
3461 
3477 
3493 
3509 
3525 


9 
8 

7 
5 
4 
3 


2373.3 
2383.1 
2392-8 
2402.6 
2412.4 
2422-3 


472 
475 
479 
483 
487 
490 


08 4385 
82 4400 
59 4416 
374431 
16 4446 
98 4462 


3 

7 
1 
4 
8 

2 


2982 
2993 
3003 
3014 
3025 
3035 


7 
3 
9 
5 
2 
8 


729 
734 
739 
744 
749 
754 


85 
76 
68 
62 
59 
57 


5291.3 
5306.1 
5320-9 
5335-6 
5350-4 
5365.1 


36° 

,10 
'20 

30 

40 

50 


1861 
1870 
1880 
1889 
1898 
1907 


7 
9 
1 
4 
6 
9 


294 
297 
300 
303 
306 
309 


86 

72 
59 
47 
37 
29 


3541 

3557. 

3572. 

3588. 

3604. 

3620. 


1 

8 
6 
5 
3 


2432.1 
2441.9 
2451.8 
2461.7 
2471.5 
2481.4 


494 
498 
502 
506 
510 
514 


82 4477 
67.4492 
54 4508 
42 4523 
33 4538 
25 4554 


5 
8 

2 
5 
8 

1 


3046 
3057 
3067 
3078 
3089 
3100 


5 
2 
9 
7 
4 
2 


759 
764 
769 
774 
779 
784 


58 
61 
66 
73 
83 
94 


5379.8 
5394.5 
5409.2 
5423.9 
5438.6 
5453.3 


37° 

10 
20 
30 
40 
50 


1917 
1926 
1935 
1945 
1954 
1963 


1 
4 
7 

3 
6 


312 
315 
318 
321 
324 
327 


22 
17 
13 
11 
11 
12 


3636. 

3651. 

3667. 

3683. 

3699 

3715 


1 
9 
7 
5 
3 



2491.3 
2501.2 
2511.2 
2521.1 
2531.1 
2541.0 


518 
522 
526 
530 
534 
538 


20 
16 
13 
13 
15 
18 


4569. 
4584. 
4599. 
4615. 
4630. 
4645. 


4 
7 
9 
2 
4 
7 


57° 
10 
20 
30 
40 
50 

58° 
10 
20 
30 
40 
50 

59° 

10 
20 
30 
40 
50 


3110 
3121 
3132 
3143 
3154 
3165 


9 
7 
6 

4 
2 

1 


790 
795 
800 
805 
810 
816 


08 
24 
42 
62 
85 
10 


5467.9 
5482.5 
5497.2 
5511.8 
5526.4 
5541.0 


38° 

10 
20 
30 
40 
50 


1972 
1982 
199] 
2000 
2010 
2019 


9 

2 

& 

2 
6 


330 
333 
336 
339 

342 
345 


15 
19 
25 
32 
41 
52 


3730 
3746 
3762 
3778 
3793 
3809 


8 

5 
3 

8 
5 


2551.0 
2561.0 
2571.0 
2581.0 
2591.1 
2601.1 


542 
546 
550 
554 
558 
562 


23 
30 
39 
50 
63 
77 


4660. 

4676. 

4691 

4706 

4721 

4736 


9 
1 
3 
5 
7 
9 


3176 
3186 
3197 
3208 
3219 
3230 



9 
8 
8 

7 
7 


821 
826 
831 
837 
842 
848 


37 
66 
98 
31 
67 
06 


5555.6 
5570.2 
5584.7 
5599.3 
5613.8 
5628.3 


39° 

10 
20 
30 
40 
50 


2029 
2038 
2047 
2057 
2066 
2076 



4 
8 
2 
6 



348 
351 
354 
358 
361 
364 


64 
78 
94 
11 
29 
50 


3825 
3840 
3856 
3872 
3888 
3903 


2 
9 
6 
3 

6 


2611.2 
2621.2 
2631.3 
2641.4 
2651.5 
2661-6 


566 
571 
575 
579 
583 
588 


d4 
12 
32 
54 
78 
04 


4752 
4767 
4782 
4797 
4812 
4827 


1 
3 
4 
5 
7 
8 


3241 
3252 
3263 
3274 
3285 
3296 


7 
7 
7 
8 
8 
9 


853 
858 
864 
869 
875 
880 


46 
89 
34 
82 
32 
84 


5642.8 
5657.3 
5671.8 
5686.3 
5700.8 
5715.2 


40° 

10 
20 
30 
40 
50 


2085 
2094 
2104 
2113 
2123 
2132 


4 
9 
3 
8 
3 
7 


367 
370 
374 
377 
380 
384 


72 
95 
20 
47 
76 
06 


3919 
3935 
3950 
3966 
3981 
3997 


3 

6 
3 
9 
5 


2671-8 
2681.9 
2692-1 
2702-3 
2712-5 
2722-7 


592 
596 
600 
605 
609 
614 


32 
62 
93 
27 
62 
00 


4842 
4858 
4873 
4888 
4903 
4918 


9 


1 
2 
2 
3 


60° 

10 
20 
30 
40 
50 

61° 


3308 
3319 
3330 
3341 
3352 
3363 



1 
.3 
.4 
.6 
-8 


886 
891 
897 
903 
908 
914 


38 
95 
54 
15 
79 
45 


5729.7 
5744.1 
5758-5 
5772.9 
5787.3 
5801.7 


41° 


2142 


2 


387 


38 


4013 


1 


2732.9 


618 


39 


4933 


_4 


3375 


A 


920 


M 


5816. 



617 



TABLE II. — TANGENTS, EXTERNAL DISTANCES, AND LONG 
CHORDS FOR A 1° CURVE. 



A 


Tang. 
T. 


Ext. 
Dist. 


Long 
Chord 


A 


Tang. 
T. 


Ext. 
Dist. 


Long 
Chord 


A 


Tang. 


Ext. 
Dist. 


Longr 
Chord 




E. 


LC. 




E. 


LC. 




E. 


LC, 


61° 


3375.0 


920-14 


5816.0 


68° 


3864.7 


1181. 


6 6408.0 


75° 


4396.5 


1492.4i6976-0 


10' 


3386 


3 


925 


85 


5830 


4 


10' 


3876 


8 


1188. 


4 6421 


8 


10' 


4409 . 8 


1500 


5 6989-2 


20 


3397 


5 


931 


58 


5844 


7 


20 


3889 





1195. 


2 6435 


6 


20 


4423 . 1 


1508 


6 


7002-4 


30 


3408 


8 


937 


34 


5859 


1 


30 


3901 


2 


1202. 


6449 


.4 


30 


4436.4 


1516 


7 


7015-6 


40 


3420 


1 


943 


12 


5873 


4 


40 


3913 


4 


1208. 


9 6463 


1 


40 


4449 . 7 


1524 


9 


7028-8 


50 


3431 


4 


948 


92 


5887 


7 


50 


3925 


6 


1215. 


8 6476 


9 


50 
76° 


4463.1 


1533 


1 


7041-9 


63° 


3442 


7 


954 


75 


5902 





69° 


3937 


9 


1222. 


7 6490 


.6 


4476.5 


1541 


4 


7055-0 


10 


3454 


1 


960 


60 


5916 


3 


10 


3950 


2 


1229. 


7 6504 


4 


10 


4489.9 


1549 


7 


7068-2 


20 


3465 


4 


966 


48 


5930 


5 


20 


3962 


5 


1236. 


7 6518 


1 


20 


4503.4 


1558 





7081-3 


30 


3476 


8 


972 


39 


5944 


8 


30 


3974 


8 


1243- 


7 6531 


8 


20 


4516.9 


1566 


3 


7094.4 


40 


3488 


2 


978 


31 


5959 





40 


3987 


2 


1250^ 


J 6545 


5 


40 


4530.4 


1574 


7 


7107.5 


50 


3499 


7 


984 


27 


5973 


3 


50 


3999 
4011 


5 
9 


1257- 


9 6559 


1 


50 


4544.0 


1583 


1 


7120.5 


63° 


3511 


1 


990 


24 


5987 


5 


70° 


1265. 


6572 


8 


77° 


4557.6 


1591 


6 


7133.6 


10 


3522 


6 


996 


24 


6001 


7 


10 


4024 


4 


1272. 


L 6586 


4 


10 


4571.2 


1600 


1 


7146.6 


20 


3534 


1 


1002 


3 


6015 


9 


20 


4036 


8 


1279. 


3 6600 


1 


20 


4584.8 


1608 


6 


7159.6 


30 


3545 


6 


1008 


3 


6030 





30 


4049 


3 


1286. 


5 6613 


■7 


30 


4598.5 


1617 


1 


7172.6 


40 


3557 


2 


1014 


4 


6044 


2 


40 


4061 


8 


1293. 


7 6627 


3 


40 


4612.2 


1625 


7 


7185.6 


50 


3568 


7 


1020 


5 


6058 


4 


50 


4074 


4 


1300. 


9 6640 


.9 


50 

78° 


4626.0 


1634 


4 


7198.6 


64° 


3580 


3 


1026 


6 


6072 


5 


71° 


4086 


9 


1308. 


2 6654 


.4 


4639.8 


1643 





7211.6 


10 


3591 


9 


1032 


8 


6086 


6 


10 


4099 


5 


1315. 


5 6668 





10 


4653.6 


1651 


7 


7224.5 


20 


3603 


5 


1039 





6100 


7 


20 


4112 


1 


1322. 


9 6681 


6 


20 


4667.4 


1660 


5 


7237.4 


30 


3615 


1 


1045 


2 


6114 


8 


30 


4124 


8 


1330. 


3 6695 


1 


30 


4681.3 


1669 


2 


7250.4 


40 


3626 


8 


1051 


4 


6128 


9 


40 


4137 


4 


1337. 


7 6708 


6 


40 


4695.2 


1678 


1 


7263.3 


50 


3638 


5 


1057 


7 


6143 





50 


4150 


1 


1345. 


1 6722 


1 


50 


4709.2 


1686 


9 


7276.1 


65° 


3650 


2 


1063 


9 


6157 


1 


73° 


4162 


8 


1352. 


6 6735 


6 


79° 


4723.2 


1695 


8 


7289.0 


10 


3661 


9 


1070 


2 


6171 


1 


10 


4175 


6 


1360. 


1 6749 


1 


10 


4737.2 


1704 


7 


7301-9 


20 


3673 


7 


1076 


6 


6185 


2 


20 


4188 


4 


1367^ 


6 6762 


.5 


20 


4751.2 


1713 


7 


7314-7 


30 


3685 


4 


1082 


9 


6199 


2 


30 


4201 


2 


1375. 


2 6776 





30 


4765.3 


17122 


7 


7327.5 


40 


3697 


2 


1089 


3 


6213 


2 


40 


4214 





1382. 


8 6789 


.4 


40 


4779.4 


1731 


7 


7340-3 


50 


3709 





1095 


7 


6227 


2 


50 


4226 


8 


1390. 


i6802 


• 8 


50 


4793.6 


1740 


8 

9 


7353-1 


66° 


3720 


9 


1102 


2 


6241 


2 


73° 


4239 


7 


1398. 


6816 


3 


80° 


4807.7 


1749 


7365-9 


10 


3732 


7 


1108 


6 


6255 


2 


10 


4252 


6 


1405. 


7 6829 


6 


10 


4822.0 


1759 





7378-7 


20 


3744 


6 


1115 


1 


6269 


1 


20 


4265 


6 


1413. 


5 6843 


.0 


20 


4836.2 


1768 


2 


7391.4 


30 


3756 


5 


1121 


7 


6283 


1 


30 


4278 


5 


1421. 


2 6856 


4 


30 


4850-5 


1777 


4 


7404.1 


40 


3768 


5 


1128 


2 


6297 





40 


4291 


5 


1429. 


6869 


7 


40 


4864.8 


1786 


7 


7416.8 


50 


3780 


4 


1134 


8 


5310 


9 


50 


4304 


6 


1436. 


8 6883 


1 


50 


4879.2 


1796 





7429-5 


67° 


3792 


4 


1141 


4 


6324 


8 


74° 


4317 


6 


1444. 


B 6896 


• 4 


81° 


4893.6 


1805 


3 


7442-2 


10 


3804 


4 


1148 





6338 


7 


10 


4330 


7 


1452. 


5 6909 


• 7 


10 


4908.0 


1814 


7 


7454-9 


20 


3816 


4 


1154 


7 


6352 


6 


20 


4343 


8 


1460. 


4 6923 


.0 


20 


4922.5 


1824 


1 


7467-5 


30 


3828 


4 


1161 


3 


6366 


4 


30 


4356 


9 


1468. 


4 6936 


.2 


30 


4937.0 


1833 


6 


7480-2 


40 


3840 


5 


L168 


1 


6380 


3 


40 


4370 


1 


1476. 


4 6949 


.5 


40 


4951.5 


1843 


1 


7492-8 


50 


3852 


6 


1174 


8 


6394 


1 


50 


4383 


3 


1484. 


4 6962 


8 


50 


4966-1 


1852 


6 


7505-4 


68° 


3864 


7 


1181 


6 


6408 





75° 


4396 


5 


1492. 


4 6976 


• 


83° 


4980.7 


1862 


2 


7518-0 


Correction Table (always additive) 




Degree of curve. 


A 


5° 


10° j 


15° 


20° 




T 


E 


LC 


T 


E 


LC 


T 


E 


LC 


T 


E 


LC 


10° 


! .03 


001 


.06 


.06 


003 


.13 


.10 . 


004 


.17 


.13 


-006 


-25 1 


20 


.06 


• 005 


.12 




13 


Oil 




25 


.19 . 


017 




8 


-26 


-022 




51 


30 


i .09 


• 012 


.18 




19 


025 




37 


.29 . 


038 




>6 


.39 


-051 




75 ! 


40 


1 .13 


022 


.24 




26 


046 




49 


.40 . 


070 




^4 


.53 


-093 




00 1 


50 


.16 


• 036 


.30 




34 


075 




61 


.51 . 


112 




)2 


.68 


-151 




23 ! 


60 


.20 


■ 054 


.35 




42 


111 




72 


.63 . 


168 


l.( 


)9 


.84 


-225 




46 ! 


70 


.24 


.077 


• 40 




50 


159 




83 


.76 . 


240 


l.i 


}5 


1-02 


321 




67 ! 


80 


• 29 


.107 


• 45 




60 


220 




93 


.91 . 


332 


1.^ 


to 


1-22 


-455 




87 


90 


.35 


.145 


• 49 




72 


298 


j^ 


02 1 


.09 . 


451 


l.f 


)4 


1-46 


-603 


2 


06 



618 



TABLE IIA. EXCESS LENGTH OF SUB CHORDS. SEE § 48. 



Nominal length of sub chord. 



10 



003 
005 
006 

008 
010 
013 
015 
018 
021 

025 
028 
032 

036 
041 
045 
050 
056 
061 
067 
073 
079 

085 
092 
099 

107 
114 



20 



006 
009 
012 

016 
020 
024 
029 
035 
041 
048 
055 
063 

071 
079 
088 
098 
108 
118 
129 
141 
153 
166 
179 
192 

207 
221 



30 



009 
012 
017 
022 
028 
035 
042 
050 
059 
068 
079 
089 
100 
113 
125 

139 
153 
168 

184 
201 
218 
236 
254 
273 

293 
314 



40 



• Oil 
.015 
.021 

.027 
.035 

• 043 

• 052 

• 062 

• 072 

• 084 
.097 
.109 
.123 
.139 
.154 

.171 
.189 
.207 
.226 
.247 
.268 
.290 
.313 

337 
.361 

387 



45 



Oil 
016 
022 

029 
037 
046 

055 
066 
077 
090 
103 
117 
132 
148 
165 

183 
202 
221 
242 
264 
286 
310 
334 
359 
386 
413 



50 



012 
017 
023 
030 
038 
048 
058 
069 
080 

094 
108 
122 

138 
155 
172 

191 
211 
231 
253 
275 
299 
324 
349 
375 

403 
431 



55 



012 
018 
024 

031 
039 
049 

059 
070 
082 

096 
110 
125 
141 
158 
176 
195 
215 
237 
259 
282 
306 
331 
357 
384 

412 
441 



60 



• 012 
.018 
.024 

031 
.039 

• 049 
.059 
.070 
.083 

.096 
.110 
.125 
.141 
.158 
.177 
.196 
.216 
.237 

.259 
.282 
.306 
.331 

• 357 
384 

.412 
.442 



65 



012 
017 
023 
030 
039 
048 

058 
069 
081 
094 
108 
122 
.138 
.155 
172 

191 
.211 

231 
.253 
.276 
.299 
.324 
.349 
.376 

.403 
.432 



70 



.011 
.016 
.022 

029 
.037 
.045 
.055 
.066 
.077 

.089 
.103 
.116 
.131 
.147 
.164 

.182 
.200 
.220 

.241 
.262 
.284 

.308 
.332 
.357 

383 
.410 



75 



010 
015 
020 

027 
034 
042 

051 
060 
071 
082 
094 
107 
120 
135 
151 
167 
184 
202 

221 
241 
261 

283 
305 



80 



328 .288 



352 
377 



.009 

• 013 
.018 
.023 

• 030 
.037 
.044 
.053 

• 062 

.072 
.083 
.094 

.106 
.119 
.132 

147 

• 162 
.177 
.194 
.211 
.229 

248 

• 268 



309 
331 



85 



.007 
.011 
.015 

.019 
.024 

• 030 

.036 

• 043 
.051 
.059 
.068 
.077 
.087 
.097 
.108 
.120 
.132 
.145 
.159 
.173 
.188 
.203 
.219 
.236 

.253 
.271 



90 



005 
008 
Oil 

014 
018 
022 
026 
031 
037 
043 
049 
056 

063 
070 
079 

087 
096 
105 
115 
125 
136 

147 
159 
171 

183 

.196 



95 



003 

• 004 
.006 
.008 
.010 

• 012 
014 

.017 
.020 

.023 
.027 
.030 
.034 
.038 
.043 

• 047 
.052 
.057 
.062 
.068 
.074 
.080 
.086 

093 
.099 
.109 



TABLE III. SWITCH LEADS AND DISTANCES. 
TRIGONOMETRICAL FUNCTIONS OF THE FROG ANGLES. 



Frog 

No. 

(n) 


Frog Angle 


Nat. 
sinF. 


Nat. 
cos F. 


Log 
sin F. 


Log 
cos F. 


Log 
cot F. 


Log 
vers F. 


Frog 
No. 
(n) 


4 
5 
6 
7 


14^ 15' 00'' 

11 25 16 

9 31 38 

8 10 16 


.24615 
.19802 
.16552 
.14213 


.96923 
.98020 
.98621 
•98985 


9.39120 
.29670 
.21884 
.15268 


9.98642 
.99131 
.99397 
.99557 


10.59522 
.69461 
.77513 
.84288 


8.4881T 
.29670 
.13966 

8.00655 


4 
5 
6 
7 


8 

10 


7 09 10 
6 21 35 
6 01 32 
5 43 29 


.12452 
.11077 
.10497 
.09975 


.99222 
.99385 
.99448 
.99501 


.09522 

.04442 

9.02107 

8.99891 


.99660 
.99732 
.99759 
.99783 


. 90138 

.95289 

.97652 

10.99892 


7.89110 
.78915 
.74232 
.69787 


8 

h 

10 


11 
12 
15 
16 


5 12 18 
4 46 19 
3 49 06 
3 34 47 


.09072 
.08319 
.06659 
.06244 


.99588 
.99653 
.99778 
.99805 


.95770 
.92007 
.82343 
.79543 


.99820 
.99849 
.99903 
.99915 


11.04050 
.07842 
.17560 
.20370 


.61527 
.53986 
.34631 
.29028 


11 
12 
15 
16 


18 
20 
24 


3 10 56 
2 51 51 
2° 23' 13" 


.05551 
.04997 
.04165 


.99846 
.99875 
.99^13 


.74438 

.69869 

8.61959 


.99933 

.99945 

9.99962 


.25494 

.30076 

11.38003 


.18807 
7.09663 
6.93834 


18 
20 
24 



619 



TABLE III. SWITCH LEADS AND DISTANCES — Continued. 



B. THEORETICAL LEADS, TTSING STRAIGHT POINT-RAILS AND 

STRAIGHT FROG RAILS) GAUGE 4' 8|". See §§ 305 and 313. 









Frog. 


Switch. 




Switch Dimensions. 




o 
















ts o w 




'A 


;3 








. 






Degree 


Oia O 


Closure. 


ta 


PQ 


to 








Angle. 


Radius. 


of 
Lead 






o 






P^ 


o 




S 


wy 






Curve. 


<ii 


Str'ght 


Curv'd 




p^ 


iS 


»H 


(^ 


>A 








Rail. 


Rail. 


(n) 




(TF) 


(i^) 


(S) 


(«) 


(r) 


(D) 


iL') 








ft. 


ft. 


in. 


ft.in. 


ft.in. 


Of// 


ft. 


o / // 


ft. 


ft. 


ft. 


4 


0.17 


3 


2 


5 4 


11 


2 36 19 


112.26 


52 53 56 


37. 22 


22-88 


23.29 


5 


021 


3 


7 


6 5 


11 


2 36 19 


183.22 


31 40 24 


42.98 


28.19 


28.55 


6 


0.25 


4 





7 


11 


2 36 19 


273.95 


21 01 58 


48.36 


33. 11 


33-38 


7 


0.29 


4 


5 


8 1 


16 6 


1 44 11 


364.88 


15 47 19 


62.23 


41.02 


41-24 


8 


0-33 


4 


9 


8 9 


16 6 


1 44 11 


488. 71 


11 44 40 


67.80 


46-22 


46-42 


9 


0-37 


6 





10 


16 6 


1 44 11 


616.27 


9 18 27 


72.61 


49-74 


49-92 


9i 


0.40 


6 





10 


16 6 


1 44 11 


699-97 


8 11 33 


75.30 


52-40 


52.58 


10 


042 


6 





10 6 


16 6 


1 44 11 


790.25 


7 15 18 


77.93 


55-01 


55.17 


11 


0.46 


6 





11 6 


22 


1 18 08 


940.21 


6 05 48 


92.52 


64.06 


64.20 


12 


0.50 


6 


5 


12 1 


22 


1 18 08 


1136. 34 


5 02 38 


97.75 


68-83 


68.96 


15 


0.62 


7 


8 


1410 


33 


52 05 


1744.45 


3 17 06 


131.12 


89-83 


89.94 


16 


0.67 


8 





16 


33 


52 05 


2005.98 


2 51 24 


136.62 


94-95 


95.05 


18 


0-75 


8 


10 


17 8 


33 


52 05 


2587.66 


2 12 52 


147.13 


104-54 


104.61 


20 


0.83 


9 


8 


19 4 


33 


52 05 


3262-98 


1 45 22 


157.18 


113- 68 


113.76 


24 


1.00 


11 


4 


23 2 


33 


52 05 


4932.77 


1 09 42 


176.09 


130-66 


130.77 



C. PRACTICAL LEADS, USING STRAIGHT POINT-RAILS AND 

STRAIGHT FROG RAILS,* GAUGE 4' SJ''; See §§ 305-307. 









4^ 


+3 
















(3 . 


^ r.^ 












o S3 




o 9 p 








Radius 
of 


Degree 
of lead 
curve. 


03 o 


c3 o 


li^o 


Closure for 


Closure for 


o 


center 
line. 




-t.3 (,N 




straight rail. 


curved rail. 


^ 






to ^ 

c3 




o m e3 






plH 






H 


H 


< 






(n) 


(r) 


{D) 


{Ts) 


{Tf) 


(LO 








ft. 


o / // 


ft. 


ft. 


ft. 






4 


110.69 


53 42 24 


1-03 


0.00 


37-94 


1-23 . 60 


1-24 


5 


174. 34 


33 19 57 


0-00 


0-82 


42-47 


1-27.68 


1-28 


6 


265.39 


21 43 04 


0-00 


0-66 


47.98 


1-32 . 73 


1-33 


7 


362.08 


15 52 29 


0.00 


0-19 


62.10 


1-13-89 1-27 


1-14.11 1-27 


8 


487.48 


11 46 27 


0-30 


0-00 


67.98 


1-1 6. 40 1-30 


1-16.60 1-30 


9 


605.18 


9 28 42 


0.00 


0-57 


72.28 


1-16.41 1-33 


1-16.59 1-33 


94 


695.45 


8 14 45 


0.76 


0-00 


75.71 


1-25. 82 1-27 


1-26 1-27 


10 


790-25 


7 15 18 


0.00 


0-00 


77.93 


1-27 1-28 


1-27.17 1-28 


11 


922-65 


6 12 47 


2.99 


0.00 


94.31 


1-32.85 1-33 


2-33 


12 


1098-73 


5 12 59 


5.33 


0.00 


100.80 


1-23. 88 2-24 


3-24 


15 


1743-80 


3 17 10 


0.09 


0.00 


131.19 


2-30 1-29.89 


3-30 


16 


1993.24 


2 52 29 


1.56 


0.00 


137.57 


1-29.90 2-33 


1-30 2-33 


18 


2546.31 


2 14 31 


0.00 


1.08 


146.51 


1-25.93 3-26 


4-26 


20 


3257-26 


1 45 32 


0.44 


0.00 


157. 42 


1-2 6. 92 2-27 1-33 


3-27 1-33 


24 


4886.16 


1 10 21 2.43 


0.00 


177. 22 


1-32.89 3-33 


4-33 



620 



TABLE IV.— FUNCTIONS OF THE TEN-CHORD SPIRAL. 
Part A, — CoeflScients of oi for deflection angles to chord points. 



Deflection 






Transit at chord-point i 


lumber. 




angle to 




















chord-point 




T. S. 




















,'?=. 


number. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


OT. S. 





2 


8 


18 


32 


50 


72 


98 


128 


162 


200 


1 


1 





5 


14 


27 


44 


65 


90 


119 


152 


189 


2 


4 


4 





8 


20 


36 


56 


80 


108 


140 


176 


3 


9 


10 


7 





11 


26 


45 


68 


95 


126 


161 


4 


16 


18 


16 


10 





14 


32 


54 


80 


110 


144 


5 


25 


28 


27 


22 


13 





17 


38 


63 


92 


125 


6 


36 


40 


40 


36 


28 


16 





20 


44 


72 


104 


7 


49 


54 


55 


52 


45 


34 


19 





23 


50 


81 


8 


64 


70 


72 


70 


64 


54 


40 


22 





26 


56 


9 


81 


88 


91 


90 


85 


76 


63 


46 


25 





29 


10 s. c. 


100 


108 


112 


112 


108 


100 


88 


72 


52 


28 








77 y 

Part B. — Values of -f and -7. 

Li Li 




<t> 


u 

L 


V 
L 


<f> 


U 
L 


V 
L 


0° 

1 

2 


.666 667 
.666 678 
.666 710 


.333333 
.333343 
.333 372 


23° 

24 

25 


.672 423 
.672 943 
• 673 486 


338^586 
339 061 
339 559 


3 
4 
5 


.666 763 
.666 838 
.666 935 


.333 421 
.333 490 
.333 578 


26 
27 
28 


.674 054 
.674 645 
.675 261 


340 078 
340 619 
341 183 


6 
7 
8 


.667 053 
.667 193 
.667 354 


.333 685 
.333 812 
.333959 


29 
30 
31 


.675 901 
.676 566 
.677 256 


341 769 

342 378 

343 Oil 


9 

10 
11 


.667 537 
.667 742 
.667 968 


.334126 
.334313 
.334519 


32 
33 
34 


.677 971 
.678 712 
• 679 478 


343 667 

344 346 

345 050 


12 
13 
14 


.668 216 
.668 487 
.668 779 


.334 746 
.334 992 
.335 259 


35 
36 
37 


.680 270 
.681089 
.681935 


345 777 

346 529 

347 307 


15 
16 
17 


.669 094 
.669 431 
.669 790 


.335 546 
.335 853 
.336 181 


38 
39 
40 


.682 808 
.683 708 
.684 636 


348 109 

348 937 

349 791 


18 
19 
20 


.670 172 
.670 576 
.671003 


.336 529 
.336 899 
.337289 


41 
42 
43 


.685 592 
.686577 
.687 590 


350 671 
351578 
352 513 


21 
22 


.671453 
.671926 


.337 700 
.338 132 


44 
45 


• 688 633 
.689 706 


353 474 

354 464 



Table IV, of which Part C is condensed, was computed by the Track 
Committee of the American Railway Engineering Association and is taken 
from the Proceedings of the Association. 

621 



TABLE IV.— FUNCTIONS OF THE TEN-CHORD SPIRAL. 

Part C. 



'Total spiral 




A. 




C 


X 


Y 


angle, ^ 








L 


L 


L 


0^ 


0' 


0= 


00' 


00" 


1.000 000 


1.000 000 


.000 000 




30 





10 


00 


.999 997 


.999 993 


• 002 909 


1 








20 


00 


.999 987 


.999 970 


.005 818 




30 





30 


00 


.999 970 


.999 932 


.008 726 


2 








40 


00 


.999 947 


.999 879 


.011635 




30 





50 


00 


.999 916 


.999 811 


• 014 542 


3 





1 


00 


00 


.999 880 


.999 727 


• 017450 




30 


1 


10 


00 


• 999 836 


.999 629 


• 020 357 


4 


00 


1 


20 


00 


.999 786 


.999515 


.023 263 




30 


1 


30 


00 


.999 729 


.999387 


.026169 


5 


00 


1 


40 


00 


.999 666 


.999 243 


.029 073 




30 


1 


50 


00 


.999 596 


.999 084 


• 031977 


6 


00 


1 


59 


59 


.999 519 


.998 910 


• 034 880 




30 


2 


09 


59 


.999 435 


.998 721 


• 037 781 


7 


00 


2 


19 


59 


.999 345 


.998 517 


.040 681 




30 


2 


29 


59 


.999 248 


.998 298 


.043 581 


8 


00 


2 


39 


58 


.999 145 


.998 063 


.046 478 




30 


2 


49 


58 


.999 035 


.997 814 


.049 374 


9 


00 


2 


59 


58 


.998 918 


.997549 


.052 269 




30 


3 


09 


57 


.998 794 


.997270 


.055 162 


10 


00 


3 


19 


57 


.998 664 


.996 975 


• 058 053 




30 


3 


29 


57 


.998 527 


.996 666 


• 060 942 


11 


00 


3 


39 


56 


.998 384 


.996341 


• 063 829 




30 


3 


49 


55 


.998 233 


.996 002 


.066 714 


12 


00 


3 


59 


55 


.998 077 


.995 647 


.069 598 




30 


4 


09 


54 


.997913 


.995 278 


.072 478 


13 


00 


4 


19 


53 


.997 743 


.994 893 


.075 357 




30 


4 


29 


53 


.997566 


.994 494 


.078 233 


14 


00 


4 


39 


52 


.997383 


.994 079 


.081106 




30 


4 


49 


51 


.997192 


.993 650 


.083 977 


15 


00 


4 


59 


50 


.996 996 


.993 206 


.086 846 




30 


5 


09 


49 


• 996 792 


.992 747 


• 089 711 


16 


00 


5 


19 


48 


.996 582 


.992 273 


• 092 574 




30 


5 


29 


47 


.996366 


.991785 


• 095 433 


17 


00 


5 


39 


45 


.996 142 


.991281 


• 098 290 




30 


5 


49 


44 


.995 912 


.990 763 


• 101143 


18 


00 


5 


59 


43 


.995 676 


.990 230 


.103 993 




30 


6 


09 


41 


.995 432 


.989 682 


.106 840 


19 


00 


6 


19 


40 


.995 183 


.989 120 


.109 683 




30 


6 


29 


36 


.994 926 


.988 543 


.112 523 


20 


00 


6 


39 


36 


.994 663 


.987951 


• 115 360 




30 


6 


49 


34 


.994393 


.987 344 


• 118 192 


21 


00 


6 


59 


32 


.994117 


.986 723 


• 121021 




30 


7 


09 


30 


.993 834 


.986 088 


• 123 846 


22 


00 


7 


19 


28 


.993 545 


.985 437 


• 126 667 


22° 


30' 


7° 


29' 


26" 


.993 248 


.984 772 


.129 483 



622 



TABLE IV.— FUNCTIONS OF THE TEN-CHORD SPIRAL. 
Part C. — Con. 



Total spiral 


A 


C 


X 


Y 


angle, <^ 


L 


L 


L 


22° 30' 


7° 29' 26" 


.993 248 


.984 772 


.129 483 


23 00 


7 39 24 


.992 946 


• 984 093 


.132 296 


30 


7 49 21 


• 992 636 


• 983 399 


• 135 105 


24 00 


7 59 19 


.992321 


• 982 691 


.137 909 


30 


8 09 16 


.991998 


• 981968 


.140 708 


25 00 


8 19 14 


• 991669 


.981231 


.143 504 


30 


8 29 11 


• 991333 


• 980 479 


• 146 294 


26 00 


8 39 08 


.990 991 


• 979 714 


• 149 080 


30 


8 49 05 


• 990 642 


• 978 933 


• 151861 


27 00 


8 59 02 


• 990 287 


• 978 139 


.154 638 


30 


9 08 58 


• 989 925 


• 977 330 


.157 409 


28 00 


9 18 55 


.989 557 


• 976 508 


.160 176 


30 


9 28 51 


.989 182 


•975 670 


.162 937 


29 00 


9 38 48 


• 988 800 


• 974 819 


.165 693 


30 


9 48 44 


.988412 


• 973 954 


.168 444 


30 00 


9 58 40 


.988018 


.973 074 


.171189 


30 


10 08 36 


.987 617 


•972 181 


.173 929 


31 00 


10 18 32 


.987209 


.971 273 


.176 664 


30 


10 28 27 


.986 795 


•970 352 


• 179 392 


32 00 


10 38 23 


.986375 


•969 417 


• 182 116 


30 


10 48 18 


.985 948 


•968 468 


.184 833 


33 00 


10 58 13 


.985 514 


• 967 504 


• 187 544 


30 


11 08 08 


.985 074 


•966 528 


.190 250 


34 00 


11 18 03 


.984 627 


•965 537 


.192 949 


30 


11 27 58 


.984 174 


•964 532 


.195 643 


35 00 


11 37 53 


.983 715 


•963 515 


.198330 


30 


11 47 47 


.983 249 


.962 483 


.201010 


36 00 


11 57 41 


.982 777 


.961 438 


.203 685 


30 


12 07 36 


• 982 298 


• 960 379 


.206 353 


37 00 


12 17 30 


• 981813 


•959 306 


.209 014 


30 


12 27 23 


.981321 


.958 221 


.211669 


38 00 


12 37 17 


.980 823 


.957 121 


.214317 


30 


12 47 11 


• 980318 


.956 009 


.216 959 


39 00 


12 57 04 


• 979 807 


.954 883 


.219 593 


30 , 


13 06 57 


• 979 290 


.953 744 


.222 221 1 


40 00 


13 16 50 


• 978 766 


•952 591 


.224 841 


30 


13 26 43 


• 978 236 


.951 426 


.227 455 


41 00 


13 36 35 


• 977 700 


.950 247 


.230 061 


30 


13 46 28 


• 977 157 


.949 055 


.232 660 


42 00 


13 56 20 


• 976 603 


• 947 850 


.235 252 


30 


14 06 12 


• 976 053 


•946 632 


.237836 


43 00 


14 16 04 


• 975 491 


.945 402 


.240 413 


30 


14 25 56 


• 974923 


.944 158 


.242 982 


44 00 


14 35 47 


• 974 348 


.942 901 


.245 544 


30 


14 45 38 


.973 768 


.941632 


.248 098 


45° 00' 


14° 55' 29" 


.973 181 


.940 350 


.250 644 



623 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 



100 

101 
102 
103 
104 
105 
106 
107 
108 
109 

110 

111 
112 
113 
114 
115 
116 
117 
118 
119 

130 

121 
122 
123 
124 
125 
126 
127 
128 
129 

130 

131 

132 
i33 
134 
135 
136 
137 
138 
139 

140 

141 
142 
143 
144 
145 
146 
147 
148 
149 

150 



00 000 

432 
860 

01 283 
703 

02 119 
530 
938 

03 342 
742 

04 139 



07 



532 
922 
308 
690 
070 
446 
818 
188 
554 



918 



08 278 
636 
990 

09 342 
691 

10 037 
380 
721 

11 059 

394 



727 

12 057 
385 
710 

13 033 
354 
672 
988 

14 301 

613 

922 

15 229 
533 
836 

16 137 
435 
731 

17 026 
318 

609 



043 087 130 



475! 518 
902, 945 
326 368 
745! 787 
160 201 
57ii 612 
973*019 
382i 422 
782 822 



178 



571 
960 
346 
728 
107 
483 
855 
225 
591 



218 



954 



314 
67l 
*026 
377 
725 
071 
414 
755 
092 



610 
999 
384 
766 
145 
520 
893 
261 
627 



990 



427 



760 
090 
418 
743 
065 
386 



350 
707 
*061 
412 
760 
106 
448 
789 
126 



461 



793 
123 
450 
775 
097 
417 



7031 73-5 
=019 *051 
332 364 



561 
987 
410 
828 
243 
653 

*060 
463 

_862 

257 



649 
*038 
423 
804 
183 
558 
930 
298 
664 



'026 



386 

742 
*096 
447 
795 
140 
483 
822 
16C 



494 



644 



955 
259 
564 



675 



983 
290 
594 
866! 896 
166, 196 
465 494 



761 
055 
348 



638 



N. O 



791 
085 
377 



826 
156 
483 
807 
130 
449 
767 
*082 
395 



667 



706 

*014 
320 
624 
926 
226 
524 
820 
114 
406 



696 



173 



604 

*030 

452 

870 

284 

694 

*100 

503 

_901 

297 

688 
*076 
461 
842 
220 
595 
967 
335 
700 



=062 



422 
778 
*131 
482 
830 
174 
517 
856 
193 



528 



859 
189 
515 
840 
162 
481 
798 
*113 
426 



736 



*045 
351 
655 
956 
256 
554 
849 
143 
435 



725 



216 



*072 
494 
911 
325 
735 

*141 
543 
941 

_336 

727 
*115 
499 
880 
258 
632 
*004 
372 
737 



=098 



457 
813 
*166 
517 
864 
209 
551 
890 
227 

561 



892 
221 
548 
872 
194 
513 
830 
*145 
457 

767 



260 



689 

*114 

536 

953 

366 

775 

*181 

583 

_981 

_375 

766 
*154 
538 
918 
296 
670 
*040 
408 
773 



= 134 



493 
849 
*202 
552 
899 
243 
585 
924 
260 



303 



8 9 



346 



732 775 

*157*199 

5781 619 

994 *036 



407 
816 

*221 
623 

*020 

415 

805 
*192 
576 
956 
333 
707 
*077 
445 
809 



448 
857 

*262 
663 

*06C 

454 



=170 



594 



925 
254 
580 
904 



529 
884 
*237 
586 
933 
277 
619 
958 
294 

627 



=206 



958 
287 
613 
937 



226 258 
545 577 
862! 893 
*176,*207 
4881 519 



564 
920 
*272 
621 
968 
312 
653 
991 
327 



661 



7981 829 860 



99l 
320 
645 
969 
290 
608 
925 
*239 
550 



817 
*241 

661 
*077 

489 

898 
*302 

703 
*100 

493 

883 

*269 
652 

*032 
408 
781 

*151 
518 
882 



*242 

600 
955 

*307 
656 

*002 
346 
687 

*025 
361 

694 



*075 
381 
685 
987 
286 
584 
879 
172 
464 



*106*137*167 
412^ 442! 473 
715 745I 776 

*017 *047 *077 
3161 346i 376 
613! 643! 672 
908| 938! 967 
202; 2311 260 
493 522 551 



753 



782, 811 840 



*024 
352 
678 

*001 
322 
640 
956 

*270 
582 

891 



*198 
503 
806 

*107 
405 
702 
997 
289 
580 



869 



8 9 



P.P. 





43 


43 


42 


• 1 


4.3 


4.3 


4.21 


• 2 


8 


7 


8 


6 


8 


4 


.3 


13 





12 


9 


12 


6 


•4 


17 


4 


17 


2 


16 


8 


.5 


21 


7 


21 


5 


21 





• 6 


26 


1 


25 


8 


25 


2 


•7 


30 


4 


30 


1 


29 


4 


8 


34 


8 


34 


4 


33 


6 


.9 


39 


1 


38 


7 


37 


8 



41 

4.1 
82 
12.3 
16. 4 
20.5 
24.6 
28. 7 
32.8 
36.9 





40 


40 


39 


38 


.1 


4.0 


4.0 


3.9 


38 


• 2 


8.1 


8 





7 


8 


7.6 


.3 


12.1 


12 





11 


7 


11.4 


• 4 


16-2 


16 


C 


15 


6 


15.2 


.5 


20.2 


20 





19 


5 


19.0 


6 


24.3 


24 


C 


23 


4 


22.8 


.7 


28.3 


28 


C 


27 


3 


26.6 


8 


32.4 


32 


C 


31 


2 


30-4 


.9 


36.4 


36 





35 


1 


34.2 





37 


37 


36 , 


.1 


3.7 


3.7 


361 


.2 


7 


5 


7 


4 


7 


2 


.3 


11 


2 


11 


1 


10 


8 


• 4 


15 





14 


8 


14 


4 


• 5 


18 


7 


18 


5 


18 


C 


• 6 


22 




22 


2 


21 


6 


.7 


26 


2 


25 


9 


25 


2 


.8 


30 





29 


6 


28 


8 


.9 


33 


7 


33 


3 


32 


4 





35 


34 


33 , 


■ 1 


3.4 


3.4 


331 


• 2 


6 9 


6.8 


6 


6 


■ 3 


10.3 


10.2 


9 


9 


.4 


13 


8 


13.6 


13 


2 


.5 


17 


2 


17-0 


16 


5 


.6 


20 


7 


20.4 


19 


8 


.7 


24 


1 


23-8 


23 


1 


8 


27 


6 


27-2 


26 


4 


.9 


31 


6 


30.6 


29 


7 



35 

3.5 
70 

10.5 
14.0 
17.5 
21.0 
24.5 
28.0 
31.5 



33 

3.2 
6.4 
9.6 
12-8 
16. 
19.2 
22.4 
25.6 
28-8 





31 


31 


30 


.1 


3.1 


3.1 


3.0 


.2 


63 


6 


2 


6-0 


.3 


9.4 


9 


3 


9.0 


.4 


12-6 


12 


4 


12.0 


.5 


15.7 


15 


5 


15-0 


.6 


18-9 


18 


6 


18.0 


.7 


22.0 


21 


7 


21.0 


.8 


25.2 


24 


8 


24.0 


.9 


28.3 


27 


9 


27.0 



29 

2.9 
5.8 

8.7 

11.6 
14.5 
17.4 
20.3 
23.2 
26.1 



P.P. 



624 









TABLE -^ 


\— LOGARITHMS 


OF NUMBERS 


. 








' 


N. 





1 
638 


2 

667 


3 

696 


4 

725 


5 

753 


6 

782 


7 
811 


8 
840 


9 

869 


P. P. 


150 


17 609 


29 38 27 
























151 


897 


926 


955 


984 


=^012 


*041 


*070 


*098 


*127 


*156 


.1 


2.9 


2.8 


2.7 


152 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


440 


• 2 


5.8 


5.6 


5.4 


153 


469 


497 


526 


554 


582 


611 


639 


667 


695 


724 


.3 


8-7 


8.4 


8.1 


154 


752 


780 


808 


836 


864 


893 


921 


949 


977 


*005 


.4 


11.6 


11.2 


10.8 


155 


19 033 


061 


089 


117 


145 


173 


201 


229 


256 


284 


.5 


14.5 


14.0 


13.5 


156 


312 


340 


368 


396 


423 


451 


479 


507 


534 


562 


.6 


17.4 


16.8 


16-2 


157 


590 


617 


645 


673 


700 


728 


755 


783 


810 


838 


.7 


20.3 


19.6 


18-9 


158 


865 


893 


920 


948 


975 


*003 


*030 


*057 


*085 


*112 


.8 


23-2 


22.4 


21.6 


159 


20 139 


107 


194 


221 


249 


276 


303 


330 


357 


385 


• 9 


26.1 


25.2 


24.3 


160 


412 


439 


466 


493 


520 


547 


574 


601 


628 


655 


26 26 




"" 




















161 


682 


709 


736 


763 


790 


817 


844 


871 


898 


924 


• 1 


2.6 


2-6 


162 


951 


978 


*005 


*032 


*058 


*085 


^112 


*139 


*165 


*192 


.2 


5 


3 


5-2 


163 


21 219 


245 


272 


298 


325 


352 


378 


405 


431 


458 


.3 


7 


• 9 


78 


164 


484 


511 


537 


564 


590 


616 


643 


669 


695 


722 


.4 


10 




10-4 


165 


748 


774 


801 


827 


853 


880 


906 


932 


958 


984 


.5 


13 


.2 


13.0 


166 


22 Oil 


037 


063 


089 


115 


141 


167 


193 


219 


245 


.6 


15 


.9 


15.6 


167 


27l 


297 


325 


349 


375 


401 


427 


453 


479 


505 


.7 


18 


• 5 


18.2 


168 


531 


557 


582 


608 


634 


660 


686 


711 


737 


763 


.8 


21 


.2 


20.8 


169 
170 


788 


814 


840 


865 


891 


917 


942 


968 


994 


*019 


.9 


23 


.8 


23.4 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 


25 25 24 
























171 


299 


325 


350 


375 


401 


426 


451 


477 


502 


527 


.1 


2.5 


2-5 


2.4 


172 


553 


578 


603 


628 


653 


679 


704 


729 


754 


779 


.2 


5 


•1 


5-0 


48 


173 


804 


829 


855 


880 


905 


930 


955 


98C 


*005 


*030 


.3 


7 


• 6 


7-5 


7.2 


174 


24 055 


080 


105 


129 


154 


179 


204 


229 


254 


279 


.4 


10 


• 2 


10.0 


9.6 


175 


304 


328 


353 


378 


403 


427 


452 


477 


502 


526 


.5 


12 


• 7 


12.5 


12.0 


176 


551 


576 


600 


625 


650 


674 


699 


723 


748 


773 


• 6 


15 


• 3 


15.0 


14.4 


177 


797 


822 


846 


871 


895 


920 


944 


968 


993 


*017 


•7 


17 


8 


17.5 


16.8 


178 


25 042 


066 


091 


115 


139 


104 


188 


212 


237 


261 


.8 


20 




20.0 


19.2 


179 
180 


285 


309 


334 


358 


382 


406 


430 


455 


479 


503 


.9 


22 


.9 


22.5 


21.6 


527 


551 


575 


599 


623 


647 


672 


696 


720 


744 


<-k77 rkf> 


181 


768 


792 


816 


840 


863 


887 


911 


935 


959 


983 


.1 


2.3 


/«0 

2.3 


182 


26 007 


031 


055 


078 


102 


126 


150 


174 


197 


22] 


.2 


4 


•7 


4.6 


183 


245 


269 


292 


316 


340 


363 


387 


411 


434 


458 


.3 


7 


.0 


6.9 


184 


482 


505 


529 


552 


576 


599 


623 


646 


670 


693 


.4 


9 


4 


9.2 


185 


717 


740 


764 


787 


.811 


834 


858 


881 


904 


928 


.5 


11 


7 


11.5 


186 


951 


^74 


998 


*021 


*044 


*068 


*091 


*114 


n37 


*161 


.6 


14 


1 


13.8 


187 


27 184 


207 


230 


254 


277 


300 


323 


346 


369 


392 


• 7 


16 


4 


16.1 


188 


416 


439 


462 


485 


508 


531 


554 


577 


600 


623 


.8 


18 


8 


18.4 


189 
190 


646 


669 


692 


715 


738 
966 


761 


784 


806 


829 


852 


.9 


21 


1 


20.7 


875 


898 


921 


944 


989 


*012 


*035 


*0&8 


*080 




191 


28 103 


126 


149 


171 


194 


217 


239 


262 


285 


307 


.1 


2.21 


2-2' 


2^1 


192 


330 


352 


375 


398 


420 


443 


465 


488 


510 


533 


.2 


4 


.5 


4-4 


4.3 


193 


555 


578 


600 


623 


645 


668 


690 


713 


735 


758 


.3 


6 


• 7 


6.6 


6.4 


i 194 


780 


802 


825 


847 


869 


892 


914 


936 


959 


981 


.4 


9 


.0 


88 


8.6 


, 195 


29 003 


025 


048 


070 


092 


114 


137 


159 


181 


203 


.5 


11 


• 2 


11.0 


10.7 


196 


225 


248 


270 


292 


314 


336 


358 


380 


402 


424 


.6 


13 


.5 


13.2 


12.9 


1 197 


446 


468 


490 


512 


534 


556 


578 


600 


622 


644 


.7 


15 


• 7 


15.4 


15.0 


198 


666 


688 


710 


732 


754 


776 


798 


820 


841 


863 


.8 


18 


.0 


17.6 


17.2 


i99 


885 


907 


929 


950 


972 


994 


*016 


*038 


*059 


*081 


.9 


20 


.2 


19.8 


19.3 


200 

1 - 


30 103 


124 


146 


168 


190 
4 


211 
5 


233 
6 


254 

7 


276 
8 


298 
9 




N. 


1 


2 


3 


P.P. 
















625 





















TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 
124 


3 

146 


3 

168 


4 

190 


5 

2ll 


6 

233 


7 
254 


8 
276 


9 

298 


P. P. 


300 


30 103 


33 31 
















201 


319 


341 


363 


384 


406 


427 


449 


470 


492 


513 


.1 2.2 


2.1 


202 


535 


556 


578 


599 


621 


642 


664 


685 


707 


728 


.2 4.4 


4.2 


203 


749 


771 


792 


813 


835 


856 


878 


899 


920 


941 


.3 6.6 


6.3 


204 


963 


984 


*005 


*027 


*048 


*069 


*090 


*112 


n33 


*154 


.4 88 


8.4 


205 


31 175 


196 


217 


239 


260 


281 


302 


323 


344 


365 


.5 11.0 


10.5 


206 


386 


408 


429 


450 


471 


492 


513 


534 


555 


576 


.613-2 


12.6 


207 


597 


618 


639 


660 


681 


702 


722 


743 


764 


785 


.7 15.4 


14.7 


208 


806 


827 


848 


869 


890 


910 


931 


952 


973 


994 


.8 17-6 


16. 8 


209 


32 014 


035 


056 


077 


097 


118 


139 


160 


180 


201 


.9 19.8 


18.9 


310 


222 


242 


263 


284 


.304 


325 


346 


366 


387 


407 


30 30 
























211 


428 


449 


469 


490 


510 


531 


551 


572 


592 


613 


.1 2.0 


2.0 


212 


633 


654 


674 


695 


715 


736 


756 


776 


797 


817 


.2 4.1 


4.0 


213 


838 


858 


878 


899 


919 


94C 


960 


980 


*C01 


*021 


.3 6.1 


6.0 


214 


33 041 


061 


082 


102 


122 


142 


163 


183 


203 


223 


.4 8.2 


8.0 


215 


244 


264 


284 


304 


324 


344 


365 


385 


405 


425 


.5 10.2 


10.0 


216 


445 


465 


485 


505 


525 


546 


566 


586 


606 


626 


.612.3 


12.0 


217 


646 


666 


686 


706 


726 


746 


766 


786 


806 


825 


.7 14.3 


14.0 


218 


845 


865 


885 


905 


925 


945 


965 


985 


*004 


*024 


.8 16.4 


16.0 


219 


34 044 


064 


084 
281 


104 
301 


123 


143 


163 


183 


203 


222 


.9 18.4 


18.0 


22Q 


242 


262 


321 


341 


360 


380 


400 


419 


19 19 ' 
























221 


439 


459 


478 


498 


518 


537 


557 


576 


596 


615 


.1 1.9 


1.9 


222 


635 


655 


674 


694 


713 


733 


752 


772 


791 


811 


.2 3.9 


38 


223 


830 


850 


869 


889 


908 


928 


947 


966 


986 


*CG5 


.3 5.8 


5.7 


224 


35 025 


044 


063 


083 


102 


121 


141 


160 


179 


199 


.4 7-8 


76 


225 


218 


237 


257 


276 


295 


314 


334 


353 


372 


391 


.5 9.7 


9.5 


226 


411 


430 


449 


468 


487 


507 


526 


545 


564 


583 


• 6 11.7 


11.4 


227 


602 


621 


641 


660 


679 


698 


717 


736 


755 


774 


• 713.6 


13.3 


228 


793 


812 


831 


850 


869 


888 


907 


926 


945 


964 


.8 15.6 


15.2 


229 


983 


*002 


*021 


*040 
229 


*059 
248 


*078 
267 


*097 
286 


*116 
305 


*135 


*154 


.9 17.5 


17.1 


330 


36 173 


191 


210 


323 


342 


-t'iS •* a 


231 


361 


380 


399 


417 


436 


455 


474 


492 


511 


530 


18_ 
.1 1.8 


1.8 


232 


549 


567 


586 


605 


623 


642 


661 


679 


698 


717 


• 2 3.7 


3.6 


233 


735 


754 


773 


791 


810 


828 


847 


866 


884 


9CS 


• 3 5.5 


5.4 


234 


92l 


940 


958 


977 


996 


*014 


*033 


*C51 


*070 


*C£8 


• 4 7-4 


7-2 


235 


37 107 


125 


143 


162 


180 


199 


217 


236 


254 


273 


.5 92 


9.0 


236 


291 


309 


328 


346 


364 


383 


401 


420 


438 


456 


.6 11.] 


lO'.S 


237 


475 


493 


511 


530 


548 


566 


584 


603 


621 


638 


.7 12.8 


12.6 


238 


657 


676 


694 


712 


730 


749 


767 


785 


803 


821 


.8 14.8 


14.4 


239 


840 
38 021 


858 
039 


876 
057 


894 
075 


912 
093 


930 
111 


948 


967 


985 


*CC3 


•9 16-6 


16.2 


340 


129 


147 


165 


183 




241 


201 


219 


237 


255 


273 


291 


309 


327 


345 


363 


.1 Tv 


1-7 1 


242 


381 


399 


417 


435 


453 


471 


489 


507 


525 


543 


.2 3-5 


3 


4 


243 


560 


578 


596 


614 


632 


65C 


667 


685 


703 


721 


.3 5-2 


5 


1 


244 


739 


757 


774 


792 


810 


828 


845 


813 


881 


899 


.4 7.C 


6 


8 


245 


916 


934 


952 


970 


987 


*005 


*023 


*040 


*058 


=^^076 


.5 8.7 


8 


5 


246 


39 093 


in 


129 


]46 


164 


181 


198 


217 


234 


252 


.6 10.5 


10 


2 


247 


269 


287 


305 


322 


340 


357 


375 


392 


410 


427 


.712.2 


11 


9 


248 


445 


462 


480 


497 


515 


532 


550 


567 


585 


602 


.814.0 


13 


6 


249 


620 


637 


655 


672 


689 


707 


724 


742 


759 


776 


.9I15.7 


15.3 1 


350 


794 


811 
1 


828 
3 


846 
3 


863 
4 


881 
5 


898 
6 


915 

7 


933 


950 


1 


N. 





8 


9 


P. I 


3, 





626 









TABLE v.— LOGARITHMS OF NUMBERS. 










N. 





1 
811 


3 

828 


3 

846 


4 

863 


5 

881 


6 

898 


7 
915 


8 
933 


9 

950 


P.P. 


350 


39 794 




251 


967 


984 


*002 


*019 


*036 


*054 


*071 


*088 


*105 


*123 




252 


40 140 


157 


174 


191 


■ 209 


226 


243 


260 


277 


295 


17 17 


253 


312 


329 


346 


363 


380 


398 


415 


432 


449 


466 


.1 


1.7 


1.7 


254 


483 


500 


517 


534 


i 551 


569 


586 


603 


620 


637 


.2 


3.5 


3 


.4 


255 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 


.3 


5.2 


5 


•1 


256 


824 


841 


858 


875 


892 


908 


925 


^942 


959 


976 


.4 


7.0 


6 


8 


257 


993 


*010 


*027 


*044 


*061 


*077*094 


*111 


*128 


*145 


-5 


8.7 


8 


5 


258 


41 162 


179 


195 


212 


229 


246 263 


279 


296 


312 


.6 10.5 


10 


2 


259 


330 


346 


363 


380 


397 


413| 430 


447 


464 


480 


.712.2 11 
.8 14.0 13 


9 
























6 


260 


497 


514 


530 


547 


564 


581 


597 


614 


631 


647 


.9 15.7 15 


3 


261 


664 


680 


697 


714 


730 


747 764 


• 780 


797 


813 




262 


830 


846 


863 


880 


896 


913 929 


946 


962 


979 




263 


995 


*012 


*028 


*045 


*0G1 


*078 *094 


nil 


*127 


*144 




264 


42 160 


177 


193 


209 


226 


242i 259 


275 


292 


308 




265 


324 


341 


357 


373 


390 


406 423 


439 


455 


472 


16 16 


266 


488 


504 


521 


537 


553 


569 586 


602 


618 


635 


.1 


1.6 


1.6 


267 


651 


667 


683 


700 


716 


732 748 


765 


781 


797 


.2 


3 


3 


3 


2 


268 


813 


829 


846 


862 


878 


894 910 


927 


943 


959 


.3 


4 


9 


4 


8 


269 


975 


991 


*007 


*023 


*040 


*056 *072 


*088 


*104 


*12d 


• 4 


6 


6 


6 


4 
























• 5 


8 


2 


8 





270 


43 136 


152 


_ 
168 


184 


200 


216 


233 


249 


265 


281 


.6 
• 7 


9 

11 


§ 


9 

11 


6 

2 






. 


















5 


271 


297 


313 


329 


345 


361 


377 


393 


409 


425 


441 


.8 


13 


2 


12 


8 


272 


457 


473 


489 


505 


520 


536 


552 


568 


584 


600 


.9 


14 


Q 


14 


4 


273 


616 


632 


648 


664 


680 


695 


711 


727 


743 


759 




274 


775 


791 


806 


822 


838 


854 


870 


886 


901 


917 




275 


933 


949 


965 


980 


996 


*012/^028 


*043 


*059 


*075 




276 


44 091 


106 


122 


138 


154 


169| 185 


201 


216 


232 




277 


248 


263 


279 


295 


310 


326 342 


357 


373 


389 




278 


404 


420 


435 


451 


467 


482 


498 


513 


529 


545 


15 15 


279 


560 


576 


591 


607 


622 


638 


653 


669 


685 


700 


.1 


1 


5 


1.5 














1 








.2 


3 


30 


380 


716 


731 


747 


762 


778 


793 809 


824 


839 


855 


.3 


4 


6 


4.5 














1 _ 








• 4 


6 


2 


60 


281 


870 


886 


901 


917 


932 


948, 963 


978 


994 


*009 


.5 


7 


7 


7.5 


282 


45 025 


040 


055 


071 


086 


1'02| 117 


132 


148 


168 


.6 


9 




9.0 


283 


178 


194 


209 


224 


240 


255i 270 


286 


301 


316 


.7 


10 


3 


10.5 


284 


332 


347 


362 


377 


393 


408 423 


438 


454 


469 


.8 


12 


4 


12.0 


285 


484 


499 


515 


530 


545 


560| 576 


591 


606 


621 


.9 


13 


9 


13.3 


286 


636 


652 


667 


682 


697 


7121 727 


743 


758 


773 




287 


788 


803 


818 


833 


848 


864' 879 


894 


909 


924 




288 


939 


954 


969 


984 


999 


*014 *029 


*044 


*059 *075| 




289 


46 090 


105 


120 


135 


150 


1651 180 


195 


210 


225 




390 


240 


255 


269 


284 


299 


314 


329 


344 


359 


374 


.1 


{^Z J*' 






















1.4 


1.4 


291 


389 


404 


419 


434 


44« 


464! 479 


493 


508 


523 


.2 


2.9 


2.8 


292 


538 


553 


568 


583 


597 


612| 627 


642 


657 


672 


.3 


4.3 


4-2 


293 


687 


701 


716 


731 


746 


76l! 775 


790 805 


820 


.4 


5.8 


5-8 


294 


834 


849 


864 


879 


894 


908 923 


938 


952 


967 


.5 


7.2 


7.0 


295 


982 


997 


*01I 


*026 


*041 


*055 *070 


*085 


*100*114l 


.6 


8.7 


8.4 


296 


47 129 


144 


158 


173 


188 


202 217 


232 


246 


261 


.7 


10.1 


98 


297 


275 


290 


305 


319 


334 


348 363 


378 


392 


407 


.8 


11.6 


11.2 


298 


42l 


436 


451 


465 


480 


494; 509 


523 


538 


552 


.9 


13.0 


12.6 


299 


567 


581 


596 


610 


625 


639 


654 


668 


683 


697 




300 


712 


726 

1 


741 
3 


755 
3 


770 
4 


784 
5 


799 
6 


813 

7 


828 
8 


842 
9 




N. 







P. 


I 


»^ 





627 



TABLE v.— LOGARITHMS OF NUMBERS 




628 



TABLE v.— LOGARITHMS OF NUMBERS, 



N. 





1 


2 


3 


4 

456 


5 

469 


6 

481 


7 
493 


8 
506 


9 
518 


P.P. 




350 


54 407 


419 


431 


444 


1^ 




























351 


530 


543 


555 


568 


580 


592 


605 


617 


629 


642 


.1 


1.2 




352 


654 


666 


679 


691 


703 


716 


728 


740 


7531 765 


.2 


2-5 




353 


777 


790 


802 


814 


826 


839 


851 


863 


876| 888 


.3 


3-7 




354 


900 


■ 912 


925 


937 


949 


961 


974 


986 


998*010 


.4 


5.0 




355 


55 023 


035 


047 


059 


071 


084 


096 


108 


1201 133 


.5 


6.2 




356 


145 


157 


169 


181 


194 


206 


218 


230 


242 


254 


.6 


75 




857 


267 


279 


291 


303 


315 


327 


340 


352 


364 


376 


.7 


8.7 




358 


388 


400 


412 


424 


437 


449 


461 


473 


485 


497 


.8 


10.0 




359 


509 


521 


533 


545 


558 


570 


582 


594 


606 


618 


.9 


11.2 




360 


630 


642 


654 


666 


678 


690 


702 


714 


726 


738 


•« t> 




361 


750 


762 


775 


787 


799 


811 


823 


835 


847 


859 


.1 


1.2 




362 


871 


883 


895 


907 


919 


931 


943 


955 


966 


978 


.2 


2 


4 




363 


990 


*002 


*014 


*026 


*038 


*050 


*062 


*074 


*086 


*098 


.3 


3 


6 




864 


56 110 


122 


134 


146 


158 


170 


181 


193 


205 


217 


.4 


4 


8 




365 


229 


241 


253 


265 


277 


288 


300 


312 


324 


336 


.5 


6 







366 


348 


360 


372 


383 


395 


407 


419 


431 


443 


455 


.6 


7 


2 




367 


466 


478 


490 


502 


514 


525 


537 


549 


561 


573 


.7 


8 


4 




368 


585 


596 


608 


620 


632 


643 


655 


667 


679 


691 


.8 


9 


6 




369 


702 


714 


726 


738 


749 


761 


773 


785 


796 


808 


.9 


10 


8 




370 


820 


832 


843 


855 


867 


879 


895 


902 


914 


925 


^T 




371 


937 


949 


961 


972 


984 


996 


*007 


*019 


*031 


*042 


.1 


ih 




372 


57 054 


066 


077 


089 


101 


112 


124 


136 


147 


159 


.2 


2.3 




373 


171 


182 


194 


206 


217 


229 


240 


252 


264 


275 


.3 


3.4 




874 


287 


299 


310 


322 


333 


345 


357 


368 


380 


391 


.4 


4-6 




375 


403 


414 


426 


438 


449 


461 


472 


484 


495 


507 


.5 


5.7 




376 


519 


530 


542 


553 


565 


576 


588 


599 


611 


622 


.6 


6.9 




377 


634 


645 


657 


668 


68C 


691 


703 


714 


726 


737 


.7 


8.(5 




878 


749 


760 


772 


783 


795 


806 


818 


829 


841 


852 


.8 


9.2 




379 


864 


875 


887 


898 


909 


921 


932 


944 


955 


967 


.9 


10.5 




380 


978 


990 


*001 


*012 


*024 
138 


*035 
~149 


*047 
161 


*058 
172 


*069 
183 


*081 
195 


■4 •< 




381 


58 092 


104 


115 


126 


.1 


XX 

1.x 




382 


206 


217 


229 


240 


252 


263 


274 


286 


297 


308 


.2 


2 


2 




S83 


320 


331 


342 


354 


365 


376 


388 


399 


410 


422 


.3 


3 


3 




384 


433 


444 


455 


467 


478 


489 


501 


512 


523 


535 


.4 


4 


4 




385 


546 


557 


568 


580 


591 


602 


613 


625 


636 


647 


.5 


5 


5 




386 


658 


670 


681 


692 


703 


715 


726 


737 


748 


760 


.6 


6 


6 




387 


771 


782 


793 


804 


816 


827 


838 


849 


861 


872 


.7 


7 


7 




388 


883 


894 


905 


916 


928 


939 


950 


961 


972 


984 


.8 


8 


8 




389 


995 


*006 


*017 


*028 


*039 


*050 


*062 


*073 


*084 


*095 


.9 


9 


9 




390 


59 106 


117 


128 


140 


151 


162 


173 


184 


195 


206 


tTi 




391 


217 


229 


240 


251 


262 


273 


284 


295 


306 


317 


.1 


ro 




392 


328 


339 


351 


362 


373 


384 


395 


4C6 


417 


428 


.2 


u 




393 


439 


450 


461 


472 


483 


494 


505 


516 


527 


538 


.3 




394 


549 


560 


571 


582 


593 


604 


615 


626 


637 


648 


.4 


^i 




395 


659 


670 


681 


692 


703 


714 


725 


736 


747 


758 


.5 


5.5 




396 


769 


780 


791 


802 


813 


824 


835 


846 


857 


868 


.6 


6.3 




397 


879 


890 


901 


912* 


923 


933 


944 


955 


966 


977 


.7 


7.§ 




S98 


988 


999 


*010 


*021 


*032 


*048 


*053 


*064 


*075 


*086 


.8 


31 




399 


60 097 


108 


119 


130 


141 


151 


162 


173 


184 


195 


.9 




400 


2C6 


217 


227 


238 
3 


249 
4 


260 
5 


271 
6 


282 

7 


293 
8 


303 
9 






N. 





1 


2 


P 


.P 







629 









TABLE v.— LOGARITHMS OF NUMBERS. 








N. 





1 


2 


3 


4 


5 


6 

271 


7 

282 
390 


8 

293 
401 


9 

303 
412 


P.P. 


400 


BO 206 


217 


227, 


238 


249 


260 






401 


314 


325 


336 347 


357 


368 


379 




402 


422 


433 


4441 


455 


466 


476 


487 


498 


509 


519 






403 


530 


541 


552 


563 


573 


584 


595 


606 


616 


627 


11 




404 


638 


649 


659 


670 


681 


692 


702 


713 


724 


735 


.1 


1.1 




405 


745 


756 


767 


777 


788 


799 


810 


820 


831 


842 


.2 


2 


2 




406 


852 


863 


874 


884 


895 


906 


916 


927 


938 


949 


.3 


3 


3 




407 


959 


970 


981 


991 


*002 


*013 


*023 *034i 


*044 


*055 


.4 


4 


4 




408 


61 066 


076 


087 


098 


108 


119 


130 


140 


151 


161 


.5 


5 


5 




409 


172 


183 


193 


204 


215 


225 


236 


246 


257 


268 


.6 
.7 


6 
7 


6 
7 




























410 


278 


289 


299 


310 


320 


331 


342 


352 


363 


373 


.8 
.9 


8 
9 


8 

9 




























411 


384 


394 


405 


416 


426 


437 


447 


458 


468 


479 






412 


489 


500 


511 


521 


532 


542 


553 


563 


574 


584 






413 


595 


605 


616 


626 


637 


647 


658 


668 


679 


689 






414 


700 


710 


721 


731 


742 


752 


763 


773 


784 


794 






415 


805 


815 


825 


836 


846 


857 


867 


878 


888 


899 


. 10 




416 


909 


920 


930 


940 


951 


961 


972 982 


993 


*003 


.1 


10 




417 


62 013 


024 


034 


045 


055 


065 


076 


086 


097 


107 


.2 


2.1 




418 


117 


128 


138 


149 


159 


169 


180 


190 


200 


211 


.3 


3-1 




419 


22l 


232 


242 


252 


263 


273 


283 


294 


304 


314 


.4 
.5 
.6 


4-2 
5-2 
6.3 




430 


325 


335 


345 


356 


366 


376 


387 


397 


407 


418 


























.7 


7.3 
























421 


428 


438 


449 


459 


469 


480 


490 


500 


510 


521 


.8 


n 




422 


531 


541 


552 


562 


572 


582 


593 


603 


613 


624 


.9 




423 


634 


644 


654 


665 


675 


685 


695 


706 


716 


726 






424 


736 


747 


757 


767 


777 


788 


798 


808 


818 


828 






425 


839 


849 


859 


869 


879 


890 


900 


910 


920 


931 






426 


941 


951 


961 


971 


981 


992 


*002 


*012 


*022 


*032 






427 


63 043 


053 


063 


073 


083 


093 


104 


114 


124 


134 


10 




428 


144 


154 


164 


175 


185 


195 


205 


215 


225 


235 


.1 


1.0 




429 


245 


256 


266 


276 


286 


296 


306 


316 


326 


336 


.2 
.3 
.4 
.5 
.6 


20 
3.0 
40 
5.0 
6.0 




430 


347 


357 


367 


377 


387 


397 


407 


417 


427 


437 




431 


447 


458 


468 


478 


488 


498 


508 


518 


528 


538 




432 


548 


558 


568 


578 


588 


598 


608 


618 


628 


639 


.7 


7.0 




433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 


.8 


80 




434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 


.9 


9.0 




435 


849 


859 


869 


879 


889 


899 


909 


919 


928 


938 






436 


948 


958 


968 


978 


988 


998 


*008 


*018 


*028 


*038 






437 


64 048 


058 


068 


078 


088 


098 


107 


117 


127 


137 






438 


147 


157 


167 


177 


187 


197 


207 


217 


226 


236 






439 


246 


256 


266 


276 


286 


296 


306 


315 


325 


335 


.1 
.2 
.3 
.4 


2*8 
3.8 




440 


345 
444 


355 


365 


375 


384 


394 


404 


414 


424 


434 




441 


453 


463 


473 


483 


493 


503 


512 


522 


532 




442 


542 


552 


562 


571 


581 


591 


601 


611 


621 


630 


.5 


4.7 




443 


640 


650 


660 


670 


679 


689 


699 


709 


718 


728 


.6 


57 




444 


738 


748 


758 


767 


777 


787 


797 


806 


816 


826 


.7 


6.6 




445 


836 


846 


855 


865 


875 


885 


894 


904 


914 


923 


.8 


7.6 




446 


933 


943 


953 


962 


972 


982 


992l*00T 


*011 


*021 


.9 


8.5 




447 


65 031 


040 


050' 060 


069 


079 


089 


098 


108 


118 






448 


128 


137 


147 


157 


166 


176 


186 


195 


205 


215 






449 


224 


234 


244 


253 


263 


273 


1 282 


292, 302 


311 






450 


321 


331 


340 
2 


350 
3 


360 
4 


369 
5 


! 37g 

6 


389 

7 


398 
8 


408 
9 




N. 





1 


P.P. 





630 







TABLE v.— LOGARITHMS i 


DF NUMBERS. 








N. 





1 

331 
"427 


2 

340 
437 


3 

350 

446 


4 

360 
456 


5 


6 


7 


8 
398 


9 

408 


P. P. 


450 


65 321 
417 


369 
466 


379 


389 






451 


475 


485 


494 


504 




452 


514 


523 


533 


542 


552 


562 


571 


581 


590 


600 


lO 




453 


610 


619 


629 


638 


648 


657 


667 


677 


686 


696 


.1 


1-0 




454 


705 


715 


724 


734 


744 


753 


763 


772 


782 


79] 


.2 


2.0 




455 


801 


810 


820 


830 


839 


849 


858 


868 


877 


887 


.3 


30 




456 


896 


906 


915 


925 


934 


944 


953 


963 


972 


982 


• 4 


4.0 




457 


991 


*001 


*010 


*020 


*029 


*039 


*048 


*058 


*067 


*077 


.5 


5.0 




458 


66 086 


096 


105 


115 


124 


134 


143 


153 


162 


172 


.6 


6.0 




459 


181 


190 


200 


209 


219 


228 


238 


247 


257 


266 


• 7 
.8 


7.0 
8.0 




























460 


276 


285 


294 


304 


313 


323 


332 


342 


351 


360 


.9 


9.0 




461 


370 


379 


389 


398 


408 


417 


426 


436 


44S 


455 






462 


464 


473 


483 


492 


502 


511 


520 


530 


539 


548 






463 


558 


567 


577 


586 


595 


605 


614 


62c 


633 


642 






464 


652 


661 


670 


680 


689 


698 


708 


717 


726 


736 






465 


745 


754 


764 


773 


782 


792 


801 


810 


820 


829 


^ 




466 


838 


848 


857 


866 


876 


885 


894 


904 


913 


922 


.1 


0.9 




467 


93l 


941 


950 


959 


969 


978 


987 


996 


*006 


*015 


.2 


1.9 




468 


67 024 


034 


043 


052 


061 


071 


080 


089 


099 


108 


.3 


2.8 




469 


117 


126 


136 


145 


154 


163 


173 


182 


191 


200 


• 4 
.5 

• 6 
.7 
.8 


11 

5-7 




470 


210 


219 


228 


237 


246 


256 


265 


274 


283 


293 




471 


302 


311 


320 


329 


339 


348 


357 


366 


376 


385 


6-6 
76 




472 


394 


403 


412 


422 


431 


440 


449 


458 


467 


477 


.9 


8.5 




473 


486 


495 


504 


513 


523 


532 


541 


550 


559 


568 






474 


578 


587 


596 


605 


614 


623 


633 


642 


651 


660 






475 


669 


678 


687 


697 


706 


715 


724 


733 


742 


751 






476 


760 


770 


779 


788 


797 


806 


815 


824 


833 


842 






477 


852 


861 


870 


879 


888 


897 


9C6 


915 


924 


933 






478 


943 


952 


961 


970 


979 


988 


997 


*0C6 


*015 


*024 


9 




479 


68 033 


042 


051 


060 


070 


079 


088 


097 


106 


115 


.1 
.2 
.3 
• 4 
.5 





9 
• 8 

7 
6 
5 




480 


124 


133 


142 


151 


160 


169 


178 


187 


196 


205 


1 
2 
3 
4 




481 


214 


223 


232 


24l 


250 


259 


268 


277 


286 


295 




482 


304 


31J 


322 


331 


340 


349 


358 


367 


376 


385 


.6 


5 


4 




483 


394 


403 


412 


421 


430 


439 


448 


457 


466 


475 


.7 


6 


3 




484 


484 


493 


502 


511 


520 


529 


538 


547 


556 


565 


.8 


7 


2 




485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


654 


.9 


8 


1 




486 


663 


672 


681 


690 


699 


708 


717 


726 


735 


744 






487 


753 


762 


770 


779 


788 


797 


806 


815 


824 


833 






488 


842 


851 


860 


868 


877 


886 


895 


904 


913 


922 






489 


931 


940 


948 


957 


966 


975 


984 


993 


*002 


*010 






490 


69 019 


028 


037 


046 


055 


064 


073 


081 


090 


099 


.1 
.2 


0^- 




491 


108 


117 


126 


134 


143 


152 


161 


170 


179 


187 




1 


o 




492 


196 


205 


214 


223 


232 


240 


249 


258 


267 


276 


• 3 


2 


5 




493 


284 


293 


302 


311 


320 


328 


337 


346 


355 


364 


.4 


3 


4 




494 


372 


381 


390 


399 


408 


416 


425 


434 


443 


451 


.5 


4 


2 




495 


460 


469 


478 


487 


495 


504 


513 


522 


530 


539 


.6 


5 


1 




496 


548 


557 


565 


574 


583 


592 


600 


609 


618 


627 


.7 


5 


9 




497 


635 


644 


653 


662 


670 


679 


688 


697 


705 


714 


.8 


6 


8 




498 


723 


731 


740 


749 


758 


766 


775 


784 


792 


801 


.9 


7 


6 




499 


810 


819 


827 


836 


845 


853 


862 


871 


879 


888 






600 


897 


905 

1 


914 
2 


923 
3 


93l 


940 


949 
6 


958 

7 


966 
8 


975 
9 




N. 





4 


5 


P. P. 





631 







TABLE V 


—LOGARITHMS Ot NUMBERS. 








N. 





1 


2 


3 

923 


4 

93] 


5 

940 


6 

949 


7 
958 


8 

966 
*053 


9 

975 
*06l 


P. P. 


500 


69 897 


905 


914 






501 


984 


992 


*001 


*010 


*018 


*027 


*036 


*044 




502 


70 070 


079 


087 


096 


105 


113 


122 


131 


139 


148 


9 




503 


157 


165 


174 


182 


191 


200 


208 


217 


226 


234 


.1 


0-9 




504 


243 


251 


260 


269 


277 


286 


294 


303 


312 


320 


.2 


1 


8 




505 


329 


337 


346 


355 


363 


372 


380 


389 


398 


406 


.3 


2 


7 




506 


415 


423 


432 


441 


449 


458 


466 


475 


483 


492 


.4 


3 


6 




507 


501 


509 


518 


526 


535 


543 


552 


560 


569 


578 


• 5 


4 


5 




508 


586 


595 


603 


612 


620 


629 


637 


646 


654 


663 


.6 


5 


4 




509 


672 


680 


689 


697 


706 


714 


723 


731 


740 


748 


• 7 
.8 
.9 


6 
7 
8 


3 

2 

1 




510 


757 


765 


774 


782 


791 


799 


808 


816 


825 


833 




511 


842 


850 


859 


867 


876 


884 


893 


90T 


910 


918 






512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*008 






513 


71 Oil 


020 


028 


037 


045 


054 


062 


071 


079 


088 






514 


096 


105 


113 


121 


130 


138 


147 


155 


164 


172 






515 


180 


189 


197 


206 


214 


223 


231 


239 


248 


256 


8 




516 


265 


273 


282 


290 


298 


307 


315 


324 


332 


340 


• 1 


o.g 




517 


349 


357 


366 


374 


382 


391 


399 


408 


416 


424 


.2 


1-7 




518 


433 


441 


449 


458 


466 


475 


48c 


491 


500 


508 


.3 


2-5 




519 


516 


525 


533 


542 


550 


558 


567 


575 


583 


592 


.43. 4 

K A n 




520 


600 


608 


617 


625 


633 


642 


650 


659 


667 
750 


675 
758 


.5 
.6 
.7 
.8 


*i5 

1:1 

68 




521 


684 


692 


700 


709 


717 


725 


734 


742 


• 


522 


767 


775 


783 


792 


800 


808 


817 


825 


833 


842 


.9 


7.6 




523 


850 


858 


867 


875 


883 


891 


900 


908 


916 


925 






524 


933 


941 


949 


958 


966 


974 


983 


991 


999 


*007 






525 


72 016 


024 


032 


040 


049 


057 


065 


074 


082 


090 






526 


098 


107 


115 


123 


131 


140 


148 


156 


164 


173 


■ 




527 


181 


189 


197 


206 


214 


222 


230 


238 


247 


255 






528 


263 


271 


280 


288 


296 


304 


312 


321 


329 


337] 




8 




529 


345 


354 


362 


370 


378 


386 


395 


403 


411 


419 


.1 
.2 


0.8 
1.6 




























530 


427 


436 


444 


452 


460 


468 


476 


485 


493 


501 


.8 
.4 
.5 


2.4 
3.2 
4.0 




531 


509 


517 


526 


534 


542 


550 


558 


566 


575 


583 




5S2 


591 


599 


607 


615 


624 


632 


640 


648 


656 


664 


.6 


48 




533 


672 


681 


689 


697 


705 


713 


721 


729 


738 


746 


• 7 


5.6 




534 


754 


762 


770 


778 


786 


795 


803 


811 


819 


827 


.8 


6.4 




535 


835 


843 


851 


859 


868 


876 


884 


892 


900 


908 


.9 


7.2 




536 


916 


924 


932 


941 


949 


957 


965 


973 


981 


989 






537 


997 


*005 


*013 


*02l 


*030 


*038 


*046 


*054 


*062 


*070 






538 


73 078 


086 


094 


102 


110 


118 


126 


134 


143 


151 






539 


159 


167 


175 


183 


191 


199 


207 


215 


223 


231 






640 


239 


247 


255 


263 


27l 


279 


287 


295 


303 


311 


.1 
.2 


0% 

1.5 




541 


319 


328 


336 


344 


352 


360 


368 


376 


384 


392 




542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 


.3 


2.2 




543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


.4 


3.0 




544 


560 


568 


576 


584 


592 


600 


608 


615 


623 


631 


.5 


3.^ 




545 


639 


647 


655 


663 


671 


679 


687 


695 


703 


711 


.6 


4.5. 




546 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 


.7 


5.2 




547 


798 


806 


814 


822 


830 


838 


846 


854 


862 


870 


.8 


6 




548 


878 


886 


894 


902 


909 


917 


925 


933 


941 


949 


9 


6.7 




549 


957 


965 


973 


981 


989 


997 


*004 


*012 


*020 


*028 






650 


74 036 


044 


052 


060 


068 


075 


083 


091 

7 


099 
8 


107 
9 




N. 





1 


2 


3 


4 


5 


6 


P. P, 





632 









TABLE v.— LOGARITHMS OF NUMBERS. 






N. 





l' 


2 


3 


4 


5 


6 

083 


7 
091 


8 

,099 
178 


9 

107 
186 


P. P. 


550 


74 036 


044 


052 


060 


068 


075 




551 


115 


123 


131 


139 


146 


154 


162 


170 




552 


194 


202 


209 


217 


225 


233 


241 


249 


257 


264 




553 


272 


280 


288 


296 


304 


312 


319 


327 


335 


343 




554 


351 


359 


366 


374 


382 


390 


398 


406 


413 


421 




555 


429 


437 


445 


453 


460 


468 


476 


484 


492 


499 




556 


507 


515 


523 


531 


538 


546 


554 


562 


570 


577 




557 
558 


585 
663 


593 
671 


601 
679 


609 
687 


616 
694 


624 
702 


632 
710 


640 
718 


648 
725 


655 
733 


.1 
.2 

• 3 
.4 
.5 
.6 
.7 

• 8 
.9 


0.8 
16 
2-4 


559 


741 


749 


756 


764 


772 


780 


788 


795 


803 


811 


560 


819 


826 


834 


842 


850 


857 


865 


873 


881 


888 


3.2 
4-0 


561 
562 
563 
564 


896 

973 

75 051 

128 


904 
981 
058 
135 


912 
989 
066 
143 


919 
997 
074 
151 


927 

*004 

081 

158 


935 

*012 

089 

166 


942 

*020 

097 

174 


950 

*027 

105 

182 


958 

*035 

112 

189 


966 

*043 

120 

197 


48 
5.6 
6.4 
7.2 


565 


205 


212 


220 


228 


235 


243 


251 


258 


266 


274 




566 


281 


289 


297 


304 


312 


320 


327 


335 


343 


350 




567 


358 


366 


373 


381 


389 


396 


404 


412 


419 


427 




568 


435 


442 


450 


458 


465 


473 


480 


488 


496 


503 




569 


511 


519 


526 


534 


541 


549 


557 


564 


572 
648 


580 
656 




570 


587 


595 


602 


610 


618 


625 


633 


641 




571 
572 


663 
739 


671 
747 


679 
755 


686 
762 


694 
770 


70l 
777 


709 
785 


717 
792 


724 
800 


732 
808 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


'l 


573 


815 


823 


830 


838 


846 


853 


861 


868 


876 


883 


1 
2 
3 
3 
4 
5 
6 
6 


2 


574 


891 


899 


906 


914 


921 


929 


936 


944 


951 


959 


575 
576 


967 
76 042 


974 
050 


982 
057 


989 
065 


997 
072 


*004 
080 


*012 
087 


*019 
095 


*027 
102 


*034 
110 


577 


117 


125 


132 


140 


147 


155 


162 


170 


178 


185 


578 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 


7, 


579 


268 


275 


283 


290 


298 


305 


313 


320 


328 


335 


580 


343 


350 


358 


365 


372 


380 


387 


.395 


402 


410 




581 


417 


425 


432 


440 


447 


455 


462 


470 


477 


485 




582 


492 


500 


507 


514 


522 


529 


537 


544 


552 


559 




583 


567 


574 


582 


589 


596 


604 


611 


619 


626 


634 




584 


641 


648 


656 


663 


671 


678 


686 


693 


700 


7C8 




585 


715 


723 


730 


738 


745 


752 


760 


767 


775 


782 




586 


790 


797 


804 


812 


819 


827 


834 


841 


849 


856 


!•( 


587 


864 


871 


878 


886 


893 


901 


908 


915 


923 


930 


.3 

.2 
.3 
.4 
• 5 
.6 
.7 
.8 

n 


4 

0.7 
1.4 
2.1 


588 


937 


945 


952 


960 


967 


974 


982 


989 


997 


*004 


589 


77 Oil 
085 
158 


019 
092 
166 


026 
100 
173 


033 


041 


048 


055 


063 


070 


078 


590 


107 


114 


122 


129 


136 


144 


151 


2.8 
3.5 


591 


181 


188 


195 


203 


210 


217 


225 


4.2 
4.9 
5.6 

ft n 


592 


232 


239 


247 


254 


261 


269 


276 


283 


291 


298 


593 


305 


313 


320 


327 


335 


342 


349 


356 


364 


371 


594 


378 


386 


39^ 


400 


408 


415 


422 


430 


437 


444 


• 9 O'O 


595 


451 


459 


466 


473 


481 


488 


49fi 


503 


510 


517 




596 


524 


532 


539 


546 


554 


561 


568 


575 


583 


590 




597 


597 


604 


612 


619 


626 


634 


641 


648 


655 


663 




598 


670 


677 


684 


692 


699 


706 


713 


721 


728 


735 




599 


742 


750 


757 


764 


771 


779 


786 


793 


800 


808 




600 


815 


822 

1 


829 
2 


837 
3 


844 
4 


851 


858 


866 


873 


880 
9 




N. 





5 


6 


7 


8 


p. p. 



ex-i 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 

829 


3 

837 


4 

844 


5 

85l 


6 

858 


7 
866 


8 
873 


9 

880 


500 


77 815 


822 


501 


887 


894 


902 


909 


916 


923 


931 


938 


945 


952 


502 


959 


967 


974 


981 


988 


995 


*003 


*010 


*017 


*024 


303 


78 031 


039 


046 


053 


060 


067 


075 


082 


089 


096 


304 


103 


111 


118 


125 


132 


139 


147 


154 


161 


168 


305 


175 


182 


190 


197 


204 


211 


218 


226 


233 


240 


306 


247 


254 


261 


269 


276 


283 


290 


297 


304 


311 


307 


319 


326 


333 


340 


347 


354 


362 


369 


376 


383 


308 


390 


397 


404 


412 


419 


426 


433 


440 


447 


454 


309 


461 
533 


469 
540 


476 
547 


483 


490 


497 


504 


511 


518 


526 


510 


554 


561 


568 


575 


583 


590 


597 
668 


511 


604 


611 


618 


625 


632 


639 


646 


654 


661 


512 


675 


682 


689 


696 


703 


710 


717 


725 


732 


739 


513 


746 


753 


760 


767 


774 


781 


788 


795 


802 


810 


514 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


515 


887 


894 


901 


908 


915 


923 


930 


937 


944 


951 


116 


958 


965 


972 


979 


986 


993 


*000 


*007 


*014 


*021 


.17 


79 028 


035 


042 


049 


056 


063 


070 


078 


085 


092 


118 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 


.19 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 


530 


239 
309 


246 
316 


253 
323 


260 


267 


274 


281 


288 


295 
365 


302 
372 


21 


330 


337 


344 


351 


358 


22 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 


23 


449 


456 


462 


469 


476 


483 


490 


497 


504 


511 


24 


518 


525 


532 


539 


546 


553 


560 


567 


574 


581 


25 


588 


595 


602 


609 


616 


622 


629 


636 


643 


650 


26 


657 


664 


671 


678 


685 


692 


699 


706 


713 


720 


27 


727 


733 


740 


747 


754 


761 


768 


775 


782 


789 


28 


796 


803 


810 


816 


823 


830 


837 


844 


851 


858 


29 


865 


872 


879 


886 


892 


899 


906 


913 


920 


927 


J30 


934 


941 


948 


954 


961 


968 


975 


982 


989 


996 
065 


31 


80 003 


010 


016 


023 


030 


037 


044 


051 


058 


32 


071 


078 


085 


092 


099 


106 


113 


120 


126 


133 


33 


140 


147 


154 


161 


168 


174 


181 


188 


195 


202 


34 


209 


216 


222 


229 


236 


243 


250 


257 


263 


270 


35 


277 


284 


291 


298 


304 


311 


318 


325 


332 


339 


33 


345 


352 


359 


366 


373 


380 


386 


393 


400 


407 


37 


414 


421 


427 


434 


441 


448 


455 


461 


468 


475 


38 


482 


489 


495 


502 


509 


516 


523 


529 


536 


543 


39 


550 


557 


563 


570 


577 


584 


591 


597 


604 


611 


540 


618 
686 


625 


63l 


638 


645 


652 


658 


685 


672 


679 


41 


692 


699 


706 


713 


719 


726 


733 


740 


746 


42 


753 


760 


767 


774 


780 


787 


794 


801 


807 


814 


43 


821 


828 


834 


841 


848 


855 


861 


868 


875 


882 


44 


888 


895 


902 


909 


915 


922 


929 


936 


942 


949 


45 


956 


962 


969 


976 


983 


989 


996 


*003 


*010 


*016 


46 


81 023 


030 


036 


043 


050 


057 


063 


070 


077 


083 


47 


090 


097 


104 


110 


117 


124 


130 


137 


144 


151 


48 


157 


164 


171 


177 


184 


191 


197 


204 


211 


218 


49 


224 


231 


238 


244 


251 


258 


264 


271 


278 


284 


550 


291 


298 

1 


304 
2 


311 
3 


318 


324 


33l 
6 


338 

7 


345 
8 


35l 
9 


N. 





4 


5 



P. p. 



1.5 
2-2 
3.0 
3.7 
4.5 
5.2 
6.0 
6.7 





7 


1 


0.7 


2 


1.4 


3 


2.1 


4 


2.8 


5 


3.5 


6 


4.2 


7 


4.9 


8 


5.6 


9 


6.3 





6 


1 


0.6 


2 


1 


3 


3 


1 


9 


4 


2 




5 


3 


2 


6 


3 


9 


7 


4 


5 


8 


5 


2 


9 


5 


8 



P. p. 



634 









TABLE v.— LOGARITHMS OF NUMBERS. 








N. 





1 


3 

304 


3 

311 


4 

318 


5 

324 


6 

33l 


7 
338 


8 


9 

351 
418 


P. P. 




650 


81 29l 


298 


345 






651 


358 


365 


371 


378 


385 


391 


398 


405 


411 




352 


425 


431 


438 


444 


451 


458 


464 


471 


478 


484 






653 


491 


498 


504 


511 


518 


524 


531 


538 


544 


551 






554 


558 


564 


571 


577 


584 


591 


597 


604 


611 


617 






655 


624 


631 


637 


644 


650 


657 


664 


670 


677 


684 






556 


690 


697 


703 


710 


717 


723 


730 


736 


•743 


750 






557 


756 


763 


770 


776 


783 


789 


796 


803 


809 


816 


.1 
.2 
.3 
.4 
.5 
.6 

• 7 
.8 

• 9 


0.7 
1.4 
2.1 
2.8 
3-5 
4.2 
4.9 
5.6 
6.3 




658 
B59 


822 
888 


829 
895 


836 
901 


842 
908 

974 


849 
915 

980 


855 
921 


862 
928 


869 
934 


875 
941 


882 
948 




660 


954 


961 


967 

033 
099 
164 
230 


987 


994 


*000 


*007 


*013 




B61 
B62 
B63 

B64 


82 020 
086 
151 
217 


026 
092 
158 

223 


040 
105 
171 
236 


046 
112 
177 
•343 


053 
118 
184 

249 


059 
125 
190 
256 


066 
131 
197 
262 


072 
138 
203 
269 


079 
145 
210 
275 




B65 


282 


288 


295 


302 


308 


315 


321 


328 


334 


341 






B66 


347 


354 


360 


367 


373 


380 


386 


393 


399 


406 






B67 


412 


419 


425 


432 


438 


445 


451 


458 


464 


471 






)68 


477 


484 


490 


497 


503 


510 


516 


523 


529 


536 






B69 


542 


549 


555 


562 


568 


575 


581 


588 


594 


601 






B70 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 




B71 


672 


678 


685 


691 


698 


704 


711 


717 


724 


730 


• 1 
.2 
.3 
.4 
.5 
.6 

• 7 
.8 
.9 


o^'' 




372 


737 


743 


750 


756 


763 


769 


775 


782 


788 


795 


u 

1 
1 

2 
3 
3 
4 
5 
5 


o 

3 

9 

9 
5 




373 


801 


808 


814 


821 


827 


834 


840 


846 


853 


859 




374 


866 


872 


879 


885 


892 


898 


904 


911 


917 


924 




375 


930 


937 


943 


949 


956 


962 


969 


975 


982 


988 




376 


994 


*001 


*007 


*014 


*020 


*027 


*033 


*039 


*046 


*052 




577 


83 059 


065 


071 


078 


084 


091 


097 


103 


110 


116 




578 


123 


129 


136 


142 


148 


155 


161 


168 


174 


180 




579 


187 
251 


193 
257 


200 


206 


212 


219 


225 


231 


238 


244 




580 


263 


270 


276 


283 


289 


295 


302 


308 






581 


314 


321 


327 


334 


340 


346 


353 


359 


365 


372 




582 


378 


385 


391 


397 


404 


410 


416 


423 


429 


435 






583 


442 


448 


455 


461 


467 


474 


480 


486 


493 


499 






584 


505 


512 


518 


524 


531 


537 


543 


550 


556 


562 






585 


569 


575 


581 


588 


594 


600 


607 


613 


619 


626 






586 


632 


638 


645 


651 


657 


664 


670 


676 


683 


689 






587 


695 


702 


708 


714 


721 


727 


733 


740 


746 


752 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


0.6 
1.2 
1.8 
2.4 
3-0 
3.6 
4.2 
4.8 
5.4 




588 


759 


765 


771 


778 


784 


790 


796 


803 


809 


815 




,39 


822 


828 


834 


841 


847 


853 


859 


866 


872 


878 




590 


885 


891 


897 


904 


910 


916 


922 


929 


935 


94l 




191 


948 


954 


960 


966 


973 


979 


985 


992 


998 


*004 




)92 


84 010 


017 


023 


029 


035 


042 


048 


054 


061 


067 




)93 


073 


079 


086 


092 


098 


104 


111 


117 


123 


129 




)94 


136 


142 


148 


154 


161 


167 


173 


179 


186 


192 




95 


198 


204 


211 


217 


223 


229 


236 


242 


248 


254 






>96 


261 


267 


273 


279 


286 


292 


298 


304 


311 


317 






»97 


323 


329 


335 


342 


348 


354 


360 


367 


373 


379 






>98 


385 


392 


398 


404 


410 


416 


423 


429 


435 


441 






>99 


447 
510 


454 
516 


460 


466 


472 


479 
541 

5 


485 
547 

6 


491 


497 


503 






roo 


522 


528 


534 
4 


553 

7 


55S 
8 


565 
9 




N. 





1 


2 


3 


P. 


P. 







635 









TABLE v.— LOGARITHMS OF NUMBERS. 




N. 





1 
516 


2 

522 


3 

528 


4 

534 


5 

541 


6 

5455 


7 
553 


8 
559 


9 

565 


P 


. P. 


700 


84 510 






701 


572 


578 


584 


590 


596 


603 


609 


615 


621 


627 




702 


633 


640 


646 


652 


658 


664 


671 


677 


683 


689 






703 


695 


701 


708 


714 


720 


726 


732 


739 


745 


751 






704 


757 


763 


769 


776 


782 


788 


794 


800 


806 


813 






705 


■ 819 


825 


831 


837 


843 


849 


856 


862 


863 


874 






706 


880 


886 


893 


899 


905 


911 


917 


923 


929 


936 




_ 


707 


942 


948 


954 


960 


966 


972 


979 


985 


991 


997 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
• 8 
.9 




708 


85 003 


009 


015 


021 


028 


034 


040 


046 


052 


058 


709 


064 


070 


077 
138 


083 

144 


089 
150 


095 


101 


107 


113 


119 


no 


126 
187 


132 
193 


156 


162 


168 


174 


181 


2.6 
32 


711 


199 


205 


211 


217 


223 


229 


236 


242 


!:§ 

5-2 
5.8 


712 


248 


254 


280 


266 


272 


278 


284 


290 


297 


303 


713 


309 


315 


321 


327 


333 


339 


345 


351 


357 


363 


714 


370 


37^ 


382 


388 


394 


400 


406 


412 


418 


424 


715 


430 


436 


443 


449 


455 


461 


467 


473 


479 


485 






716 


491 


497 


503 


509 


515 


521 


527 


533 


540 


546 






717 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 






718 


612 


618 


624 


630 


636 


642 


648 


655 


661 


667 






719 


673 
733 


679 
739 


685 

745 


691 
751 


697 


703 


709 


715 


721 


727 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 




720 


757 


763 


769 


775 


781 


787 
847 




721 


793 


799 


805 


811 


817 


823 


829 


835 


841 


6 

0.6 
1.2 
1-8 
2.4 
3.0 
3.6 
4.2 
4.8 
5.4 


^22 


853 


859 


865 


872 


878 


884 


890 


896 


902 


908 


723 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


^24 


974 


980 


986 


992 


998 


*004 


*010 


*016 


*022 


*028 


^25 


86 034 


040 


046 


052 


058 


063 


069 


075 


081 


087 


^26 


093 


099 


105 


111 


117 


123 


129 


135 


141 


147 


^27 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 


^28 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


^29 


273 


278 


284 


290 


296 


302 


308 


314 


320 


326 


730 


332 
391 


338 


344 


350 


356 


362 
421 


368 

427 


374 


380 


386 






731 


397 


403 


409 


415 


433 


439 


445 




732 


451 


457 


463 


469 


475 


481 


486 


492 


498 


504 






733 


510 


516 


522 


528 


534 


540 


546 


552 


558 


563 






734 


569 


575 


581 


587 


593 


599 


605 


611 


617 


623 






735 


628 


634 


640 


646 


652 


658 


664 


670 


676 


682 






736 


688 


693 


699 


705 


711 


717 


723 


729 


735 


741 






737 


746 


752 


758 


764 


770 


776 


782 


788 


794 


800 


• 1 
.2 

• 3 

• 4 
.5 

• 6 
.7 
.8 
.9 


738 


805 


811 


817 


823 


829 


835 


841 


847 


852 


858 


739 


864 


870 


876 


832 

941 


888 

946 


894 


899 


905 


911 


917 


740 


923 


929 


935 


952 


958 


964 


970 


976 


2.2 
2-7 


741 


982 


987 


993 


999 


*005 


*011 


*017 


*023 


*028 


*034 


3.3 
3.8 


742 


87 040 


046 


052 


058 


064 


069 


075 


081 


087 


093 


743 
744 


099 
157 


104 
163 


110 
169 


116 
175 


122 
180 


128 
186 


134 
192 


140 
198 


145 
204 


151 
210 


745 


215 


221 


227 


233 


239 


245 


250 


256 


262 


268 






746 


274 


279 


285 


291 


297 


303 


309 


314 


320 


326 






747 


332 


338 


343 


349 


355 


361 


367 


372 


378 


384 






748 


390 


396 


402 


407 


413 


419 


425 


431 


436 


442 






749 


448 


454 


460 


465 


471 


477 


483 


489 


494 


500 






750 


506 


512 
1 


517 
2 


523 
3 


529 
4 


535 
5 


541 
6 


546 


552 


558 




N. 





7 


8 


9 


P 


. P. 



636 









TABLE V 


.—LOGARITHMS 


OF NUMBERS. 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


750 


8? 506 
564 


512 


517 


523 


529 


535 


541 
598 


546 
604 


552 
610 


558 
616 




751 


570 


575 


581 


587 


593 




752 


622 


627 


633 


639 645 


650 


656 


662 


668 


673 




753 


679, 685 


691 


697| 702 


708 


714 


720 


725 


731 




754 


737 743 


748 


754 


760 


766 


! 771 


777 


783 


789 




755 


794 800 


806 


812 


817 


823 


1 829 


835 


840 


846 




756 


852 858 


863 


869 


875 


881 


1 886 


892 


898 


904 


^^ 


757 


9091 915 


921 


927 


932 


938 


944 


949 


955 


961 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


o 


758 


967 


972 


978 


984 


990 


995 


*001 


*007 


*012 


*0]8 




1 
1 
2 
3 
3 
4 
4 


•0 

.2 
8 


759 


88 024 


030 


035 


041 


047 


053 


1 058 


064 


070 


075 


760 


081 


087 


093 


098 


104 


110 


115 


121 


127 


133 


4 



761 


138 144 


150 


155 


161 


167 


172 


178 


184 


190 


6 
2 
8 


762 


1951 201 


207 


21^ 


218 


224 


229 


235 


241 


247 


763 


252 


258 


264 


269 


275 


281 


286 


292 


298 


303 


764 


309 


315 


320 


326 


332 


337 


343 


349 


355 


360 


0-* 


765 


366 


372 


377 


383 


389 


394 


400 


406 


411 


417 




766 


423! 428 


434 


440 


445 


451 


457 


462 


468 


474 




767 


479! 485 


491 


496 


502 


508 


513 


519 


525 


530 




768 


536 


542 


547 


553 


558 


564 


570 


575 


581 


587 




769 


592 


598 


604 


609 


615 


621 


626 


632 


638 


643 




770 


649 


654 


660 


666 


671 


677 


683 


688 


694 


700 




771 


705 


711 


716 


722 


728 


733 


739 


745 


750 


756 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


0^' 


772 


76l 


767 


773 


778 


784 


790 


795 


801 


806 


812 




1 
1 

2 
2 
3 
3 
4 
4 




I 


773 


818 


823 


829 


835 


840 


846 


851 


857 


863 


868 


774 


874 


879 


885 


891 


896 


902 


907 


913 


919 


924 


775 


930 


936 


941 


947 


952 


958 


964 


969 


975 


9R0 


776 


986 


992 


997 


*003 


*008 


^014 


*0]9 


*025 


*031 


*036 


i 

4 
9 


777 
778 


89 042 
098 


047 
103 


053 
109 


059 
114 


064 
120 


070 
126 


075 
131 


081 
137 


087 

142 


092 
148 


779 


153 


159 


165 


170 


176 


181 


187 


193 


198 


204 


780 


209 


215 


220 


226 


231 


237 


243 


248 


254 


259 




781 


265 


270 


276 


282 


287 


293 


298 


304 


309 


315 




782 


320 


326 


332 


337 


343 


348 


354 


359 


365 


370 




783 


376 


381 


387 


393 


398 


404 


409 


415 


420 


426 




784 


431 


437 


442 


448 


454 


459 


465 


470 


476 


481 




785 


487 


492 


498 


503 


509 


514 


520 


525 


531 


536 




786 


542 


548 


5531 


559 


564 


570 


575 


581 


586 


592 


p 


787 


597 


603 


608; 


614 


619 


625 


630 


636 


641 


647 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


0.5 
1-0 
1.5 


788 


652 


658 


663 


669 


674 


680 


685 


691 


696 


702 


789 


707 


713 


718| 


724 


729 


735 


740 


746 


751 


757 


790 


765 


768 


773 


779 


784 


790 


795 


801 


806 


812 


2.0 

2.5 


791 

792 


817 
872 


823 
878 


828| 
883 


834 
889 


839 
894 


845 
900 


850 
905 


856 
911 


861 

916 


867 
922 


3.0 
3.5 
4.0 
4.5 


793 


927 


933 


988! 


943! 949 


954 


960 


965 


971 


976 


794 


982 


987 


993' 


998 *004 


*009 


*015 


*020 


*026 


*031 


795 


90 036 


042 


0471 


053 


058 


064! 


069 


075 


080 


086 




796 


091 


097 


102! 


107 


113 


118! 


124 


129 


135 


140 




797 


146 


151 


156 


162 


167 


173! 


178 


184 


189 


195 




798 


200 


205 


211 


216 


222 


2271 


233 


238 


244 


249 




799 


254 


260 


265 


271 


276 


282j 


287 


292 


298 


303 




800 


809 


314 
1 


320 
2 


325 
3 


330 
4 


336 
5 


341 
6 


347 

7 


352 
8 


358 
9 




N. 





P. P. 



637 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 

330 


5 

336 


6 

341 


7 


8 


9 

358 


F. P. 


800 


90 309 


314 


320 


325 


347 


352 




801 
802 
803 
804 
805 
806 
807 
808 
809 


363 
417 
471 
525 
579 
633 
687 
741 
795 


368 
423 
477 
531 
585 
639 
692 
746 
800 


374 
428 
482 
536 
590 
644 
698 
752 
805 


379 
433 
488 
542 
596 
649 
703 
757 
811 


385 
439 
493 
547 
601 
655 
709 
762 
816 


390 
444 
498 
552 
606 
660 
714 
768 
821 


396 
450 
504 
558 
612 
666 
719 
773 
827 


401 
455 
509 
563 
617 
671 
725 
778 
832 


406 
460 
515 
569 
622 
676 
730 
784 
838 


412 
466 
520 
574 
628 
682 
736 
789 
843 




810 


848 

902 
955 
91 009 
062 
116 
169 
222 
275 
328 


854 

907 
961 
014 
068 
121 
174 
227 
280 
333 


859 


864 


870 


875 


880 


886 


891 


896 




811 
812 
813 
814 
815 
816 
817 
818 
819 


913 
966 
019 
073 
126 
179 
233 
286 
339 


918 
971 
025 
078 
131 
185 
238 
291 
344 


923 
977 
030 
084 
137 
190 
243 
296 
349 


929 
982 
036 
089 
142 
195 
249 
302 
355 


934 
987 
041 
094 
147 
201 
254 
307 
360 


939 
993 
046 
100 
153 
206 
259 
312 
365 


945 
998 
052 
105 
158 
211 
264 
318 
371 

423 


95n 
*0iJ3 

'ill 

163 
217 
270 
323 
376 

429 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


0^5 

1.1 

1.6 

i:f 

3.3 
3.8 

11 


830 


381 


386 


392 


397 


402 


408 


413 


418 




821 
822 
823 
824 
825 
826 
827 
828 
829 


434 
487 
540 
592 
645 
698 
750 
803 
855 


439 
492 
545 
598 
650 
703 
756 
808 
860 


445 
497 
550 
603 
656 
708 
761 
813 
866 


450 
503 
556 
608 
661 
714 
766 
819 
871 


455 
508 
561 
614 
666 
719 
771 
824 
876 


461 
513 
566 
619 
671 
724 
777 
829 
881 


466 
519 
571 
624 
677 
729 
782 
834 
887 


471 
524 
577 
629 
682 
735 
787 
839 
892 


476 
529 
582 
635 
687 
740 
792 
845 
897 


482 
534 
587 
640 
692 
745 
798 
850 
902 


V 


830 


908 


913 


918 


923 


928 


934 


939 


944 


949 


955 




831 
832 
833 
834 
835 
836 
837 
838 
839 


960 
92 012 
064 
116 
168 
220 
272 
324 
376 


965 
017 
069 
122 
174 
226 
277 
329 
381 


970 
023 
075 
127 
179 
231 
283 
335 
386 


976 
028 
080 
132 
184 
236 
288 
340 
391 

443 


981 
033 
085 
137 
189 
241 
293 
345 
397 

448 


986 
038 
090 
142 
194 
246 
298 
350 
402 


99l 
043 
096 
148 
200 
252 
303 
355 
407 


996 
049 
101 
153 
205 
257 
309 
360 
412 


*002 
054 
106 
158 
210 
262 
314 
366 
417 


*007 
059 

nl 

163 
215 
267 
319 
371 
423 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


5 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


840 


428 

479 
531 
583 
634 
685 
737 
788 
839 
891 


433 


438 


454 


459 


464 


469 


474 




841 
842 
843 
844 
845 
846 
847 
848 
849 


485 
536 
588 
639 
691 
742 
793 
844 
896 


490 
54l 
593 
644 
696 
747 
798 
850 
901 


495 
546 
598 
649 
701 
752 
803 
855 
906 


500 
552 
603 
655 
706 
757 
809 
860 
911 


505 
557 
608 
660 
711 
762 
814 
865 
916 


510 
562 
613 
665 
716 
768 
819 
870 
921 


515 
567 
619 
670 
721 
773 
824 
875 
926 


521 
572 
624 
675 
727 
778 
829 
880 
931 


526 
577 
629 
680 
732 
783 
834 
885 
937 




850 


942 


947 



1 


952 


957 


962 
4 


967 
5 


972 


977 


982 
8 


988 
9 




N. 





2 


3 


6 


7 


P. P. 



638 







TABLE V 


.—LOGARITHMS OF NUMBERS. 








N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


P. 




850 


92 942 


947 


952 


957 


962 


967 


972 


977 


982 


988 








851 


993 


998 


*003 


*008 


*013 


*018 


*023 


*028 


*034 


*039 




852 


93 044 


049 


054 


059 


064 


069 


074 


079 


084 


090 








853 


095 


100 


105 


110 


115 


120 


125 


130 


135 


H40 








854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


191 








855 


196 


201 


207 


212 


217 


222 


227 


232 


237 


242 








856 


247 


252 


257 


262 


267 


272 


278 


283 


288 


293 




O^'^ 




857 


298 


303 


308 


313 


318 


323 


328 


333 


338 


343 


.1 

.2 

3 

.4 
• 5 
.6 
•7 
.8 
.9 




858 
859 


348 
399 


354 
404 


359 
409 


364 
414 


369 
419 


374 
424 


379 
429 


384 
434 


389 

439 


394 
445 


1 
1 
2 
2 
3 
3 
4 

A 


• il 

1 

• 6 

• 2 
.7 

3 
.8 

• 4 

n 




860 


450 


455 


460 


465 


470 


475 


480 


485 


490 


495 




861 


500 


505 


510 


515 


520 


525 


530 


535 


540 


545 




862 
863 


550 
601 


556 
606 


561 
611 


566 
616 


571 
621 


576 
626 


581 
631 


586 

636 


591 
641 


596 
646 




864 


651 


656 


661 


666 


671 


676 


681 


686 


691 


696 


4-» 




865 


701 


706 


711 


716 


721 


726 


731 


736 


742 


747 








866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 








867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 








868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 








869 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 


■ 1 
.2 

• 3 
.4 
.5 
.6 
.7 
.8 

• 9 


5 

0.5 
1.0 
1.5 
2.0 
2.5 
30 
3.5 
4.0 
4.5 




870 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 




871 
872 
873 


94 002 
051 
lOl 


007 
056 
106 


012 
061 
111 


017 
066 
116 


022 
07l 
121 


026 
076 
126 


03l 
081 
131 


036 
086 
136 


041 
091 

141 


046 
096 
146 




874 
875 


151 
201 


156 
206 


161 
210 


166 

215 


171 
220 


176 
225 


181 
230 


186 
235 


191 
240 


196 
245 




876 
877 


250 
300 


255 
305 


260 
310 


265 
315 


270 
320 


275 

324 


280 

329 


285 
334 


290 
339 


295 
344 




878 
879 


349 
399 


354 
404 


359 
409 


364 

413 


369 
418 


374 
423 


379 
428 


384 
433 


389 
438 


394 
443 




880 


448 


453 


458 


463 


468 


473 


478 


483 


487 


492 




881 


497 


502 


507 


512 


517 


522 


527 


532 


537 


542 




882 


547 


552 


556 


561 


566 


571 


576 


581 


586 


591 








883 


596 


601 


606 


611 


615 


620 


625 


630 


635 


640 








884 


645 


650 


655 


660 


665 


670 


674 


679 


684 


689 








885 


694 


699 


704 


709 


714 


719 


724 


728 


733 


738 








886 


743 


748 


753 


758 


763 


788 


773 


777 


782 


787 




0^^ 




887 


792 


797 


802 


807 


812 


817 


821 


826 


831 


836 


•1 
.2 
.3 
.4 
.5 
.6 
• 7 
.8 
.9 




888 


841 


846 


851 


856 


861 


865 


870 


875 


880 


885 


0. 
1- 
1. 
2. 
2. 
3- 
3. 

A 


9 

3 

f 




889 


890 
939 


895 


900 


905 


909 


914 


919 


924 


929 


934 




890 


944 


949 


953 


958 


963 


963 


973 


978 


983 




891 


988 


992 


997 


*002 


*007 


*012 


*017 


*022 


'^026 


031 




892 


95 036 


041 


046 


051 


058 


061 


065 


070 


075 


080 




893 


085 


090 


095 


099 


104 


109 


114 


119 


124 


129 




894 


134 


138 


143 


148 


153 


158 


163 


167 


172 


177 






895 


182 


187 


192 


197 


201 


206 


211 


216 


221 


226 








896 


231 


235 


240 


245 


250 


255 


260 


264 


269 


274 








897 


279 


284 


289 


294 


298 


303 


308 


313 


318 


323 








898 


327 


332 


337 


342 


347 


352 


356 


361 


366 


371 








899 


376 


381 


385 


390 


395 


400 


405 


410 


414 


419 








900 


424 


429 


434 


438 


443 


448 


453 


458 


463 


467 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. 


P. 





639 







TABLE V 


.—LOGARITHMS OF NUMBERS. 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


900 


95 424 


429 


434 


438 


443 


448 


453 


458 


463 


467 




901 


472 


477 


482 


487 


492 


496 


501 


506 


511 


516 




902 


520 


525 


530 


535 


540 


544 


549 


554 


559 


564 




903 


569 


573 


578 


586 


588 


593 


597 


602 


607 


612 




904 


617 


621 


626 


631 


636 


641 


645 


650 


655 


660 




905 


665 


669 


674 


679 


684 


689 


693 


698 


703 


708 




906 


713 


717 


722 


727 


732 


737 


741 


746 


751 


756 




907 


760 


765 


770 


775 


780 


784 


789 


794 


799 


804 




908 


808 


813 


818 


823 


827 


832 


837 


842 


847 


851 




909 


856 


861 


866 


870 


875 


880 


885 


890 


894 


899 




910 


904 


909 


913 


918 


923 


928 


933 


937 


942 


947 




911 


952 


956 


96l 


966 


971 


975 


980 


985 


990 


994 


5 

line 


912 


999 


*004 


*009 


*014 


*018 


*023 


*G28 


*G38 


*G37 


*042 


•1 

.2 
.3 
.4 
.5 
.6 
•7 
.8 
• 9 


u 

1 
1 
2 
2 
3 
3 
4 
4 


.0 
5 


• 5 


.5 

5 


913 
914 
915 
916 
917 
918 
919 


96 047 
094 
142 
189 
237 
284 
331 


052 
099 
147 
194 
241 
289 
336 


056 
104 
151 
199 
246 
293 
341 


061 
109 
156 
204 
251 
298 
345 


066 
113 
161 
208 
256 
303 
350 


071 
118 
166 
213 
260 
308 
355 


075 
123 
170 
218 
265 
312 
360 


080 
128 
175 
222 
27C 
317 
364 


085 
132 
180 

227 
275 
322 
369 


090 
137 
185 
232 
279 
327 
374 


930 


379 


383 


388 


393 


397 


402 


407 


412 


416 


421 




921 


426 


430 


435 


440 


445 


449 


454 


459 


463 


468 




922 


473 


478 


482 


487 


492 


496 


501 


506 


511 


515 




923 


520 


525 


529 


534 


539 


543 


548 


553 


558 


562 




924 


567 


572 


576 


581 


586 


590 


595 


600 


605 


609 




925 


614 


619 


623 


628 


633 


637 


642 


647 


651 


656 




926 


661 


666 


670 


675 


680 


684 


689 


694 


698 


703 




927 


708 


712 


717 


722 


726 


731 


736 


741 


745 


750 




928 


755 


759 


764 


769 


773 


778 


783 


787 


792 


797 




929 


801 


806 


811 


815 


820 


825 


829 


834 


839 


843 




930 


848 


853 


857 


862 


867 


87l 

918 
965 


876 


881 


885 


890 




931 
932 


895 
94j 


899 

946 


904 
951 


909 
955 


913 
960 


923 
969 


927 
974 


932 
979 


937 
983 


.1 
.2 
.3 
.4 
.5 
.6 
• 7 
.8 
.9 





1 
1 
2 
2 
3 
3 
4. 


1 • 

3 

I 

a 


933 
934 
935 
936 
937 


988 
97 034 
081 
127 
174 


993 
039 
086 
132 
178 


997 
044 
090 
137 
183 


*002 
048 
095 
141 
188 


*007 
053 
099 
146 
192 


*011 
058 
104 
151 
197 


*016 
062 
109 
.155 
202 


*020 
067 
113 
160 
206 


*025 
072 
118 
164 
211 


*030 
076 
123 
169 
215 


938 
939 


220 
266 


225 
271 


229 
276 


234 
280 


239 
285 


243 
289 


248 
294 


252 
299 


257 
303 


262 
308 


940 


313 


317 


322 


326 


331 


336 


340 


345 


349 


354 




941 


359 


363 


368 


373 


377 


382 


386 


39l 


396 


400 


942 


4.05 


409 


414 


419 


423 


428 


432 


437 


442 


446 




943 


. 451 


456 


460 


465 


469 


474 


479 


483 


488 


492 




944 


497 


502 


506 


511 


515 


520 


525 


529 


534 


538 




945 


543 


548 


552 


557 


561 


566 


570 


575 


580 


584 




946 


589 


593 


598 


603 


607 


612 


616 


621 


626 


630 




947 


635 


639 


644 


649 


653 


658 


662 


667 


671 


676 




948 


681 


685 


690 


694 


699 


703 


708 


713 


717 


722 




949 


726 


731 


736 


740 


745 


749 


754 


758 


763 


768 




950 


772 


777 
1 


781 
2 


786 


790 
4 


795 


800 
6 


804 

7 


809 
8 


813 


N. 





3 


5 


, 9 


P. P. 



640 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 

78l 


3 

786 


4 

790 


795 


6 

800 


7 
804 


8 
809 


9 

813 


P.P. 


950 


97 772 


777 




951 


818 


822 


827 


831 


836 


841 


845 


850 


854 


859 




952 


863 


868 


873 


877 


882 


886 


891 


895 


900 


904 




953 


909 


914 


918 


92S 


927 


932 


936 


941 


945 


950 




954 


955 


959 


964 


968 


973 


977 


982 


986 


991 


996 




955 


98 000 


005 


009 


014 


018 


023 


027 


032 


036 


041 




956 


046 


050 


055 


059 


064 


068 


073 


077 


082 


086 


s 


957 

958 

. 959 


091 
136 
182 


095 
141 
186 


100 
145 
191 


105 
150 
195 


109 
154 
200 


114 
159 
204 


118 
163 
209 


123 
168 
213 


127 
173 
218 


132 
177 
222 


.1 

.2 
.3 
.4 
.5 
.6 
.7 
• 8 
.9 


0.3 
1.0 

1.5 


960 


227 


231 


236 


240 


245 


249 


254 


259 


263 


268 


2-0 

2.5 


961 
962 
963 
964 


272 
317 
362 
407 


277 
322 
367 
412 


28l 
326 
371 
416 


286 
331 
376 
421 


290 
335 
380 

425 


295 
340 
385 
430 


299 
344 
389 

434 


304 
349 
394 
439 


308 
353 
398 
443 


313 
358 
403 
448 


30 
3.5 
4.0 

4»5 


965 


452 


457 


461 


466 


470 


475 


479 


484 


488 


493 




966 


497 


502 


506 


511 


515 


520 


524 


529 


533 


538 




967 


542 


547 


551 


556 


560 


565 


569 


574 


578 


583 




968 


587 


592 


596 


601 


605 


610 


614 


619 


623 


628 




969 


632 


637 


641 


646 


650 


655 


659 


663 


668 


672 




970 


677 


68l 


686 


690 


695 


699 


704 


708 


713 


717 




971 


722 


726 


731 


735 


740 


744 


749 


753 


757 


762 


.1 

:f 

.4 
.5 
.6 
.7 
.8 
.9 


0^ 
0.9 
1.3 

2J 

i:f 
U 


972 
973 


766 
8ll 


771 
815 


775 
820 


780 
824 


784 
829 


789 
833 


793 
838 


798 
842 


802 
847 


807 
85l 


974 
975 
976 


856 
900 
945 


860 
905 
949 


865 
909 
954 


869 
914 
958 


873 
918 
963 


878 

922 
967 


882 
927 
97l 


887 
931 
976 


891 
936 
980 


896 
940 
985 


977 


989 


994 


998 


*003 


*007 


*011 


*016 


*020 


*025 


*029 


978 


99 034 


038 


043 


0x7 


051 


056 


060 


065 


069 


074 


979 


078 


082 


087 


091 


096 


100 


105 


109 


113 


118 


980 


122 


127 


13l 


136 


140 


145 


149 


153 


158 


162 




981 


167 


171 


176 


180 


184 


189 


193 


198 


202 


206 




; 982 


211 


215 


220 


224 


229 


233 


237 


242 


246 


251 




; 983 


25rj 


280 


264 


268 


273 


277 


282 


286 


290 


295 




984 


299 


304 


308 


312 


317 


321 


326 


330 


335 


339 




985 


343 


348 


352 


357 


36x 


365 


370 


374 


379 


383 




986 


387 


392 


396 


401 


405 


409 


414 


418 


423 


427 


a 


'987 


431 


436 


440 


445 


449 


453 


458 


462 


467 


47] 


.1 

.2 
.3 
•4 
.5 
.6 
.7 
.8 


0.4 
0.8 
1-2 


988 


475 


480 


484 


489 


493 


497 


502 


506 


511 


515 


939 


519 


524 


528 


533 


537 


541 


546 


550 


554 


559 


990 


563 


568 


572 


576 


581 


585 


590 


594 


598 


603 


].6 
20 


1 991 


607 


611 


616 


620 


625 


629 


633 


638 


642 


647 


2.4 
2.8 
3.2 


992 


651 


655 


660 


664 


668 


673 


677 


682 


686 


690 


993 


695 


699 


703 


708 


712 


717 


721 


725 


730 


734 


, 994 


738 


743 


747 


751 


756 


760 


765 


769 


773 


778 


.g j3»o 


995 


782 


786 


791 


795 


80n 


804 


808 


813 


817 


82T 




996 


826 


830 


834 


839 


843 


847 


852 


856 


861 


865 




997 


869 


874 


878 


882 


887 


891 


895 


900 


904 


908 




998 


913 


917 


922 


926 


930 


935 


939 


943 


948 


952 




999 


956 


961 


965 


969 


974 


978 


982 


987 


991 


995 




^tooo 


00 000 


004 
1 


008 
2 


013 
3 


017 
4 


021 
5 


026 
6 


030 

7 


034 
8 


039 
9 




]V. 





p. p. 














6^ 


il 















TABLE V- 


-LOGARITHMS OF 


NUMBERS. 










N. 




000 000 


1 
043 


2 

087 


3 

130 


4 

173 


5 

217 


6 

260 


7 
304 


8 
347 


9 

390 


P. P. 


1000 




01 


434 


477 


521 


564 


607 


651 


694 


737 


781 


824 




02 


867 


911 


954 


997 


*041 


*084 


n27 


*171 


*214 


*257 




03 


001 301 


344 


387 


431 


474 


517 


560 


604 


647 


690 




04 


733 


777 


820 


863 


906 


950 


993 


*036 


*079 


*123 




05 


002 166 


209 


252 


295 


339 


382 


425 


468 


511 


555 




06 
07 
08 
09 


598 

003 029 

460 

891 


641 
072 
503 
934 


684 
115 
546 
977 


727 

159 

590 

*020 


770 

202 

633 

*063 


814 

245 

676 

*106 


857 

288 

719 

*149 


900 

331 

762 

*192 


943 

374 

805 

*235 


986 

417 

848 

*278 


.1 
.2 
.3 
.4 
• 5 
.6 
.7 
.8 
.9 


43 
4. 
8. 

13. 

17- 

21. 

26. 

30. 

34 


3 

7 

4 
7 

1 
4 
8 

T 


4i 

4. 
8. 

12 
17 
21 
25 
30 
34 
S8 


3 

6 
9 


1010 


004 32l 


364 


407 


450 


493 


536 


579 


622 


665 


708 


2 
5 


11 


751 


794 


837 


880 


923 


966 


*009 


*05l 


*094 


*137 


8 

1 
4 

7 


12 


005 180 


223 


266 


309 


352 


395 


438 


481 


523 


566 


13 


609 


652 


695 


738 


781 


824 


866 


909 


952 


995 


14 


006 038 


081 


123 


166 


209 


252 


295 


337 


380 


423 






15 


466 


509 


551 


594 


637 


680 


722 


765 


808 


851 




16 


893 


936 


979 


*022 


*064 


*107 


*150 


*193 


*235 


*278 




17 


007 321 


363 


40C 


449 


491 


534 


577 


620 


662 


705 




18 


748 


790 


833 


875 


918 


961 


*003 


*046 


*089 


131 




19 


008 174 


217 


259 


302 


344 


387 


430 


472 


515 


557 




1030 


600 


642 


685 


728 


770 


813 


855 


898 


940 


983 




21 


009 025 


068 


111 


153 


196 


238 


281 


323 


366 


408 


.1 
.2 
.3 
.4 


f-n 


42 

4- *> 


22 


451 


493 


536 


578 


621 


663 


706 


748 


790 


833 


8 


5 
7 



8 

1 ^ 


4 


23 


875 


918 


960 


*003 


*045 


*088 


*130 


*172 


*215 


*257 


24 


010 300 


342 


385 


427 


469 


512 554 


596 


639 


681 


17 
•^1 


16 
''I 


8 



25 


724 


766 


808 


851 


893 


935 


978 


*020 


*062 


*105 


2 


26 


Oil 147 


189 


232 


274 


316 


359 


401 


443 


486 


528 


.6 


25 


5 


25 


2 


27 


570 


612 


655 


697 


739 


782 


824 


866 


908 


951 


.7 
.8 


29 
34 


7 


29 
33 


.4 


28 


993 


*035 


*077 


^^120 


*162 


*204 


*246 


*288 *331 


*373 


6 


29 


012 415 


457 


500 


542 


584 


626 


668 


710 


753 


795 


.9 


38 


2 


37 


• 8 


1030 


837 


879 


92l 


963 


*006 


*048 


*090 


*132 


174 


216 




31 


013 258 


301 


343 


385 


427 


469 


511 


553 


595 


637 




32 


679 


722 


764 


806 


848 


890 


932 


974 


*016 


*058 




33 


014 100 


142 


184 


226 


268 


310 


352 


394 


436 


478 




34 


520 


562 


604 


646 


688 


730 


772 


814 


856 


898 




35 


940 


982 


*024 


*066 


*108 


'^•150 


*192 


*234 


*276 


*318 




36 


015 360 


401 


443 


485 


527 


569 


611 


653 


695 


737 


aT A-i 


37 


779 


820 


862 


904 


946 


988 


*030 


*072 


*113 


Ibb 


.1 


4 T 


41 


38 


016 197 


239 


281 


323 


364 


406 


448 


490 


532 


578 


• 2 


8 


3 


82 


39 


615 


657 


699 


741 


782 


824 
242 


866 
284 


908 


950 


991 


.3 
= 4 
.5 
.6 
.7 


12 
16 
20 
24 
29 


.1 
i 


12.3 


1040 


017 033 


075 


117 


158 


200 


325 


367 


409 
826 


16.4 
20.5 
24-6 
28-7 


41 


450 


492 


534 


576 


617 


659 


701 


742 


784 


42 
43 


867 
018 284 


909 
326 


951 
367 


992 
409 


*034 
451 


*076 
492 


*117 
534 


*159 
575 


*201 
617 


•*-242 
659 


.8 
,9 


33 
q7 




32.8 


44 


700 


742 


783 


825 


867 


908 


950 


991 


•^033 


-^074 




45 


019 116 


158 


199 


241 


282 


324 


365 


407 


448 


490 




46 


531 


573 


614 


656 


697 


739 


780 


822 


863 


905 




47 


946 


988 


*029 


*071 


ni2 


*154 


*195 


*237 


*278 


^320 




48 


020 361 


402 


444 


485 


527 


568 


610 


651 


692 


734 




49 


775 


817 


858 


899 


941 


982 


*024 


*065 


'i^ioe 


^148 




1050 


021 189 



230 
1 


272 
2 


313 
3 


354 
4 


396 
5 


437 
6 


478 
7 


520 
8 


561 
9 


N. 


P.P. 

V 1 



642 









TABLE V.- 


-LOGARITHMS OF NUMBERS. 






N. 





1 


3 

272 


3 

313 


4 


5 


6 

437 


7 
478 


8 
520 


9 

561 


P 


.P. 


1050 


021 


189 


230 


354 


396 


.1 


fi 


51 


602 


644 


685 


726 


768 


809 


850 


892 


933 


974 


, 52 


022 


015 


057 


098 


139 


181 


222 


263 


304 


346 


387 


.2 


83 


1 53 




428 


469 


511 


552 


593 


634 


676 


717 


758 


799 


• 3 


12-4 


54 




840 


882 


923 


964 


*005 


*046 


*088 


n29 


*170 


*211 


.4 


16.6 


55 


023 


252 


293 


335 


376 


417 


458 


499 


540 


581 


623 


.5 


20-7 


56 




664 


705 


746 


787 


828 


869 


910 


951 


993 


*034 


.6 


24.9 


57 


024 


075 


116 


157 


198 


239 


280 


321 


362 


403 


444 


.7 


29.0 


58 




485 


526 


568 


609 


650 


691 


732 


773 


814 


855 


.8 


33.2 


59 
1060 

61 


025 


896 


937 


978 


*019 


*060 


*101 


*142 


*183 


*224 


*265 


.9 
.1 


37.3 


306 


347 


388 


429 


469 


510 


551 


592 


633 


674 


41 

4.1 


715 


756 


797 


838 


879 


920 


961 


*002 


*042 


*083 


62 


026 


124 


165 


206 


247 


288 


329 


370 


410 


451 


492 


.2 


8.2 


63 




533 


574 


615 


656 


696 


737 


778 


819 


860 


901 


.3 


12.3 


! 64 




941 


982 


*023 


*064 


*105 


*145 


*186 


*227 


*268 


*309 


.4 


16.4 


65 


027 


349 


390 


431 


472 


512 


553 


594 


635 


675 


716 


.5 


20.5 


66 




757 


798 


838 


879 


920 


961 


*001 


*042 


*083 


*123 


.6 


24.6 


1 67 


028 


164 


205 


246 


286 


327 


368 


408 


449 


490 


530 


• 7 


28.7 


1 68 




571 


612 


652 


693 


734 


774 


815 


856 


896 


937 


• 8 


32-8 


' 69 
1070 

71 


029 


977 
384 


*018 
424 


*059 
465 


*099 


*140 


••^181 


'^22] 


*262 


*302 


*343 


.9 
.1 


36-9 


505 


546 


586 


627 


668 


708 


749 


fo 


789 


830 


870 


911 


951 


992 


*032 


*073 


*114 


*154 


72 


030 


195 


235 


276 


316 


357 


397 


438 


478 


519 


559 


.2 


8 


1 


' 73 




599 


640 


680 


721 


761 


802 


842 


883 


923 


964 


• 3 


12 


1 


74 


031 


004 


044 


085 


125 


166 


206 


247 


287 


327 


368 


.4 


16 


2 


75 




408 


449 


489 


529 


570 


610 


651 


691 


731 


772 


.5 


20 


2 


76 




812 


852 


893 


933 


973 


*014 


*054 


*094 


*135 


*175 


.6 


24 




77 


032 


215 


256 


296 


336 


377 


417 


457 


498 


538 


578 


.7 


28 


3 


78 




619 


659 


699 


739 


780 


820 


860 


900 


941 


981 


.8 


32 


4 


79 
1080 

81 


033 


021 


061 


102 


142 


182 


222 


263 


303 


343 


383 


.9 

.1 


36 


4 


424 
825 


464 
866 


504 


544 


584 


625 


665 


705 


745 


785 


40 

40 


906 


946 


986 


*026 


*066 


*107 


147 


187 


; 82 


034 


227 


267 


307 


347 


388 


428 


468 


508 


548 


588 


.2 


8 





83 




628 


668 


708 


748 


789 


829 


869 


909 


949 


989 


.3 


12 





84 


035 


029 


069 


109 


149 


189 


229 


269 


309 


349 


389 


.4 


16 





85 




429 


470 


510 


550 


590 


630 


670 


710 


750 


790 


.5 


20 





86 




830 


870 


910 


950 


990 


*029 


*069 


*109 


*149 


*189 


.6 


24 





87 


036 


229 


269 


309 


349 


389 


429 


469 


509 


549 


589 


.7 


28 





88 




629 


669 


708 


748 


788 


828 


868 


908 


948 


988 


.8 


32 





89 
1090 

1 91 


037 


028 


068 


107 


147 


187 


227 


267 


307 


347 


386 


.9 

• 1 


36 





426 


466 


506 


546 


586 


625 


665 


705 


745 


785 


39. 

3.9 


825 


864 


904 


944 


984 


*023 


*063 


*103 


143 


183 


92 


038 


222 


262 


302 


342 


381 


421 


461 


501 


540 


580 


.2 


7 


9 


93 




620 


660 


699 


739 


779 


819 


858 


898 


938 


977 


.3 


11 


8 


, 94 


039 


017 


057 


096 


136 


176 


216 


255 


295 


335 


374 


.4 


15 


8 


, 95 




414 


454 


493 


533 


572 


612 


652 


691 


731 


771 


.5 


19 


7 


96 




810 


850 


890 


929 


969 


*008 


*048 


*088 


*127 


*167 


.6 


23 




97 


040 


206 


246 


286 


325 


365 


404 


444 


483 


523 


563 


.7 


27 


. 6 


98 




602 


642 


681 


721 


760 


800 


839 


879 


918 


958 


.8 


31 


• 6 


99 
jllOO 


041 


997 


*037 


*076 


*116 


*155 


*195 


*234 


*274 


*313 


*353 


.9 


35 


• 5 


392 


432 


471 
3 


511 
3 


550 
4 


590 
5 


629 
6 


669 

7 


708 


748 




N. 





1 


8 


9 


F 


.P. 






r 










( 


343 













TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin (f) = log (f)" + S. 




0° 


log<5&' 


' = log 


sin <A + ,S'. 


Log tan = log <f>'' + T. 




log^' 


' = log tan '4> + T\ 


It 


f 


S 


T L< 


>§:. Sin. 


S' 


T' 


t.og. Tan. 








4.685 57 


57 


— eo 


5.314 42 


42 


00 


60 


1 


57 


57 6 


.46 372 


42 


42 


6. 46 372 


120 


2 


57 


57 


.76 475 


42 


42 


.76 475 


180 


3 


57 


57 


.94 084 


42 


42 


.94 084 


240 


4 
5 


57 


57 7 


.06 578 


42 
5.314 42 


42 


7-06 578 


300 


4.685 57 


57 7 


-16 269 


42 


7.16 269 


360 


6 


57 


57 


.24 187 


42 


42 


.24 188 


420 


7 


57 


57 


.30 882 


42 


42 


.30 882 


480 


8 


57 


57 


.36 681 


42 


42 


.36 681 


540 


9 
10 


57 


57 


■ 41 797 


42 


42 


• 41 797 


600 


4-685 57 


57 7 


.46 372 


5.314 42 


42 


7-46 372 


660 


11 


57 


57 


.50 512 


42 


42 


.50 512 


720 


12 


57 


57 


.54 290 


42 


42 


.54 291 


780 


13 


57 


57 


.57 767 


42 


42 


-57 767 


840 


14 


57 


57 


-60 985 


42 


42 


-60 985 


900 


15 


4.685 57 


58 7 


-63 981 


5.314 42 


42 


7.63 982 


960 


16 


57 


58 


.66 784 


42 


42 


.66 785 


1020 


17 


57 


58 


.69 417 


45 


42 


.69 418 


1080 


18 


57 


58 


.71 899 


42 


42 


.71900 


1140 


19 
20 


57 


58 


- 74 248 


42 


42 


. 74 248 


1200 


4.685 57 


58 7 


-76 475 


5.314 43 


42 


7.76 476 


1260 


21 


57 


58 


.78 594 


43 


42 


• 78 595 


1320 


22 


57 


58 


.80 614 


43 


42 


.80 615 


1380 


23 


57 


58 


.82 545 


43 


42 


.82 546 


1440 


24 
25 


57 


58 


-84 393 


43 


42 
41 


• 84 394 


1500 


4.685 57 


58 7 


-86 166 


5.314 43 


7-86 167 


1560 


26 


57 


58 


.87 869 


43 


41 


.87 871 


1620 


27 


57 


58 


.89 508 


43 


41 


• 89 510 


1680 


28 


57 


58 


.91088 


43 


4l 


-91089 


1740 


29 


57 


58 


92 612 


43 


41 


-92 613 


1800 


30 


4.685 57 


58 7 


94 084 


5.314 43 


4" 


7-94 086 


1860 


31 


57 


58 


95 508 


43 


4" 


-95 510 


1920 


32 


57 


58 


96 887 


43 


4 


.96 889 


1980 


33 


57 


59 


98 223 


43 


41 


• 98 225 


2040 


34 


57 


59 


99 520 


43 


41 
41 


-99 522 


2100 


35 


4.685 56 


59 8- 


00 778 


5.314 43 


8-00 78 


2160 


36 


56 


59 


02 002 


43 


41 


• 02 00-; 


2220 


37 


56 


59 


03 192 


43 


41 


• 03 194 


2280 


38 


56 


59 


04 350 


43 


40 


-04 352 


2340 


39 


56 


59 


05 478 


43 


40 


05 481 


2400 


40 


4.685 56 


59 8. 


06 577 


5.314 43 


40 


806 580 


2460 


41 


56 


59 


07 650 


43 


40 


-07 653 


2520 


42 


56 


59 


08 696 


43 


40 


-08 699 


2580 


43 


56 


60 


09 718 


43 


40 


-09 721 


2640 


44 


56 


60 


10 716 


43 


40 


-10 720 


2700 


45 


4-685 56 


60 8. 


11 692 


5.314 44 


40 


8-11 696 


2760 


46 


56 


60 


12 647 


44 


40 


.12 651 


2820 


47 


56 


60 


13 581 


44 


40 


.13 585 


2880 


48 


56 


60 


14 495 


44 


39 


• 14 499 


2940 


49 


56 


60 


15 390 


44 


39 


• 15 395 


3000 


50 


4.685 56 


60 8- 


16 268 


5.314 44 


39 


8-16 272 


3060 


51 


56 


60 


17 128 


44 


39 


.17 133 


3120 


52 


56 


61 


17 971 


44 


39 


.17 976 


3180 


53 


56 


61 


18 798 


44 


39 


.18 80c: 


3240 


54 


55 


61 


19 610 


44 


39 


.19 615 


3300 


55 


4.685 55 


61 8- 


20 407 


5.314 44 


39 


8^20 412 


3360 


56 


55 


6:, 


21 189 


44 


38 


.21 195 


8420 


57 


55 


61 


21958 


44 


38 


.21 964 


3480 


58 


55 


6:. 


22 713 


44 


38 


.22 719 


3_540 


59 


55 


62 


23 455 


44 


38 


.23 462 



644 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin (f) = log <f>'' + S. 




r 


log0' 


' = log 


sin d> + S\ 


Log tan (f> = log 0" + T. 




log<;6' 


= log tan + T\ 


ff 


/ 


S 


T 


Logf. Sin. 


S' 


T' 


Log. Tan. 


3600 





4.685 55 


62 


8-24 185 


5.314 44 


38 


8.24 192 


3660 


1 


55 


62 


.24 903 


45 


38 


-24 910 


3720 


2 


55 


62 


.25 609 


45 


38 


-25 616 


3780 


3 


55 


62 


.26 304 


45 


37 


-26 311 


3840 


4 


55 


62 


-26 988 


45 


37 


-26 995 


3900 


5 


4.685 55 


62 


8-27 661 


5-31445 


37 


. 8-27 669 


3960 


6 


55 


63 


.28 324 


45 


37 


-28 332 


4020 


7 


54 


63 


.28 977 


45 


37 


-28 985 


4080 


8 


54 


63 


.29 620 


45 


37 


.29 629 


4140 


9 


54 


63 


-30 254 


45 


36 


-30 263 


4200 


10 


4.685 54 


63 


8-30 879 


5-31445 


36 


8. 30 888 


4260 


11 


54 


63 


-31 495 


45 


36 


.31 504 


4320 


12 


54 


64 


.32 102 


45 


36 


.32 112 


4380 


13 


54 


64 


• 32 701 


46 


36 


-32 711 


4:440 


14 


54 


64 


-33 292 


46 


36 
35 


.33 302 


4500 


15 


4-685 54 


64 


8-33 875 


5-314 46 


8. 33 885 


4560 


16 


54 


64 


-34 450 


46 


35 


.34 461 


4620 


17 


54 


65 


.35 018 


46 


35 


.35029 


4680 


18 


54 


65 


.35 578 


46 


35 


.35 589 


4740 


19 


53 


65 


-36 131 


46 


35 


• 36 143 


4800 


20 


4-685 53 


65 


8-36 677 


5-314 46 


34 


8-36 689 


4860 


21 


53 


65 


.37 217 


46 


34 


.37 229 


4920 


22 


53 


65 


.37 750 


46 


34 


.37 762 


4980 


23 


53 


66 


.38 276 


46 


34 


.38 289 


5040 


24 
25 


53 


66 


-38 796 


47 


34 
33 


.38 809 


5100 


4-685 53 


66 


8-39 310 


5-314 47 


8-39 323 


5160 


26 


53 


66 


.39 818 


47 


33 


.39 831 


5220 


27 


53 


67 


.40 320 


47 


33 


.40 334 


5280 


28 


52 


67 


.40 816 


47 


33 


.40 830 


5340 


29 


52 


67 


-41 307 


47 


33 


-41 32l 


5400 


30 


4-685 52 


6Z 


8-41 792 


5-31447 


32 


8-41 807 


5460 


31 


52 


67 


.42 271 


47 


32 


.42 287 


5520 


32 


52 


68 


.42 746 


47 


32 


.42 762 


5580 


33 


52 


68 


.43 215 


48 


32 


.43 231 


5640 


34 


52 


68 
68 


.43 680 


48 


31 


-43 696 


5700 


35 


4.685 52 


8-44 139 


5-31448 


31 


8-44 156 


5760 


36 


52 


69 


• 44 594 


48 


31 


.44 611 


5820 


37 


51 


69 


.45 044 


48 


31 


.45 061 


5880 


38 


51 


69 


.45 489 


48 


3';' 


.45 507 


5940 


39 


51 


69 


-45 930 


48 


30 


-45 948 


6000 


40 


4.685 51 


69 


8-46 366 


5.314 48 


30 


846 385 


6060 


41 


51 


70 


-46 798 


49 


30 


.46 817 


6120 


42 


51 


70 


.47 226 


49 


30 


.47 245 


6180 


43 


51 


70 


-47 650 


49 


v29 


.47 669 


6240 


44 


51 


70 


.48 069 


49 


29 


.48 089 


6300 


45 


4-685 50 


71 


8-48 485 


5-314 49 


29 


8-48 505 


6360 


46 


50 


71 


-48 896 


49 




.48 917 


6420 


47 


50 


71 


-49 304 


49 


28 


.49 325 


6480 


48 


50 


72 


-49 708 


49 


28 


.49 729 


6540 


49 


50 


72 


• 50 108 


50 


28 

27 


-50 130 


6600 


50 


4.685 50 


72 


8-50 504 


5-314 50 


■ 8-50 526 


6660 


51 


50 


72 


.50 897 


50 


27 


.50 920 


6720 


52 


50 


73 


.51 286 


50 


27 


.51 310 


6780 


53 


49 


73 


-51 672 


50 


27 


.51 696 


6840 


54 


49 


73 


.52 055 


50 
5-314 50 


26 


.52 079 


6900 


55 


4.685 49 


73 


8-52 434 


26 


8.52 458 


6960 


56 


49 


74 


-52 810 


51 


26 


.52 835 


7020 


57 


49 


74 


-53 183 


51 


25 


.53 208 


7080 


58 


49 


74 


.53 552 


51 


25 


.53 578 


7140 


59 


49 


75 


-53 918 


51 


25 


.53 94i 



645 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES 



Log sin (f) = log <^'' -f S. 




2° 


\o^4>' 


' = log 


ain 6 4- S\ 


Log tan (f> = log 9!)" + T. 




logos' 


= log tan <f> -¥ T\ 


tf 


/ 


S 


T 


Log. Sin. 


S' 


T' 


Log. Tan. 


7200 





4.685 48 


75 


8-54 282 


5.314 5l 


25 


8-54 308 


7260 


1 


48 


75 


.54 642 


51 


24 


.54 669 


7320 


2 


48 


75 


.54 999 


51 


24 


.55 027 


7380 


3 


48 


76 


.55354 


52 


24 


.55 381 


7440 


4 
5 


48 


76 


.55 705 


52 


23 


.55 733 


7500 


4.686 48 


76 


8- 56 054 


5.314 52 


23 


8.56 083 


7560 


6 


48 


77 


.56 400 


52 


23 


.56 429 


7620 


7 


47 


77 


.56 743 


52 


22 


.56 772 


7680 


8 


47 


77 


.57 083 


52 


22 


.57 113 


7740 


9 


47 


78 


-57 421 


52 


22 


.57 452 


7800 


10 


4-685 47 


78 


8-57 756 


5.314 53 


22 


8.57 787 


7860 


11 


47 


78 


.58 089 


53 


21 


.58 121 


7920 


12 


47 


79 


.58 419 


53 


21 


.58 451 


7980 


13 


46 


79 


.58 747 


53 


21 


.58 779 


8040 


14 


46 


79 
80 


.59 072 


53 


20 


.59 105 


8100 


15 


4.685 46 


8.59395 


5.314 53 


20 


8-59 428 


8160 


16 


46 


80 


.59 715 


54 


20 


.59 749 


8220 


17 


46 


80 


.60 033 


54 


19 


.60 067 


8280 


18 


46 


81 


.60 349 


54 


19 


.60 384 


8340 


19 


45 


81 


.60 662 


54 


19 


-60 698 


8400 


30 


4.685 45 


81 


8.60 973 


5.314 54 


18 


8-61009 


8460 


21 


45 


82 


.61 282 


54 


18 


.61319 


8520 


22 


45 


82 


.61 589 


55 


18 


.61 626 


8580 


23 


45 


82 


.61893 


55 


17 


.61931 


8640 


24 


45 


83 


.62 196 


55 


17 


.62 234 


8700 


25 


4.685 44 


83 


8.62 496 


5-314 55 


16 


8.62 535 


8760 


26 


44 


83 


.62 795 


55 


16 


.62 834 


8820 


27 


44 


84 


.63 091 


55 


16 


.63 131 


8880 


28 


44 


84 


.63 385 


56 


15 


.63 425 


8940 


29 


44 


84 


.63 677 


56 


15 


-63 718 


9000 


30 


4.685 43 


85 


8.63 968 


5.314 56 


15 


8-64 009 


9060 


31 


43 


85 


.64 256 


56 


14 


.64 298 


9120 


32 


43 


86 


.64 543 


56 


14 


.64 585 


9180 


33 


43 


86 


.64 827 


57 


14 


.64 870 


9240 


34 


43 


86 


.65 110 


57 


13 


-65 153 


9303 


35 


4.685 43 


87 


8-65 391 


5.314 57 


13 


8.65 435 


9360 


36 


42 


87 


.65 670 


57 


12 


.65 715 


9420 


37 


42 


87 


.65 947 


57 


12 


.65 993 


9480 


38 


42 


88 


.66 223 


58 


12 


.66 269 


9540 


39 


42 


88 


-66 497 


58 


ll 


-66 543 


9600 


40 


4.685 42 


89 


8.66 769 


5.314 58 


11 


8.66 816 


9660 


41 


41 


89 


.67 039 


58 




.67 087 


9720 


42 


41 


89 


.67 308 


58 


10 


.67 356 


9780 


43 


41 


90 


.67 575 


59 


10 


.67 624 


9840 


44 


41 


90 


-67 840 


59 


09 


-67 890 


9900 


45 


4.685 41 


91 


8-68 104 


5-314 59 


09 


8.68 154 


9960 


46 


40 


91 


.68 366 


59 


08 


.68 417 


10020 


47 


40 


91 


.68 627 


59 


08 


.68 678 


10080 


48 


40 


92 


.68 886 


60 


08 


.68 938 


10140 


49 


40 


92 


-69 144 


60 
5.314 60 


07 


-69 196 


10200 


50 


4.685 40 


93 


8-69 400 


07 


8.69 453 


10260 


51 


39 


93 


.69 654 


60 


06 


.69 708 


10320 


52 


39 


93 


.69 907 


60 


06 


.69 961 


10380 


53 


39 


94 


.70 159 


61 


06 


.70 214 


10440 


54 
55 


39 


94 


- 70 409 


61 


05 


.70 464 


10500 


4.685 38 


95 


8.70 657 


5.314 61 


05 


8-70 714 


10560 


56 


38 


95 


.70 905 


61 


04 


.70 962 


10620 


57 


38 


96 


.71 150 


6l 


04 


.71208 


10680 


58 


38 


96 


.71 395 


62 


03 


.71453 


10740 


59 


38 


97 


-71638 


62 


03 


.71697 



646 





TABLE VII.— LOGARITHMIC 


SINES, COSINES, TANGENTS, 




0** 






AND COTANGENTS. 




179^ 


/ 


Log. Sin. 


D 


log. Tan. 


Com. D. 


Log. Cot. 1 


Log. Cos. 






1 

2 
3 

4 

5 

8 
9 

10 

11 
12 
13 
14 

15 

1 16 

17 

,18 

'19 


6 
6 
6 
7 


— 00 

46 372 
76 475 
94 084 
06 578 


30103 
17609 
12494 

9691 
7918 
6695 
5799 
5115 

4575 
4139 
3778 
3476 
3218 

2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 

1772 
1703 
1639 
1579 
1524 

1472 
1424 
1379 
1336 
1296 

1258 
1223 
1190 
1158 
1128 

1099 

1072 

1046 

1022 

998 

976 

954 

934 

914 

895 

877 
860 
843 
827 
811 
797 
782 
768 
755 
742 

730 


6 
6 
6 
7 


— CX) 

46 372 
76 475 
94 084 
06 578 


30103 
17609 
12494 

9691 
7918 
6694 
5799 
5115 

4575 
4139 
3779 
3476 
3218 

2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 

1773 
1703 
163& 
1579 
1524 

1472 
1424 
1379 
1336 
1296 

1259 
1223 
1190 
1158 
1128 

1099 
1072 
1046 
1022 
999 

976 
954 
934 
914 
895 

877 
860 
843 
827 
812 

797 
783 
768 
755 
742 
730 


3 
3 
3 

9 


4-00 

53 627 

23 524 

.05 915 

93 421 









00 000 
00 000 
00 000 
00 000 
00 000 


60 

59 
58 
57 
5b 


7 
7 
7 
7 
7 


16 269 
24 187 
30 882 
36 681 
41 797 


7 
7 
7 
7 
7 


16 269 
24 188 
30 882 
36 681 
41 797 


2 
2 
2 
2 
2 


.83 730 
.75 812 
.69 117 
.63 318 
.58 203 









00 000 
00 000 
00 000 
00 000 
00 000 


55 
54 
53 
52 
51 


7 
7 
7 
7 
7 


46 372 
50 512 
54 290 
57 767 
60 985 


7 
7 
7 
7 
7 


46 372 
50 512 
54 291 
57 767 
60 985 


2 
2 
2 
2 
2 


.53 627 
.49 488 
.45 709 
.42 233 
.39 014 




9 
9 
9 


00 000 
00 000 
99 999 
99 999 
99 999 


50 

49 
48 
47 
46 


7 
7 
7 
7 
7 


63 981 
66 784 
69 417 
71 899 
74 248 


7 
7 
7 
7 
7 


63 982 
66 785 
69 418 
71 900 
74 248 


2 
2 
2 
2 
2 


• 36 018 

• 33 215 
.30 582 
.28 099 

• 25 751 


9 
9 
9 
9 
9 


99 999 
99 999 
99 999 
99 999 
99 999 


45 
44 
43 
42 
41 


30 

21 
,22 
'23 

24 

25 
26 
127 
28 
29 

30 

31 
32 
33 
34 


7 
7 
7 
7 
7 


76 475 
78 594 
80 614 
82 545 
84 393 


7 
7 
7 
7 
7 


• 76 476 
78 595 
.80 615 
.82 546 
.84 394 


2 
2 
2 
2 
2 


.23 524 
.21405 
.19 384 
.17 454 
.15 605 


9 
9 
9 
9 
9 


99 999 
99 999 
99 999 
99 999 
99 999 


40 

39 
38 
37 
36 


7 
7 
7 
7 
7 


86 166 

87 869 
89 508 

91 088 

92 612 


7 
7 
7 
7 
7 


• 86 167 
87 871 

• 89 510 

91 089 

92 613 


2 
2 
2 
2 

2 


.13 832 
.12 129 

• 10 490 

• 08 910 
.07 386 


9 
9 
9 
9 
9 


99 999 
99 999 
99 998 
99 998 
99 998 


35 
34 
33 
32 
31 


7 
7 
7 
7 
7 


94 084 

95 508 

96 887 

98 223 

99 520 


7 
7 
7 
7 
7 


.94 086 

.95 510 

96 889 

.98 225 

.99 522 


2 
2 
2 
2 
2 


.05 914 
.04 490 
.03 111 
.01 774 
.00 478 


9 
9 
9 
9 
9 


99 998 
99 998 
99 998 
99 998 
• 99 998 


30 

29 
28 
27 
26 


35 
,36 
'37 

38 
1 39 


8 
8 
8 
8 
8 


00 778 

02 002 

03 192 

04 350 
on 478 


8 
8 
8 
8 
8 


00 781 
.02 004 
.03 194 
.04 352 
.05 481 




.99 219 
.97 995 

• 96 805 

• 95 647 

• 94 519 


9 
9 
9 
9 
9 


• 99 997 
99 997 
99 997 
99 997 
99 997 


25 
24 
23 
22 
21, 


40 

41 

42 

43 

1 44 


8 
8 
8 
8 
8 


06 577 

07 650 

08 696 

09 718 

10 716 


8 
8 
8 
8 
8 


06 580 

07 653 

08 699 

09 72T 

10 720 




• 93 419 

• 92 347 

• 91 300 

• 90 278 
•89 279 


9 
9 
9 
9 
9 


99 997 
99 997 
99 997 
99 996 
99 996 


30 

19 
18 
17 
16 


. 45 

1 46 

47 

48 

49 


8 
8 
8 
8 
8 


11 692 

12 647 

13 581 

14 495 

15 390 


8 
8 
8 
8 
8 


11 696 

12 651 

13 585 

14 499 

15 395 




• 88 303 

• 87 349 

• 86 415 

• 85 500 

• 84 605 


9 
9 
9 
9 
9 


99 996 
99 996 
99 996 
99 996 
99 995 


15 
14 
13 
12 
11 


^ 50 

51 
52 
53 
54 


8 
8 
8 
8 
8 


16 268 

17 128 

17 971 

18 798 

19 610 


8 
8 
8 
8 
8 


16 272 

17 133 

17 976 

18 803 

19 615 




•83 727 

• 82 867 
-82 023 

• 81 196 
80 384 


9 
9 
9 
9 
9 


99 995 
99 995 
99 995 
99 995 
99 994 


10 

9 
8 
7 
6 


55 

56 

57 

, 58 

^ 59 


8 
8 
8 
8 
8 


20 407 

21 189 

21 958 

22 713 

9.^ 455 


8 

8 

8 

8. 

8 


20 412 

21 195 

21 964 

22 719 

23 462 




79 587 
78 804 
78 036 
77 280 
76 538 


9 
9 
9 
9 
9 


99 994 
99 994 
99 994 
99 994 
99 993 


5 
4 
3 

2 

1 


60 


8 


24 185 


8. 


24 192 




75 808 


9 


99 993 





i 


Log. Cos. 


D 


Log. Cot. 


Com. D. 


Log. Tan. | 


Log. Sin. 


/ 



647 



89** 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



178' 



log. Sin. 



8.24 185 

8.24 903 

8.25 609 

8.26 304 
8-26 988 



8.27 661 

8.28 324 

8.28 977 

8.29 620 
8-30 254 



30 879 

31 495 

32 102 

32 701 

33 292 



8.33 875 
8-34 450 
8. 35 018 

8.35 578 

8.36 131 



36 677 

37 217 

37 750 

38 276 
38 796 



8.39 310 

8.39 818 

8.40 320 

8.40 816 

8.41 307 



8.41792 
8.42 271 

8.42 746 

8.43 215 
8-43 680 



8.44 139 

8.44 594 

8.45 044 
8.45 489 
8.45 930 



8.46 366 

8.46 798 

8.47 226 
8.47 650 
8-48 069 



8.48 485 

8.48 896 

8.49 304 
8.49 708 
8-50 108 



8.50 504 
8-50 897 

8.51 286 

8.51 672 

8.52 055 



8.52 434 
8-52 810 

8.53 183 
8.53 552 
8. 53 918 



60 



91* 



8.54 282 



Log. Cos. 



718 
706 
694 
684 

673 
663 
653 
643 
634 

625 
616 
607 
599 
591 

583 
575 
567 
560 
553 

546 
539 
533 
526 
520 

514 
508 
502 
496 
491 

485 
479 
474 
469 
464 

459 
454 
450 
445 
440 

436 
432 
428 
423 
419 
415 
411 
407 
404 
400 

396 
393 
389 
386 
382 

379 
375 
373 
369 
366 

363 



Log. Tan. 



8.24 192 
8-24 910 

8.25 616 

8.26 311 
8.26 995 



8.27 669 

8.28 332 

8.28 985 

8.29 629 
8-30 263 



8. 30 888 

8.31 504 
8-32 112 
8-32 711 
8. 33 302 



8. 33 885 

8.34 461 

8.35 029 
8-35 589 

8. 36 143 



8-36 689 
8-37 229 
8-37 762 
8-38 289 
8-38 809 



8-39 323 

8.39 831 

8.40 334 
8.40 830 
8.41321 



8.41807 
8.42 287 

8.42 762 

8.43 23l 
8-43 696 



8-44 156 
8.44 611 
8-45 061 
8-45 507 
8-45 948 



8-46 385 
8.46 817 
8-47 245 
8-47 669 
8. 48 089 



8-48 505 
8-48 917 
8-49 325 
8-49 729 
8. 50 130 



8-50 P16 

8.50 20 

8.51 310 
8.51 696 
8-52 079 



8-52 458 
8-52 835 
8.53 208 
8.53 578 
8. 53 944 



8-54 308 



Log. Cot. 



Com, D. 



718 
706 
695 
684 

673 
663 
653 
643 
634 

625 
616 
607 
599 
591 
58Q 
575 
568 
560 
553 

546 
539 
533 
527 
520 

514 
508 
502 
496 
491 

485 
480 
475 
469 
464 
460 
455 
450 
445 
441 

437 
432 
428 
424 
419 

416 
412 
408 
404 
400 
396 
393 
390 
386 
383 
379 
376 
373 
370 
366 
364 



Log. Cot. 



1.75 808 
1.75 090 
1.74 383 
1.73 688 
1-73 004 



1.72 331 
1.71 667 
1.71014 
1.70 371 
1.69 736 



1.69 111 
1.68 495 
1.67 888 
1.67 288 
1.66 697 



1.66 114 
1.65 539 
1.64 971 
1.64 410 
1-63 857 



1.63 310 
1.62 771 
1.62 238 
1.61 711 
1-61 191 



1.60 676 
1-60 168 
1.59 666 
1.59 169 
1-58 678 



Com. D. 



1-58 193 
1.57 713 
1.57 238 
1.56 768 
1-56 304 



1.55 844 
1.55 389 
1.54 93§ 
1.54 493 
1-54 052 



1-53 615 
1.53 183 
1.52 754 
1.52 330 
1-51 911 



1.51495 
1.51083 
1.50 675 
1.50 270 
1-49 870 



1.49 473 
1.49 080 
1.48 690 
1.48 304 
1-47 921 



1.47 541 
1.47 165 
1.46 792 
1.46 422 
1-46 055 



1.45691 



Log. Tan. 



Log. Cos. 



9.99 993 
9.99 993 
9.99 993 
9.99 992 
9. 99 992 



9-99 992 
9-99 992 
9-99 992 
9-99 991 
9.99 991 



9.99 991 
9.99 990 
9.99 990 
9.99 990 
9-99 990 



9-99 989 
9.99 989 
9.99 989 
9.99 989 
9-99 988 



9.99 988 
9.99 988 
9.99 987 
9.99 987 
9.99 987 



9.99 986 
9.99 986 
9.99 986 
9.99 886 
9.99 985 



9-99 985 
9.99 985 
9.99 984 
9.99 984 
9-99 984 



9.99 983 
9.99 983 
9.99 982 
9.99 982 
9.99 982 



9.99 981 
9.99 981 
9.99 981 
9.99 980 
9. 99 980 



999 979 
9.99 979 
9.99 97? 
9.99 978 
9. 99 978 



9.99 978 
9-99 977 
9.99 977 
9.99 976 
9-99 976 



9-99 975 
9-99 975 
9-99 975 
9-99 974 
9. 99 974 



9. 99 973 



Log. Sin. 



648 



88' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



Log. Sin. 

8-54 282 
8.54 642 

8.54 999 

8.55 354 
8.55 705 



8.56 054 
8.56 400 

8.56 743 

8.57 083 
8.57 421 



8.57 756 

8.58 039 
8.58 419 

8.58 747 

8.59 072 



8.59 395 

8.59 715 

8.60 033 
8.60 349 
8.60 662 



8.60 973 

8.61 282 
8.61 589 

8.61 893 

8.62 196 



8.62 496 

8.62 795 

8.63 091 
8.63 385 
8.63 677 



8.63 968 

8.64 256 
8.64 543 

8.64 827 

8.65 110 



8.65 391 
8.65 670 

8.65 947 

8.66 223 
8.66 497_ 



8.66 769 

8.67 039 
8.67 308 
8- 67 575 
8.67 840 



8.68 104 
8.68 366 
8.68 627 

8.68 886 

8.69 144 . 
8.69 400 
8.69 654 

8.69 907 

8.70 159 
8 . 70 409 



8.70 657 

8.70 905 

8.71 150 
8.71 395 
8-71 638 



8-71 880 



Log. Cos. 



360 
357 
354 
35l 

348 
346 
343 
340 
338 

335 
332 
330 
327 
325 

323 
320 
318 
316 
313 

311 
309 
306 
304 
302 

300 
298 
296 
294 
292 

290 
288 
286 
284 
282 

281 
279 
277 
275 
274 

272 
270 
268 
267 
265 

264 
262 
260 
259 
257 

256 
254 
253 
25l 
250 

248 
247 
245 
244 
243 
241 



Log. Tan. 



8.54 308 

8.54 669 

8.55 027 
8.55 381 
8.55 733 



8.56 083 
8.56 429 

8.56 772 

8.57 113 
8.57 452 



8.57 787 

8.58 121 
8.58 451 
8.58 779 
8-59 105 



8.59 428 

8.59 749 

8.60 067 
8-60 384 
8.60 698 



8.61 009 
8.61 319 
8.61 626 
8.61 931 
8-62 234 



8-62 535 

8.62 834 

8. 63 131 
8.63 425 
8.63 718 



8.64 009 
8.64 298 
8.64 585 

8.64 870 

8.65 153 



8. 65 435 
8.65 715 
8. 65 993 
8-66 269 
8-66 543 



8.66 816 

8.67 087 
8-67 356 
8.67 624 
8. 67 890 



8. 68 154 
8.68 417 
8. 68 678 

8.68 938 

8.69 196 



Com. D. 



8-69 453 

8.69 708 
8-69 96l 

8.70 214 
8 . 70 464 



8.70 714 

8.70 962 

8.71 208 
8.71 453 
8.71 697 



8-71 939 



Log. Cot, 



360 
358 
354 
352 

349 
346 
343 
341 
338 

335 
333 
330 
328 
325 
323 
320 
318 
316 
314 

311 
309 
307 
305 
303 

300 
299 
297 
294 
293 

291 
288 
287 
285 
283 

28l 
280 
278 
276 
274 

272 
271 
269 
267 
266 

264 
262 
26T 
259 
258 

256 
255 
253 
252 
250 

249 
248 
246 
245 
243 
242 



Log. Cot. 



Com. D. 



1.45 691 
1.45 331 
1.44 973 
1.44 618 
1.44 266 



1.43 917 
1.43 571 
1.43 227 
1.42 886 
1.42 548 



Log. Cos. 



1.42 212 
1.41 879 
1.41 548 
1.41 220 
1.40 895 



1.40 571 
1.40 251 
1.39 932 
1.39 616 
1.39 302 



1.38 990 
1.38 681 
1.38 374 
1.38 068 
1.37 765 



1.37 465 
1.37 166 
1.36 869 
1.36 574 
1.36 28l 



1.35 990 
1.35 702 
1.35 414 
1.35 129 
1.34 846 



1.34 565 
1.34 285 
1.34 007 
1.33 731 
1.33 456 



1.33 184 
1.32 913 
1.32 643 
1.32 376 
1.32 110 



31 845 
31 583 
31 32l 
31 062 
30 803 



1.30 547 
1.30 292 
1.30 038 
1.29 786 
1.29 535 



1.29 286 
1.29 038 
1.28 791 
1.28 546 
1.28 303 



1.28 060 



Log. Tan. 



9.99 973 
9.99 973 
9.99 972 
9.99 972 
9.99 97l 



9.99 971 
9.99 971 
9.99 970 
9.99 970 
9.99 969 



9.99 969 
9.99 968 
9.99 968 
9.99 967 
9.99 967 



9.99 966 
9.99 966 
9.99 965 
9.99 965 
9.99 964 



9.99 964 
9.99 963 
9.99 963 
9.99 962 
9.99 962 



9.99 961 
9.99 961 
9.99 960 
9.99 959 
9.99 959 



9.99 958 
9.99 958 
9.99 957 
9.99957 
9.99 956 

9.99 956 
9.99955 
9.99954 
9.99 954 
9.99 953 



99 953 
99 952 
99 952 
99 951 
99 950 



9.99 950 
9.99 949 
9.99 948 
9.99 948 
9.99 947 



9.99 947 
9.99 946 
9.99 945 
9.99 945 
9.99 944 



99 943 
99 943 
99 942 
99 942 
99 941 



9.99 940 



Log. Sin. 



649 



87* 



TABLE VIL— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



176^ 



Log. Sin, 



71 
72 
72 
72 
72 



880 
120 
359 
597 
833 

069 
302 
535 
766 
997 

^26 
453 
680 
905 
129 



75 
75 
75 
76 
76 



353 
574 
795 
015 
233 



76 
76 
76 
11 
11 



451 
667 
883 
097 
310 



77 
77 
77 
78 
78 



522 

733 

943 

152 

360 
^.207 
206 
205 
204 



240 
239 
237 
236 
235 
233 
233 
231 
230 

229 

227 
226 
225 
224 

223 
221 
221 
219 
218 

217 
216 
215 
214 
213 
212 
211 
210 
209 
208 



567 
773 
978 
183 
386 



79 
79 
79 
80 
80 



203 
202 



78§201 



189 
387 



80 
80 
80 
81 
81 



585 
782 
977 
172 
366 



81 
81 
81 
82 
82 



560 
752 
943 
134 

324 



83 
83 
83 
83 
84 



84 



513 
701 
888 
075 
260 

445 
629 
813 
995 
177 
358 



Log. Cos 



198 

197 
197 
195 
195 
194 

193 
192 
191 
191 
189 
189 
188 
187 
186 
185 

185 
184 
183 
182 
182 

181 



Log. Tan. 



8.71 
8-72 
8.72 
8.72 
8.72 



8.73 
8.73 
8. 73 
8.73 
8.74 



8.74 
8.74 
8.74 
8.74 
8.75 



8.75 
8. 75 
8-75 
8.76 
8.76 



8.76 
8.76 
8.76 
8.77 
8.77 



8.77 
8.77 
8. 78 
8.78 
8. 78 



8.78 
8.78 
8.79 
8-79 
8.79 



8.79 
8.79 
8.80 
8.80 
8.80 



8-80 
8.80 
8.81 
8.81 
8.81 



8.81 

8. 81 

8. 82 
8. 82 
8 82 

8.82 
82 
8.82 
8.83 

8. 83 



8.83 
8. 83 

8. 83 

8. 84 
8. 84 



8. 84 



Log. 



939 
180 
420 
659 
896 

131 
366 
599 
831 
062 

292 
520 
748 
974 
199 
422 
645 
867 
087 
306 

524 
74l 
958 
172 
386 
599 
8ll 
022 
232 
441 

648 
855 
061 
266 
470 

673 
875 
076 
276 
476 

674 
871 
068 
264 
459 

653 
846 
038 
230 
420 

610 
799 
987 
175 
361 

547 
732 
916 
100 
282 
464 
Cot 



241 
240 
238 
237 

235 
235 
233 
232 
231 
229 
228 
227 
226 
225 

223 
223 
221 
220 
219 

218 
217 
216 
214 
214 

213 
212 
210 
210 
209 

207 
207 
206 
204 
204 

203 
202 
201 
200 
199 
198 
197 
197 
195 
195 

194 
193 
192 
191, 
190 

190 
188 
188 
187 
186 

185 
185 
184 
183 
182 

182 



Log. Cot, 



28 060 
27 819 
27 579 
27 341 
27 104 



26 868 
26 633 
26 400 
26 168 
25J37 
25 708 
25 479 
25 252 
25 026 
24 801 

24 577 
24 354 
24 133 
23 913 
23 693 



23 475 
23 258 
23 042 
22 827 
22 613 



22 400 
22 188 
21 978 
21 768 
21 559 



21351 
21 144 
20 938 
20 734 
20 530 



20 327 
20 125 
19 923 
19 723 
19 524 



19 326 
19 128 
18 931 
18 736 
18 541 



18 347 
18 154 
17 961 
17 770 
17 579 



17 389 
17 201 
17 012 
16 825 
16 638 



16 453 
16 268 
16 083 
15 900 
15 717 



15 535 



Lo^:, Tan. 



Log. Cos, 



99 940 
99 940 
99 939 
99 938 
99 938 



99 937 
99 936 
99 935 
99 935 
99 934 



99 933 
99 933 
99 932 
99 931 
99 931 



99 930 
99 929 
99 928 
99 928 
99 927 



99 926 
99 925 
99 925 
99 924 
99 923 



99 922 
99 922 
99 921 
99 920 
99 919 



99 919 
99 918 
99 917 
99 9] 6 
99 916 



99 915 
99 914 
99 913 
99 912 
99 912 



99 911 
99 910 
99 909 
99 908 
99 907 



99 907 
99 906 
99 905 
99 904 
99 903 
99 902 
99 902 
99 901 
99 900 
99 899 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 

42 
41, 

40 

39 
38 
37 
36 



35 
34 
33 
32 
31 
30 
29 
28 
27 
26 



30 

19 
18 
17 
16 



10 

9 
8 
7 
6 



99 898 
99 897 
99 896 
99 896 
99 895 



99 894 
Log. Sin 



P. P. 





330 


330 


310 


6 


33.0 


32-0 


31-0 


7 


38.5 


37.3 


36.1 


8 


44.0 


42.6 


41.3 


9 


49.5 


48.0 


46.5 


10 


55.0 


53.3 


51.6 


20 


110.0 


106.6 


103.3 


30 


165.0 


160.0 


155.0 


40 


220.0 


213.3 


206.6 


50 


275.0 


266.6 


258.3 





290 


280 


270 


6 


29.0 


28.0 


27.0 


7 


33.8 


32.6 


31.5 


8 


38.6 


37.3 


36.0 


9 


43.5 


42.0 


40.5 


10 


48.3 


46.6 


45.0 


20 


96.6 


93.3 


90.0 


30 


145.0 


140.0 


135.0 


40 


193.3 


186.6 


180.0 


50 


241.6 233.3 


225.0 



250 

25.0 

29.1 

33.3 

37.5 

41.6 

83.3 

125.0 

166.6 

208.3 

210 

21.0 

24.5 

28.0 

31.5 

35.0 

70.0 

105.0 

140.0 

175.0 



240 

24.0 

28.0 

32.0 

36.0 

40.0 

80.0 

120.0 

160.0 

200.0 

200 

20.0 

23.3 

26.6 

30.0 

33.3 

66.6 

100.0 

133.3 

166.6 



230 

23.0 

26.8 

30.6 

34.5 

38.3 

76.6 

115.0 

153.3 

191.6 

190 

19.0 
22.1 
25.3 
28.5 
31.6 
63 = 3 
95.0 
126.6 
158.3 



300 

30.0 

35.0 

40.0 

45.0 

50.0 

100.0 

150.0 

200.0 

250.0 

260 

26.0 

30.3 

34. § 

39.0 

43.5 

86.6 

130.0 

173.3 

216.6 

220 

22.0 

25.6 

29.3 

33.0 

36.6 

73.3 

110.0 

146.6 

183.3 



180 

18.0 

21 



24 
27 
30 
60 
90 
120 
150 





9 


9 


8 


7 


6 


6 


0.9 


0.9 


0.8 


0.7 


0.6 


7 


1.1 


1 .0 


0.9 


08 


0.7 


8 


1.2 


1.2 


1 .0 


0.9 


0.8 


9 


1.4 


1.3 


1.2 


1.0 


0.9 


10 


1.6 


1.5 


1.3 


1.1 


1.0 


20 


3.1 


3.0 


2.6 


2.3 


2.0 


30 


4.7 


4.5 


4.0 


3.5 


3.0 


40 


6.3 


6.0 


5.3 


4.6 


4.0 


50 


7.9 


7.5 


6.6 


5.8 


5.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



0.4 
0.5 
0.6 
0.7 
0.7 
1.5 
2.2 
3.0 
3.7 



4 

0.4 
0.4 
0.5 
0.6 
0.6 
1.3 
2.0 
2.6 
33 



3 

0.3 
0.3 
0.4 
0.4 
0.5 
1.0 
1.5 
2.0 
2.5 



2 

0.2 
0.2 



0.2 
03 
0-3 
0.6 
1.0 
1.3 
1.6 



1 

0.1 
0.1 
O.I 
O.I 
0.1 
0.3 
0.5 
0.6 
0.8 



5 

0.5 
0.6 
0.6 
0.7 
0.8 
1.6 
2.5 
3.3 
4.1 

O 

0.0 
0.0 
0.0 
0.1 
0.1 
0.1 
0.2 
0.3 
0.4 



P. P. 



93° 



650 



86' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 

AND COTANGENTS. ' 175* 



Log, Sin. 



84 358 
84 538 
84 718 

84 897 

85 075 



85 252 
85 429 
85 605 
85 780 

85 954 

86 128 
86 301 
86 474 
86 645 
86 816 



86 987 

87 156 
87 325 
87 494 
87 661 



87 828 

87 995 

88 160 
88 326 
88 490 



88 654 
88 817 

88 980 

89 142 
89 303 



89 464 
89 624 
89 784 

89 943 

90 101 



90 259 
90 417 
90 573 
90 729 
90 885 



91 040 
91 195 
91 349 
91 502 
91 655 



91807 

91 959 

92 110 
92 261 
92 411 



92 561 
92 710 

92 858 

93 007 
93 154 



93 301 
93 448 
93 594 
93 740 
93 885 



8-94 029 



Log. Cos, 



d. 

180 
180 
178 
178 

177 
176 
176 
175 
174 

174 
173 
172 
17l 
171 
170 
169 
169 
168 
167 
167 
166 
165 
165 
164 

163 
163 
162 
162 
161 

161 
160 
159 
159 
158 

158 
157 
156 
156 
156 

155 
154 
154 
153 
153 

152 
151 
151 
150 
150 

150 
149 
148 
148 
147 

147 
146 
146 
146 
145 

144 



Log. Tan 



8.84 464 
8.84 645 

8.84 826 

8.85 005 
8.85 184 

8.85 363 
8.85 540 
8.85 717 

8.85 893 

8.86 068 



8.86 243 
8.86 417 
8.86 590 
8.86 763 
8. 86 935 



8.87 106 
8.87 277 
8.87 447 
8.87 616 
8.87 785 



87 953 

88 120 
8.88 287 
8.88 453 
8. 88 618 



88 783 
88 947 
8.89 111 
8.89 274 
8. 89 436 



8.89 598 
8.89 759 

8.89 920 

8.90 080 
8 90 240 

8.90 398 
8.90 557 
8.90 714 
8.90 872 
8-91 028 



8-91 184 
8.91 340 
8.91 495 
8.91 649 
8-91 803 



8.91 957 

8.92 109 
8 92 262 
8.92 413 
8. 92 565 



8.92 715 
92 866 

8-93 015 

8.93 164 
8-93 313 



8.93 461 
8.93 609 
8.93 756 

8.93 903 

8.94 049 



8.94195 



Log. Cot. 



c.d. 

181 
180 
179 
179 
178 
177 
176 
176 
175 
175 
174 
173 
172 
172 

171 
170 
170 
169 
169 

168 
167 
167 
166 
165 

165 
164 
163 
163 
165 

162 
161 
161 
160 
159 

158 
158 
157 
157 
156 

156 
155 
155 
154 
154 

153 
152 
152 
151 
151 

150 
150 
149 
149 
149 

148 
148 
147 
146 
146 
145 



C.d. 



Log. Cot. 



1.15 535 
1.15 354 
1.15 174 
1.14 994 
1.14815 

1.14 637 
1.14 459 
1.14 283 
1.14 107 
1.13 931 



1.13 756 
1.13 582 
1.13 409 
1.13 237 
1.13 065 



1.12 893 
1.12 723 
1.12 553 
1.12 384 
1.12 215 



1.12 047 
1.11 880 
1.11713 
1.11 547 
1.11 38l 



1.11 216 
1.11 052 
1.10 889 
1.10 726 
1.10 563 



1.10 401 
1.10 24C 
1.10 079 
1.09 919 
1.09 760 



1.09 601 
1.09 443 
1.09 285 
1.09 128 
] .08 971 



1.08 815 
1.08 660 
108 505 
1.08 350 
1.08 196 



1.08 043 
1.07 890 
1.07 738 
1.07 586 
1.07 435 



1.07 284 
1.07 134 
1.06 984 
1.06 835 
1.06 686 



1.06 538 
1.06 390 
1.06 243 
1.06 097 
1^05 950 

1.05 805 



Log. Tan, 



Log. Cos, 



99 894 
99 893 
99 892 
99 891 
99 890 



99 889 
99 888 
99 888 
99 887 
99 886 



99 885 
99 884 
99 883 
99 882 
99 881 



99 880 
99 879 
99 878 
99 877 
99 876 



99 875 
99 874 
99 874 
99 873 
99 872 



99 871 
99 870 
99 869 
99 868 
99 867 



99 866 
99 865 
99 864 
99 863 
99 862 



99 861 
99 860 
99P59 
99 858 
99 857 



99 856 
99 855 
99 853 
99 852 
99 85l 



99 850 
99 849 
99 848 
99 847 
99 846 



99 845 
99 844 
99 843 
99 842 
99 841 



99 840 
99 839 
99 837 
99 836 
99 835 



9. 99 834 



Log. Sin. 
651 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21 
20 
19 
18 
17 
16 



10 

9 
8 

7 
6 



O 



P. P. 





181 


180 


178 


6 


18.1 


18.0 


17.8 


y 


21 


1 


21.0 


20.7 


8 


24 


1 


24.0 


23.7 


9 


27 


1 


27.0 


26.7 


10 


30 


1 


30.0 


^.6 


20 


60 


3 


60.0 


59.3 


30 


90 


5 


90.0 


89.0 


40 


120 


5 


120.0 


118.6 


50 


150 


8 


150.0 


148-3 





174 


172 


170 


6 


17.4 


17.2 


17.0 


7 


20.3 


20.0 


19-8 


8 


23.2 


22.9 


22.6 


9 


26.1 


25.8 


25.5 


10 


29.0 


28.6 


28-3 


20 


58.0 


57.3 


56.6 


30 


87.0 


86.0 


85.0 


40 


116.0 


114.6 


113.3 


50 


145-0 


143.3 


141.6 



176 

17.6 
20.5 
23.4 
26-4 
29-3 
58.6 
88. 
117.3 
146-6 

168 

16.8 
19.6 
22.4 
25-2 
28.0 
56.0 
84.0 
112.0 
140.0 





166 


164 


163 


160 


6 


16.6 


16.4 


16.2 


16.0 


7 


19 


3 


19.1 


18.9 


18 


Q 


8 


22 


1 


21.8 


21.6 


21 


3 


9 


24 


9 


24.6 


24-3 


24 





10 


27 


6 


27.3 


27.0 


26 


g 


20 


55 


3 


54.6 


54.0 


53 


3 


30 


83 





82.0 


81.0 


80 





40 


110 


6 


109.3 


108.0 


106 


g 


50 


138 


3 


136.6 


135.0 


133 


3 





158 


156 


154 


152 


6 


15.8 


15.6 


15.4 


15.2 


7 


18 


4 


18.2 


17 


9 


17 


. 7 


8 


21 





20.8 


20 


5 


20 


2 


9 


23 


7 


23.4 


23 


1 


22 


8 


10 


26 


3 


26.0 


25 


6 


25 


3 


20 


52 


Q 


52.0 


51 


3 


50 


g 


30 


79 





78.0 


77 





76 





4C 


105 


3 


104.0 


102 


6 


101 


3 


50 


131 


6 


130.0 


128 


3 


126 


6 





150 


149 


148 


6 


15.0 


14.9 


14.81 


7 


17 


5 


17 


4 


17 


2 


8 


20 





19 


8 


19 


7 


9 


22 


5 


22 


3 


22 


2 


10 


25 





24 


3 


24 


6 


20 


50 





49 


6 


49 


3 


30 


75 





74 


5 


74 





40 


100 





99 


3 


98 


g 


50 


125 





124 


1 


123 


3 





146 


145 


T 


1 


6 


14.6 


14.5 


0.1 


0.1 


7 


17.0 


16-9 


0.2 


0.1 


8 


19.4 


19.3 


0.2 


0.1 


9 


21.9 


21.7 


0.2 


0.1 


10 


24.3 


24.1 


0.2 


0.1 


20 


48.6 


48.3 


0.5 


0.3 


30 


73-0 


72.5 


0.7 


0.5 


40 


97.3 


96.6 


1.0 


0.6 


50 


121.6 


120.8 


1.2 


0.8 



147 

14.7 
17.1 
19. 6 
22.0 
24.5 
49. 
73.5 
98. 
122-5 



O 

0.0 
0.0 
0.0 
0.1 
0.1 
O-I 
0.2 
0.3 
0.4 



P. P. 



85^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
E^ AND COTANGENTS. 17^ 



Logt Sin. 



8-84 
8.94 
8.94 
8.94 
8-94 

894 
8.94 
895 
8.95 
8-95 



029 
174 
317 
430 
603 



8.95 
8.95 
8.95 
8.95 
8.93 



450 
589 
728 
867 

005 



8.98 
8-96 
8.98 
8.98 
8.98 



8.96 
8.96 
8.97 
8.97 
897 



3-98 
8-98 
898 
898 
898 



8-98 
8-98 
8-99 
399 
99 



157 
288 
419 
549 
67g 

803 
937 
066 
194 

322 



3-99 
8-99 
8-99 
8-99 
8-99 



9-00 
9-00 
9-00 
9-00 
9-00 

9-00 
9-00 
9-00 
9-01 
9-01 



449 
577 
703 
830 
958 

081 
207 
332 
456 
580 



9.01 
9.01 
9.01 
9.01 
9.01 

9-01 
[Log- 



704 
828 
951 
073 
196 

318 
440 
56l 
682 
803 



8.94 
8.94 
8.94 
8.94 
8.94 

8-94 
895 
8-95 
8.95 
895 



8.95 
8.95 
8.95 
8.96 
8.96 



8.96 
8.96 
8.96 
8.96 
8.96 



Tan. c.d. Log. Cot. Log. Cos. 



1.05 805 
1.05 659 
1.05 515 
1.05 370 
1.05 226 



1.U5 083 
1.04 940 
1.04 798 
1.04 656 
1.04 514 



1.04 373 



9-99 834 
9.99 833 
9.99 832 
9.99 831 
9.99 830 

9.99 829 
9.99 827 
9.99 826 
9.99 825 
9.9S 824 



9.99 823 



8.99 



9.00 
9.01 
9.01 
9.01 
9.01 



1.04 232 9.99 822 
, -• — 9.99 g21 

9.99 819 
9. 99 8ia 



1.04 092 
1.03 952 
1.03 813 

1.03 674 
1.03 536 
1.03 398 
1.03 260 
1.03 123 



9.99 817 
9.99 816 
9.99 815 
9.99 814 
9.99 813 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



P.P. 





145 


144 


143 


143 


6 


14.5 


14.4 


14.3 


14.2 


7 


16.9 


16.8 


16.7 


16.5 


8 


19-3 


19.2 


19.0 


18.9 


9 


21.7 


21.6 


21.4 


21.3 


10 


24.1 


24.0 


23.8 


23.6 


20 


48.3 


48.0 


47.6 


47.3 


30 


72.5 


72.0 


71.5 


71.0 


40 


96.6 


96.0 


95.3 


94-6 


50 


120.8 


120.0 


119.1 


118.31 



1.02 98619.99 811 
1.02 850 
1.02 714 
1.02 579 
1.02 444 



1.02 309 
1.02 175 
1.02 041 
1.01 908 
1.01 775 

1.01 642"^ 
1.01 510 
1.01 378 
1.01 247 
1.01 116 



1.00 935 
1.00 855 
1.00 725 
1.00 595 
1.00 436 

1.00 337 
1.00 209 
1.00 081 
0.99 953 
0-99 826 



0.99 699 
0.99 573 
0.99 446 
0.99 321 
0-99 195 



9^2 

Log." 



162 



Cot, 



0-99 070 
0-98 945 
0.98 821 
0.98 697 
0-98 573 

0.98 450 
0.98 327 
0-98 204 
0.98 081 
0-97 959 



9.99 810 
9.99 809 
9.99 808 
9 .99 807 

9-99 805 
9.99 804 
9.99 803 
9.99 802 
9.99 801 

9.99 799 
9.99 798 
9.99 797 
9.99 796 
9 .99 794 

9.99 793 
9.99 792 
9.99 791 
9.99 789 
9.99 788 



9.99 787 
9.99 786 
9.99 784 
9.99 783 
9.99 782 

9.99 781 
9.99 779 
9.99 778 
9.99 777 
9.99 776 



9.99 774 
9.99 773 
9.99 772 
9.99 770 
9.99 769 



c.d, 



9.99 768 
9.99 766 
9.99 765 
9-99 764 
9-99 763 



40 

39 
38 
37 
36 



140 

14.0 
16-3 
18.6 
21.0 
23.3 
46.6 
70.0 
93.3 
116.6 



139 

13.9 
16.2 
18-5 
20-8 
23.1 
46-3 
69.5 
92.6 
115.8 



138 

13.8 
16.1 
18.4 
20.7 
23.0 
46.0 
69.0 
92.0 
115.0 



137 

13.7 
16.0 
18.2 
20.5 
22.8; 
45.6, 
68.5 
91.3 
114.1113-3 



141 

14.1 
16.4 
18.8 
21.1 
23-5 
47.0 
70.5 
94-0 
117.5 

136 

13.6 
15.8 
18-1 
20.4 
22.6 
45-3 
68-0 
90-6 



30 

29 
28 
27 
26 



30 

19 
18 
17 
li 
15 
14 
13 
12 
11 



10 

9 
8 
7 
6 



097 838 



Log. Tan, 



9-99 761 



95^ 



Log. Sin 
652 





135 


134 


133 


6 


13-5 


13.4 


13.3 


7 


15-7 


15.6 


15-5 


8 


18.0 


17.8 


17-7 


9 


20.2 


20-1 


19-9 


10 


22-5 


22-3 


22-1 


20 


45-0 


44-6 


44-3 


30 


67.5 


67.0 


66-5 


40 


90.0 


89.8 


88-6 


50 


112.5 


111.6 


110-8 



133 

13-2 
15.4 
17.6 
19.8 
22.0 
44.0 
66.0 
88.0 
110.0 





131 


130 


139 


6 


13-1 


13.0 


12.9 


7 


15-3 


15.1 


15-0 


8 


17-4 


17.3 


17-2 


9 


19.6 


19.5 


19-3 


10 


21.8 


21.6 


21-5 


20 


43.6 


43.3 


43-0 


30 


65.5 


65.0 


64-5 


40 


87.3 


86.6 


86-0 


50 


109.1 


108.3 


107.5 



138 

12.8 
14.9 
17.0 
19.2 
21.3 
42.6 
64.0 
85.3 
106.6 



8 { 

9 I 





137 


136 


135 


134 


6 


12.7 


12-6 


12-5 


12.4 


7 


14-8 


14-7 


14-6 


14.4 


8 


16-9 


16-8 


16.6 


16-5 


9 


19-0 


189 


18-7 


18-6 


10 


21.1 


21-0 


20.8 


20-6 


20 


42-3 


42-0 


41.6 


41-3 


30 


63-5 


63-0 


62.5 


62-0 


40 


84-6 


84-0 


83. c 


82-6 


50 


105.8 


105.0 


104.1 


103-3 



133 

12-3 
14-3 
16-4 
18.i 
20.5 
41-0 
61.5 
82.0 
102.5 





133 


131 


130 


T 


1 , 


6 


12.2 


12.1 


12.0 


0-1 


o.il 


7 


14.2 


14.: 


14.0 


0-2 





1 


8 


16.2 


16.1 


16.0 


0-2 







9 


18.3 


18.1 


18-0 


0-2 





■ 


10 


20.3 


20.1 


20.0 


0.2 







20 


40.6 


40.3 


40.0 


0.5 





3 


30 


61.0 


60.5 


60.0 


0.7 





5 


40 


81.8 


80.6 


80.0 


1.0 





^1 


50 


101.6 


100.8 


100.0 


±1 





8i 



4 

0.1 
0.1 

0.3 



P.P. 



84* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 173° 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
ii 
15 
16 
17 
18 
19. 

20 
21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34. 

35 9 

36 9 
37 
38 
39 



Log. Sin. 



40 

41 

' 42 
43 
41 
45 

h 47 
li 48 

ill 49 

\\^ 
51 

III 52 
53 
54_ 
55 
56 
57 
58 
^ 
60 



01 923 

02 043 
02 163 
02 282 
02 401 



02 520 
02 638 
02 756 
02 874 
02 992 



03 109 
03 225 
03 342 
03 458 
03 574 



03 689 
03 805 

03 919 

04 034 
04 1^.8 



04 262 
04 376 
04 489 
04 602 
04 715 



04 828 

04 940 

05 052 
05 163 
05 275 



05 386 
05 496 
05 607 
05 717 
05 827 



05 936 

06 046 
06 155 

06 264 
06 372 

06 480 
06 588 
06 696 
06 803 
06 910 



07 017 
07 124 
07 230 
07 336 
07 442 



07 548 
07 653 
07 758 
07 863 
07 967 



08 072 
08 176 
08 279 
08 383 
08 486 



9.08 589 
Log. Cos. 



120 
119 
119 
119 
119 
118 
118 
118 
117 
117 
116 
116 
116 
116 

115 
115 
114 
114 
114 

114 
113 
113 
113 
113 
112 
112 
112 
111 
111 

111 
110 
110 
110 
110 
109 
109 
109 
109 
108 
108 
108 
107 
107 
107 

107 

106 
106 
106 
106 

105 
105 
105 
104 
104 
104 
104 
103 
103 
103 

103 



9-02 162 
9.02 283 
9.02 404. 
9.02 525 
9.02 645 



9-02 765 
9-02 885 
9.03 004 
9.03 123 
9.03 242 



9.03 361 
9.03 479 
9.03 597 
9.03 714 
9.03 83l 



d. Log. Tan. c.d. Log. Cot.jLog. Cos. 

0.97 838 9. 99 761^ 
0.97 716 9.99 76G 
0.97 595 9.99 759 



d. 



9.03 948 

9.04 065 
9.04 181 
9.04 297 
9.04 413 



9.04 528 
9.04 643 
9.04 758 

9.04 872 
9-04 987 

9.05 101 
9.05 214 
9.05 327 
9.05 440 
9.05 55 3 

9.05 666 
9.05 778 

9.05 890 

9.06 OOl 
9.06 113 
9-06 224 
9.06 335 
9.06 445 
9-06 565 
9.r6 665 



9-06 775 
9.06 884 

9.06 994 

9.07 102 
? 07, 211 
9.07 319 
9.07 428 
9.07 635 
9.07 643 
9.07 7 55 

9.07 857 

9.07 962 

9.08 071 
9-08 177 
9.08 283 

9.08 889 
9.08 494 
9.08 600 
9.08 705 
908 810 



9.08 914 



121 

121 

120 

120 

120 

119 

119 

119 

119 

118 

118 

118 

117 

117 

117 

116 

116 

116 

115 

115 

116 

114 

111 
114 

114 
113 
113 
113 
113 
112 
112 
112 
111 
111 

111 
111 
IliO 
liC 
liO 
109 
1C9 
109 
108 
109 

108 

108 
107 
107 
107 

107 

107 
106 
106 
106 

105 
105 
105 
105 
105 
104 



0.97 475 
0.97 354 



0.97 234 
0»97 115 
0.96 995 
0.96 876 
0.96 757 

0.96 639 
0.96 521 
0.96 403 
0.96 285 
0.96 168 



0-96 051 
0.95 935 
0.95 818 
0.95 702 
0.95 587 

0.95 471 
0.95 356 
0.95 242 
0-95 127 
0:95,013 

0.94 898 
0.94 785 
0-94 672 
0.94 559 
0-94 446 
0-94 334 



9.99 757 
9;9_9 756 

9-99 754 
9.99 753 
9.99 752 
9-99 750 
9 -99 74 9 

9-99 748 
9-99 746 
9-99 745 
9.99 744 
9-99 742 

9.99 741 
9-99 739 
999 738 
9-99 737 
f-P9 735 

S.99 734 
9-99 732 
9. 99 731 
9-99 73C 
9-99 728 



60 

59 
58 
57 
56 



P.P. 



9.99 727 
999 725 
9.99 724 
9.99 723 
9.99 721 

9.99 72C 
0.94 222 9.99 718 



0.94 110 
0.93 998 
QJH.887 
0.93 776 
0-93 665 
0.93 55i 
0.93 4*4 
0.93 334 



9.99 717 
9.99 715 
9-99 714 
9-99 712 
9-99 711 
G.99 71G 
9-99 708 
9.99 70-7 



0.93 225 9.99 705 
0.93 li5 9.99 704 
0.93 006 9-99 702 
0.92 897 9.99 70i 
0j12.788 9.99 699 

0.92 685 



log. Cot. 



c.d. 



0-92 572 
0.92 464 
0.92 357 
0:92 249 

0.92 142 
0.92 035 
0.91 929 
0.91 822 
0r91_716 

0-91 611 
0.91 505 



9.99 698 
9.99 696 
9.99 695 
9.99 693 
9.99_69j 

9.99 690 
9.99 689 
9.99 687 
9.99 686 
9.99 684 



50 

49 
48 
47 
J6 
45 
44 
43 
42 
Ji 
40 
39 
38 
37 
36 

35 
34 
33 
32 

M. 

30 
29 
28 
27 

_26 
25 
24 
23 
22 
21 



9.99 683 
9-99 681 



0.91 400«9-99 679| 
0.91 295 9-99 678 
0.91 190 9-99 67 6 

0-91 085(9.99 675 



Log. Tan, I Leg, Sin. 



20 

19 

18 

17 

M 

15 

14 

13 

12 

il 

10 

9 

8 

7 

6 

5 
4 
3 
2 
1 
O 



121 

12.Il 
14.2: 
16.21 
18.2i 
20.2 
40.5 
6O.7 
81.0 
101.2 



121 

12.1 
14.1 
16.1 
18.1 
20.1 
40-3 
60. 5 
80.6 
100.8 



120 

12.0 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
80.0 
100.0 



119 

11.9^ 
13.9 
15.8 
17-8 
19-8 
39-6 
59.5 
79.3 
99.1 



118 

11.8 
13.7 
15.7 
17.7 
19.6 
39.3 
59.0 
78.6 
98.3 



117 117 

6 11.7:11.7 
13.7il3.6 
15.615.6 
I7.6I7.5 



116 115 



11.6 
13.5 
15.4 



11.5 
134 
15.3 



17.4 17.2 
10:19. 6il9.5 19.3119-1 
20 39-1 39.0 38.6 38.3 
80 58.7 58.5 58.0 57.5 
40 78.3 78.0 77.3 76.6 
60 97.9]97.5 96.6 95.8 



115 

eiii.i 

7113.3 
8115.2 
9 17.2 



19.1 
38.i 
57.2 
76.3 
95.4] 



114 

11.4 
13.3 
15.2 
17.1 
190 
38.0 
57.0 
760 
95.0 



113 112 111 

11-3111-2 11.1 
13-2|13-0 12-9 
15.5jl4-9 14.8 
16. 9|l6. 8116.6 
18-8|18-6I18.5 
37.6137.3:37.0 
56.5|56-0i55.5 
75.3 74-6174.0 
94.I]93.3!92.5 



6 

7 
8 

9 
10 
20 
30 
40 
50 



110 

11.5! 

12.9 
14.7I 

16.6: 

18-4 
36.8 
5o.2 
73.6 
92.il 



110 109 108 

11.0110.9 10-8 
12.8 12.7 12.6 
14.6 14. 
16-5 16. 
18.3 18. 
36.6 3b. 
5o.Oi5y:. 
73.3i7z. 



14.4 
162 
180 
36.0 
5^0 



.6 72.0 
9x.6i9u.8 9u.O 



6 
7 
8 

9 
10 
20 
30 
40 
50 



107 


107 


106 105 


10.7 


10.7 


10.6 


10.5 


12.5 


12.5 


12-3 


12-2 


14.3 


14.2 


14.1 


14-0 


16.1 


16.0 


15.9 


15-7 


17.9 


17.8 


17.6 


17.5 


35-8 


35.6 


35.3 


35-0 


53.7 


53.5 


53.0 


52-5 


71.6 


71.3 


70.6 


70-0 


89.6 


89.1 


883 


87.5 



10.4 

12.1 
13.8 
15-6 
17J- 
34.6 
52.0 
69.3 
86.6 





105 


103 


2 


6 


10.3 


10.3 


0.2 


7 


12.1 


12.C 


0-2 


8 


13.8 


13.7 


0-2 


9 


15.5 


15-4 


0-3 


10 


17.2 


17.1 0.3 


20 


34.5 


34.3 


0.6 


30 


51.7 


51.5 


1.0 


40 


690 


68. 


].3 


50 


MM. 


85-^' 





1 


1 


0.1 


0.1 


0.2 


0.1 
0.1 


0.2 


0.2 


0.: 


0.2 


0.:: 


0.5 


0.3 


0.7 


0.5 


1.0 


0.6 


-1^ 


0.8 . 



P.?. 



w 



8^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
7° . AND COTANGENTS. 17^ 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19_ 

30 

21 
22 
23 
24_ 

25 
26 
27 
28 
29_ 

30 

31 

32 

33 

31 

35 

36 

37 

38 

39. 

40 

41 

42 

43 

44 

45 
46 
47 
48 
49_ 

50 

51 
52 
53 
54 

55 

56 

57 

58 

59^ 

60 



log. Sin. I d. Log, Tan. c.d. Log. Cot 



9-08 589 



102 



9.10 006 
9.10 105 
9.10 205 
9.10 303 
9.10 402 
9.10 501 



10 599 
9.10 697 

10 795 
9.10 892 
9.10 990 



9.11087 
9.11184 

11 281 
9.11377 

11473 



9.08 692||^f 
9.08 794!|n^ 
9.08 897|tn9 
9-08 999r"'^ 
102 



09 lOll^Xf 
09 202itm 

09 303!|ni 
09 40i'{^^ 
09 505^ „ 
100 



9.09 608 tnH 

9-09 706 Inn 

9.09 806||nn 
9.09 906 ^^ 



9.08 914 

9.09 018 
9 09 123 
9.09 226 

.09 330 



9.09 433 
9.09 536 
9.09 639 
09 742 
9.09 844 



9.11570 
9.11 665 
9.11761 
9.11 856 
9.11 952 



12 047 
12 141 
9.12 236 
9.12 330 
12 425 



.12 518 
•12 612 
.12 706 
•12 799 
• 12 892 



9.12 985 
13 078 

9.13 170 
9 13 263 
8.13 355 



13 447 
9.13 538 
9-13 630 
9^13 721 

13 813 



13 903* 
13 994 
19.14 085 
9.14 175 
9.14 265 



914355 



Log. Cos 



9r 



99 
99 
99 
98 



98 
98 
97 
97 
97 

97 
96 
97 
96 
96 
96 
95 
98 
95 
95 

95 
94 
94 
94 
94 

93 
94 
93 
93 
93 

93 
92 
92 
92 
92 

92 
91 
92 

91 
91 

90 
91 
90 
90 
90 

90 



9.09 947 
9 . 10 048 

10 150 

9.10 252 
9.10 353 



9.10 454 
10 555 

9.10 655 
10 756 

9.10 856 

10 956 
9.11055 
9.11155 
9.11254 
9-11353 



9.11452 
11550 

9.11649 
11747 

9.11 845 



9.11943 
9.12 040 
9.12 137 
9.12 235 
9.12 331 



9.12 428 
9.12 525 
9.12 621 
9.12 717 
9.12 813 



9.12 908 

9.13 004 
9.13 099 
9.13 194 
9.13 28 9 



13 384 
9 13 478 
9.13 572 
9.13 666 
9.13 760 



9.13 854 

9.13 947 

9. 14 041 
14 134 

9.14 227 



9.14319 
9.14 412 
9.14 504 
9-14 596 
9.14 688 



104 
104 
103 
103 

103 
103 
103 
102 
102 
102 
101 
102 
101 
101 

101 
101 
100 
100 
100 

100 
99 
99 



0.90 566 
0.90 463 
0.90 360 
0.90 258 
0-90 155 



0-91 085 
0-90 981 
0.90 877 
0.90 773 
0.90 670 



0.90 053 
0.89 951 
0.89 849 
. 89 748 
0.89 647 



Log. Cos 



9-99 
9-99 
9-99 
9.99 
9.99 



675 
673 
672 
670 
669 



9.99 
9.99 
9.99 
9.99 
9-99 



667 
665 
664 
662 
661 



0.89 546 
0.89 445 
0.89 344 
0.89 244 
0-89 144 

0.89 044 
0.88 944 
0.88 845 
0.88 745 
0.88 64 



9-99 
9-99 
99 
9.99 
9.99 

9.99 
9.99 
9-99 
9.99 
9-99 

9.99 
9.99 
99 
99 
9.99 



9.14 780 



98 
98 
98 
98 
97 
97 
97 
96 
97 



95 
95 
95 
95 
95 
94 
94 
94 
94 
94 

93 
93 
93 
93 
93 
92 
92 
92 
92 
92 
92 



0.88 548 
0.88 449 
0.88 351 
0.88 253 
0.88 155 



9.99 
9.99 
99 
9.99 
9.99 



0.88 057 
0.87 959 
0.87 862 
0.87 765 
0.87 668 



9.99 
99 
9-99 
9-99 
9-99 



0.87 571 
0.87 475 
0.87 379 
0.87 283 
0.87 187 



Log. Cot. c.d. 



0.87 091 
0.86 996 



0.86 900 3.99 



0.86 805 
0-86 710 



0.86 616 
0.86 521 
0.86 427 
0. 86 333 
0.86 239 



= 86 146 
0.86 052 
0.85 959 
0-85 866 
0-85 773 



0-85 680 
0-85 588 
0.85 495 
0-85 403 
0^85 311 



0-85 219 



Log, Tan 



659 
658 
656 
654 
653 



651 
650 
648 
646 
645 



643 40 

641 39 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 



50 

49 
48 
47 
46 

45 
44 
43 
42 
41 



38 
37 
36 



640 
638 
637 

635 35 
63"^ 34 
632 33 



630 
628 



627 

625 
623 
622 
620 



9-99 
9-99 
9.99 
9-99 
9-99 



618 
617 
615 
613 
611 



9-99 
9.99 



9-99 
9. 99 



610 
608 
606 
605 
603 



9.99 
9.99 
9.99 
9.99 



601 
600 
598 
596 
5.94 



99 
9.99 
9.99 
9.99 
9.99 



9.99 
9-99 
9.99 
9.99 
9.99 
9-99 



Log. 



593 
591 
589 
587 
586 

584 
582 
580 

579 
577 
575 
Sin. 



30 

29 
28 

27 
26 



30 

19 
18 
17 
16 

15 
14 
.13 
12 
11 



10 

9 
8 

7 
6 

5 
4 
3 
2 
1 




654 



P. P. 



104 



6 

7 
8 
9 
10 
20 
30 
40 
50 



103 

10.3 
12.0 
13.7 
15-4 
17.1 
34-3 
51-5 
68-6 
85.8 



103 101 



10.2 
11.9 
13.6 
15-3 
170 
34.0 



10.1 
11.8 
13.4 
15.1 
16.8 
33.6 



51.050.5 
68.0J67.3 
85.0184.1 





100 


100 


99 


98 


6 


10.0 


10-0 


9.9 


9.8 


7 


11.7 


11-6 


11.5 


11. 4 1 


8 


13-4 


13.3 


13.2 


13.(5 


9 


15.1 


15.0 


14.8 


14.7 


10 


16.7 


16.6 


16.5 


16.3 


20 


33.5 


33.3 


33.0 


32.6 


3^ 


50-2 


50.0 


49.5 


49.0 


\0 


67.0 


66.6 


66.0 


65.3 


50 


83.7 


83.8 


82.5 


81.6 



6 
7 
8 

9 
10 
20 
30 
40 
50 



6 
7 
8 

9 
10 
20 
30 
40 



97 


97 


96 


9.7 


9.7 


9.6 


11-4 


11-3 


11.2 


13-0 


12-9 


12-81 


14.6 


14-5 


14.41 


16-2 


16-1 


16-01 


32-5 


32.3 


32 Oi 


48-7 


48.5 


48.0 


65.0 


64-6 


64-0 


81.2 


80.8 


80. Oi 



95 

9.5 
11.1 
12.6 
14.2 
15.8 
31.6 
47.5 
63. 
79. 











9f 94 93 9S 1 


9.4 


9.4 


9-3 


9.2 : 


11. 


10-9 


10.8 


10. Z i 


12.6 


12.5 


12.4 


12.2 ; 


14-2 


u.i 


13.9 


13. 


15-7 


15-6 


15.5 


15.3 i 


31.5 


31 3 


31.0 


30.6 ; 


47-2 


47-0 


46. 5 


46. Q i 
61. S 


63-062. 6 


62.0 


78.7 


78.3 


77.5 


76.6 I 





9T 


91 


90 


2 


6 


9.1 


9.1 


9.0 


0.2 


7 


10.7 


10.6 


10.5 


0.2 


8 


12.2 


12.1 


12-0 


0.2 


9 


13-7 


13.6 


13.5 


0.3 


10 


15.2 


15.1 


15.0 


0.3 


20 


30.5 


30.3 


30.0 


0.6 


30 


45.7 


45.5 


45.0 


1.0 


40 


61.0 


60.6 


60-0 


1 • 3 


50 


76.2 


75.8 


75.0 


1.6 



t 

0.1 
0.2 
0.2 
0.2 
0-2 
0.5 
0.7 
10 
1.2 



P. P. 



8»'t 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
8P AND COTANGENTS. l^l* 



' iLoff. Sin. d 



O 

1 
2 
3 

^1 

5 

6 

7 

8 
J_ 

10 

11 
12 
13 
11 
15 
16 
17 
18 

il 
30 

21 

22 

23 

2£ 

25 

26 

27 

28 

£1 

30 

31 
32 
33 
31 

35 

36 
37 
38 
S9_ 

40 

41 
42 
43 
44 

45 

46 

47 

48 

49L 

50 

51 

52 

)3 

)± 

15 

16 

)7 

18 

^ 

50 



9.14 355 
9 . 14 445 
9.14 535 
9-14 624 
9 .14 713 
14 802 
14 891 

14 980 

15 068 
15 157 



9. 



15 245 
15 333 
15 421 
15 508 
15 595 



15 683 
15 770 
15 857 

15 943 

16 030 



16 116 
16 202 
16 288 
16 374 
16 460 

16 545 
16 630 
16 716 
16 801 
_16_885 

16 970 

17 054 
17 139 
17 223 
17 307 



17 391 
9.17 474 
9.17 558 
9.17 641 
9.1772A 



17 807 
17 890 

17 972 

18 055 
18 137 



18 219 
18 301 
18 383 
18 465 
18 546 



18 628 
18 709 
18 790 
18 871 
18 952 



9 
9 
9 
9 
9_ 

9-19 032 
9-19 113 
9 19 193 
19 273 
9.19 353 



9.19 433 



Log. Cos. 



90 
89 
89 
89 
89 
89 
88 
88 
88 

88 
88 
88 
87 
87 

87 
87 
87 
86 
86 

86 
86 
86 
86 
85 

85 
85 
85 
85 
84 
84 
84 
84 
84 
84 
84 
83 
83 
83 
83 

83 
83 
82 
82 
82 

82 
82 
82 
81 
81 

8l 
81 
81 
80 
81 
80 
80 
80 
80 
80 

79 



Log. Tan. c.d. 



9.14 
9.14 
9-14 
9-15 
9.15 



780 

872 
963 
054 
145 



9.15 
9.15 
9.15 
9.15 
9-15 



236 
327 
417 
507 
598 



9.17 
9.17 
9.17 
9.17 
9.17 



9.15 
9.15 
9.15 
9.15 
9.16 

9.16 
9 16 
9.16 
9.16 
9.16 



9.16 
9.16 
16 
9.16 
9.16 



9.17 
9.17 
9.17 
9.17 
9.17 



687 
777 
867 
956 
_045 

134 
223 
312 
401 
489 

577 
665 
753 
841 
928 
015 
103 
190 
276 
363 
450 
536 
622 
708 
794 



9.178-0 

9.17 965 

9.18 051 
9.18 136 
9-18 221 



18 306 
18 390 
18 475 
18 559 
18 644 



18 728 
18 812 
18 896 

9.18 979 

9.19 063 



19 146 

19 229 

19 312 

9-19 395 

9.19 478 



19 560 
19 643 
19 725 
19 807 



9-19 889 



9.19 971 



Log. Cot. 



91 
91 
91 
91 
91 
90 
90 
90 
90 

89 
90 
89 
89 
89 

89 
89 
89 
88 
88 

88 
88 
87 
88 
87 
87 
87 
87 
86 
87 

86 
86- 
86 
86 
85 
86 
85 
85 
85 
85 

85 

84 
84 
84 
84 
84 
84 
84 
83 
83 

83 
83 
83 
83 
82 

32 
82 
82 
82 
82 
82 



Log. Cot. Log. Cos. 



0.85 219 
0.85 128 
0.85 037 
0.84 945 
0^84854 

0-84 76§ 
0.84 673 
0.84 582 
0-84 492 
0.84 402 



0-84 312 
0.84 222 
0.84 133 
0.84 043 
P_^83^954 

0.83 865 
0-83 776 
0.83 687 
0-83 599 
0-83 51 



0-83 422 
0-83 334 
0.83 247 
0.83 159 
0-83 071 



0-82 
0-82 
0.82 
0-82 
0.82 



0-82 
0.82 
0.82 
0.82 
0.82 



0.82 
0.82 
0.81 
0.81 
0.81 



984 
897 
810 

723 
636 

550 
464 
377 
291 
206 

120 
034 
949 
864 

779 



0.81 
0.81 
0.81 
0.81 
0.81 



0.81 
0.81 
0.81 
0.81 
0.80 



694 
609 
525 
440 
356 
272 
188 
104 
020 
937 9 



9.99 575 
9-99 573 
9-99 571 
9-99 570 
9-99 568 



99 566 
9-99 564 
9-99 563 
9-99 561 

99 559 



99 557 
99 555 
99 553 
99 552 
99 55C 



99 548 
99 546 
99 544 
99 542 
99 541 



99 539 
99 537 
99 535 
99 533 
99 531 



9- 



99 529 
90 528 
99 526 
99 524 
99 522 



99 520 
99 518 
99 516 
99 514 
99 512 



99 511 
99 509 
99 507 
99 505 
99 503 



99 501 
99 499 
99 497 
99 495 
99 493 



80 854 
80 770 
0-80 687 
0-80 604 
0-80 522 

0-80 439 
0-80 357 
0-80 274 
0.80 192 
0.80 110 



C.d, 



99 491 
99 489 
99 487 
99 485 
99 484 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 



30 

29 
28 
27 
26 



25 
24 
23 
22 
21_ 

20 

19 
18 
17 
16 



9. 99 482 
9. 99 480 
9. 99 478 
9.99 476 
9-99 474 



0-80 028 



99 472 
99 470 
99 468 
99 466 
99^464 
9-99 462 



10 

9 
8 

7 
6 



Log, Tan.|Log. Sin. 



P. P. 





91 


91 


00 


6 


9-1 


9.1 


9-0 


7 


10.7 


10.6 


10-5 


8 


12.2 


12-1 


12-0 


9 


13-7 


13-6 


13-5 


10 


15.2 


15-1 


15-0 


20 


30-5 


30-3 


30-0 


30 


45-7 


45-5 


45-0 


40 


61-0 


60.6 


60-0 


50 


76.2 


75-8 


75.0 



89 
8-9 

10 

11 

13 

14 

29 

44 

59. 

74 





88 


88 


87 


6 


8 8 


88 


8.7 


7 


10.3 


10-2 


10.1 


8 


11-8 


11-7 


11-6 


9 


13-3 


13-2 


13-0 


10 


14-7 


14-6 


14-5 


20 


29-5 


29-3 


29-0 


30 


44-2 


44-0 


43-5 


40 


59.0 


58.6 


58.0 


50 


73.7 


73.3 


72.5 





85 


85 


84 


6 


8.5 


8.5 


8.41 


7 


10.0 


9.9 


9 


8 


8 


11-4 


11.3 


11 


2 


9 


12.8 


12.7 


12 


6 


10 


14.2 


14-1 


14 





20 


28.5 


28-3 


28 





30 


42-7 


42-5 


42 





40 


57.0 


56-6 


56 





50 


71.2 


70.8 


70 






86 
8.6 

10.0 

11.5 

12-9 

14-3 

28-6 

43-0 

57- 

71. 



83 

8.3 

9.7 

11-0 
12. i 
13.8 
27-6 
41-5 
55-3 
69.1 





83 


83 


81 


80 


6 


8-2 


8-2 


8-1 


8-0 


7 


9 


6 


9 


5 


9.4 


9 


3 


8 


11 





10 


g 


10.8 


10 


Q 


9 


12 


4 


12 


3 


12.1 


12 


• 


10 


13 


7 


13 


6 


13-5 


13 


3 


20 


27 




27 


3 


27-0 


26 


5 


30 


41 


2 


41 





40-5 


40 





40 


55 





54 


6 


54-0 


53 


3 


50 


68 


7 


68 


3 


67.5 


66 


6 





79 


1 


3 


6 


7-9 


0-2 


7 


9-3 





2 


8 


10-6 





2 


9 


11-9 







10 


13-2 





3 


20 


26-5 





5 


30 


39-7 


1 





40 


53-0 


1 




50 


66.2 


1 


6 



0^1 

0.2 
0.2 
0-2 
0-2 
0-5 
0-7 
1-0 
1.2 



P.P. 



^' 



655 



81^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



■fviifi 



Log. Sin. 



19 433 
19 513 
19 592 
19 672 
19 75] 



19 830 
19 909 

19 988 

20 066 
20 145 



20 223 
20 301 
20 379 
20 457 
20 535 



20 613 
20 690 
20 768 
20 845 
20 922 



20 999 

21 076 
21 152 
21 229 
21 305 



21 382 
21458 
21 534 
21 609 
21 685 





1 

2 

3 
_± 

5 

6 

7 

8 
JL 
10 
11 
12 
13 
14 
15 
16 
17 
18 

11 
20 

21 

22 

23 

24_ 

25 

26 

27 

28 

29L 

30 

31 

32 

33 

31 

35 

36 

37 

38 

39. 

40 

41 
42 
43 

45 
46 
47 
48 
49 

50 

51 
52 
53 

55 
56 
57 
58 
5i 
6019.23 967 



21 761 
21 836 
21911 

21 987 

22 062 



22 136 
22 211 
22 285 
22 380 
22 435 



22 509 
22 583 
22 657 
22 731 
22 805 



22 878 

22 952 

23 025 
23 098 
23 171 



23 244 
23 317 
23 390 
23 462 
23 535 



23 607 
23 679 
23 751 
23 823 
23 895 



JLog. Cos. 



80 
79 
79 
79 

79 
79 
79 
78 
78 

78 
78 
78 
78 
78 

77 
77 
77 
77 
77 

77 
77 
76 
76 
76 

76 
76 
76 
75 
76 

75 
75 
75 
75 
75 
74 
75 
74 
74 
74 

74 
74 
74 
73 
74 

73 
73 
73 
73 
73 

73 
72 
73 
72 
72 

72 
72 
72 
72 
72 

71 



9.19 971 
9-20 053 

9.20 134 
20 216 

9.20 297 



9.20 378 
20 459 
20 540 
20 620 
20 701 



21 181 
21 261 
9.21 340 
9.21 420 
9.21 499 



9.21 578 
21 657 
9.21 735 
9.21 814 
9.21 892 



Log. Tan, 



20 781 
20 862 

20 942 

21 022 
21 102 



9.21 971 
22 049 

9-22 127 

9.22 205 
9.22 283 



9.22,360 
22 438 
9.22 515 
9.22 593 
22 670 



9-22 747 
9 22 824 
9.22 900 
9.22 977 
9 - 23 054 



9 23 130 
9-23 206 
9. 23 282 
9 23 358 
9.23 434 



9.23 510 
9.23 586 
9.23 66l 
9.23 737 
9.23 812 



9.23 887 

9.23 962 

9.24 037 
9.24 112 
9.24 186 



9.24 261 
9-24 335 
9-24 409 
9.24 484 
9. 24 558 



9.24 632 
Log. Got 



c.d. 

81 
81 
81 
81 

81 
81 
81 
80 
81 

80 
80 
80 
80 
80 

79 
79 
79 
79 
79 

79 
79 
78 
78 
78 

78 
78 
78 
78 
78 

77 
77 
77 
77 
77 
77 
77 
76 
77 
76 

76 
76 
76 
76 
78 

76 

75 
75 
75 
75 

75 
75 
75 
75 
74 

74 
74 
74 
74 
74 
74 



Loo:. Cot, 



79 218 
79 138 
79 058 
78 978 
78 898 



78 818 
78 739 
78 659 
78 580 
78 501 



78 422 
78 343 
78 264 
78 188 
78 107 



78 029 
77 951 
77 873 
77 795 
77 717 



c.d. 



80 028 
79 947 
79 865 
79 784 
79 703 



79 622 
79 541 
79 460 
79 379 
79 298 



9.99 452 
9.99 450 
99 448 
99 446 
99 444 



77 639 
77 562 
77 484 
77 407 
77 330 

77 253 
77 176 
77 099 
77 022 
76 946 



Log. Cos. 



■ 99 462 

• 99 460 
99 458 

• 99 456 

• 99 454 



99 442 
99 440 
9-99 437 
9.99 435 
9.99 433 

9.99 431 
99 429 
99 427 
99 425 

99 423 



99 421 
99 419 
99 417 
99 415 
9.99 413 



9.99 411 
9.99 408 
9.99 406 
9.99 404 
9.99 402 

9.99 400 
99 398 
9.99 396 
19 99 394 
9-99 392 



9-99 389 
9.99 387 
9.99 385 
9-99 383 
9 99 381 



60 

59 
58 
57 
56 

55 
54 
53 
52 
11 
50 
49 
48 
47 
M. 
45 
44 
43 
42 
41 



40 

39 
38 
37 
36. 
35 
34 
33 
32 
II 
30 
29 
28 
27 
_26 

25 
24 
23 
22 
21 



76 870 
76 793 
76 717 
76 641 
76 565 



999 379 
99 377 
9. 99 374 
9-99 372 
9-99 370 



76 489 
76 414 
76 338 
76 263 
76 188 



76 113 
76 038 
75 963 
75 888 
75 813 



75 739 
75 664 
75 590 
75 516 
75 ^^2 



75 368 



Log. Tan 



99 368 
9. 99 366 
9. 99 364 
9- 99 361 
9-99 359 



99 357 
99 355 

9. 99 353 
99 350 

9-99 348 

9-99 346 
9. 99 344 
9. 99 342 
9. 99 339 
9.99 337 



9. 99 335 



99*^ 



Log. Sin. 
656 



30 

19 
18 
17 

15 
14 
13 
12 
11 

10 

9 
8 

7 

" 5 
4 
3 
2 

_1 





P. P. 



81 


81 


80 


79 


81 


8-1 


80 


7.9 


9 


5 


9-4 


9 


3 


9 


2 


10 


8 


10-8 


10 


6 


10 


5 


12 


2 


12-1 


12 





11 


g 


13 


6 


13.5 


13 


3 


13 


■ 


27 


1 


27.0 


26 


6 


26 


V 1 


40 


7 


40.5 


40 





39 




54 


3 


54.0 


53 


3 


52 


i i 
O 


67.9 


67.5 


66.6 


65 




7^ 


I 7 


H 


7 


7 





6 


7 


8 


78 


7 


9 


1 


9.1 


8 


10 


4 


10.4 


9 


11 


8 


11.7 


10 


13 


1 


13.0 


20 


26 


1 


26 


30 


39 


2 


390 


40 


52 


3 


52.0 


50 


65 


4 


65.0 



7-7 
9-0 

10.2 
11.5 
12-8 
25-6 
38-5 
51-3 



64 





76 


•^6 


75 


74 


6 


76 


7.G 


7 5 


7. 


7 


89 


8.8 


8 


7 


8- 


8 


10.2 


lO-IllO 





9- 


9 


11.5 


11.411 


2 


11- 


10 


12.7 


12.6112 


5 


12- 


20 


25.5 


25.3 


25 





24. 


30 


38-2 


38. 


37 


5 


37- 


40 


51.0 


50-6 


50 





49. 


50 


63.7 


63-3 


62 


5 


61. 



73 

7-3 
86 
98 



11 
12 
24 
36 
40 
61-2 



73 

7 3 



• 5 

• 7 
9 

• 1 
.3 
•5 

48 6 48 
60-8160 



72 
7.2 



71 

7-1 
8 



71 

7-1 



8 23 
35 
47 
59 



2 

0.2 



P P. 



? 



8ft 



10^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



169' 



Log. Sin, 



o 

1 

2 
3 

5 
6 
7 
8 
_9 

10 

11 

12 

13 

lA 

15 

16 

17 

18 

11 

20 

21 

22 

23 

24 

25 
26 
27 
28 
29_ 
30 9 



9.23 967 

9.24 038 
9.24 110 
9.24 181 
9. 24 252 



24 323 
9.24 394 
9.24 465 
9.24 536 
9-24 607 



9-24 677 
9 • 24 748 
9-24 818 
9.24 888 
9-24 958 



9-25 028 
9-25 098 
9.25 167 
9.25 237 
9-25 306 
9-25 376 
9- 25 445 
9-25 514 
9-25 583 
9 25 652 



25 721 

25 790 

9-25 858! 

9-25 927 

9-25 995 



31 
32 
33 
34 

35 
36 
37 
38 

39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49_ 
60 
51 
52 
53 
'1 
!5 
16 
!7 
% 

50 



.26 063 
•26 131 
•26 199 
•26 267 
• 26 335 

26 402 
26 470 
26 537 
26 605 
36 672 



26 739 
26 806 
26 873 

26 940 

27 007 



27 073 
27 140 
27 206 
27 272 
27 339 



27 405 
27 471 
27 536 
27 602 
27 688 



27 733 
27 799 
27 864 
27 929 
27 995 



9-28 060 



Log. Cos. 



71 
7l 
7l 
71 

71 
71 
71 
71 
70 

70 
70 
70 
70 
70 
69 
70 
69 
70 
69 

69 
69 
69 
69 
69 

68 
69 
68 
68 
68 

68 
68 
68 
68 
67 

67 
68 
67 
67 
67 

67 
67 
67 
66 
67 

66 
66 
66 
66 
66 

66 
66 
65 
66 
65 
65 
65 
65 
65 
65 
65 



Log. Tan. c. d. Log> Cot 



9-24 632 
9-24 705 
9.24 779 
9-24 853 
9-24 926 



9-25 000 
9-25 073 
9- 25 146 
9.25 219 
9-25 292 



9-25 365 
9-25 437 
9.25 510 
9-25 582 
9-25 654 



9-25 727 
9.25 799 
9.25 871 
9-25 943 
9-26 01 



26 086 
26 158 
9.26 229 
9.26 300 
9-26 371 



27 148 
9-27 218 

27 287 
9.27 357 

27 427 



9.26 443 
9-26 514 
26 584 
9-26 655 
9-26 726 



9-26 796 

9.26 867 

9-26 937 

27 007 

27 078 



27 496 
27 566 
27 635 
27 704 
27 773 



9.27 842 
27 911 
27 980 

9-28 049 

9.28 117 



28 186 
28 254 
28 322 
28 390 
28 459 



9.28 527 
9.28 594 
9.28 662 
9-28 730 
9.28 797 



9-28 865 
Log. Cot. 



73 
74 
73 
73 

73 
73 
73 
73 
73 

73 
72 
72 
72 
72 
72 
72 
72 
72 
71 
72 
71 
71 
71 
71 
71 
71 
70 
71 
70 
70 
70 
70 
7C 
70 

70 
70 
69 
70 
69 

69 
69 
69 
69 
69 

69 
69 
68 
69 
68 

68 
68 
68 
68 
68 

68 
67 
68 
67 
67 
67 

cTI 



75 368 
75 294 
75 220 
75 147 
75 073 



75 COG 
74 927 
74 854 
74 781 
74 708 

74 635 
74 562 
74 490 
74 417 
74 345 



74 273 
74 201 
74 129 
74 057,- 
73 985 J9 

9 



Log. Cos 



99 335 
99 333 
99 330 
99 328 
99 326 



99 324 
99 321 
99 319 
99 317 
99 315 



•73 913 
•73 842 
73 771 
73 699 
73 628 

73 557 
73 486 
73 415 
73 344 
73 274 

73 203 
73 133 
73 062 
72 992 
72 922 



•99 312 
.99 310 

• 99 308 

• 99 306 

• 99 303 
•99 301 

• 99 299 

• 99 296 
•99 294 

• 99 292 
•99 29C 
-99 28? 
•99 285 
•99 283 
■ 99 280 



60 

59 
58 
57 

55 
54 
53 
52 
51 



72 852 
72 782 
:'2 712 
72 642 
72 573 



72 503 
72 434 
72 365 
'72 295 
72 226 



72 157 
72 088 
72 020 
71 951 
71 882 



■71814 
71 746 
•71 677 
■ 71 609 
71 541 



0.71 473 
0.71 405 
0.71 337 
0.71 270 
0.71 202 



0.71 135 



Log. Tan 



•99 278 
.99 276 
.99 273 
•99 271 
^^69 
• 99 266 
99 264 
99 262 
99 259 
99 257 



99 255 
99 252 
99 250 
99 248 
99 245 



50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



35 
34 
33 
32 
li 

30 

29 
28 
27 
26 



99 243 
99 240 
99 238 
99 236 
99 233 



99 231 
99 228 
99 226 
99 224 
99 221 



99 219 
99 216 
99 214 
99 212 
99 209 



99 207 
99 204 
99 202 
99 199 
99 197 



9-99 194 



100' 



Log. Sin. 
657 



25 
24 
23 
22 
-21 
20 
19 
18 
17 
16 



!4 

;5 



10 

9 
8 
7 
6 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 





1 


73 


7.4 


7.3 


8 


6 


8.6 


9 


8 


9.8 


11 


1 


11.0 


12 


3 


12.2 


24 


6 


24.5 


37 





36.7 


49 


3 


49.0 


61 


6 


61.2 



73 

7.3 
5 
7 
9 
1 
3 
5 
6 
8 



8 
9 

10 
12 
24 
36 
48. 
60. 



73 

7.2 

8 4 

9 = 6 
10-9 

10 12.1 
20 24 1 
3036. 2 
40 48-3 
50 60.4 



72 

7.2 
8 4 
9.6 
10.8 
12.0 
24.0 
36 
48.0 
60.0 



71 

7.1 
8 3 



59 6 



71 

7.1 





70 


70 


69 


69 


e 


7.0 


7.0 


6.9 


6.9 


7 


8.2 


8 


.1 


81 


8 


■^ 


8 


9.4 


9 


3 


9-2 


9 


2 


9 


10.6 


10 


5 


10. 4 


10 


3 


10 


11.7 


11 


Q 


11.6 


11 


5 


20 


23.5 


23 


3 23-1 


23 





30 


35.2 


35 


34.7 


34 


5 


40 


47.0 


46 


6 46.3 


46 





50 


58.7 


58 


3 


57.9 


57.5 





68 


68 


67 


67 


6 


6.8 


68 


6^7 


6-7 


7 


8 





7 


g 


1 


9 


7 


• 8 


8 


9 


1 


9 





9 





8 


9 


9 


10 


3 


10 


2 


10 


] 


10 





10 


11 


4 


11 


3 


11 


2 


11 


1 


20 


22 


8 


22 


6 


22 


5 


22 


CJ 


30 


34 


2 


34 





33 


7 


33 


5 


40 45 


6 


45 


3 


45 





44 


6 


50 


57. 


1 


56 


6 


56 


2 


55 


8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



66_ 

6 6 



66 

6 
7 
8 
9 


C 




65 

6 E 

7 

8 

9 
10 
21 
32 
43 
54 



65 

65 



54.1 





2 


6 


0.21 


7 





3 


8 





3 


9 





4 


10 





4 


20 





8 


30 


1 


2 


40 


1 


6 


50 


2 


1 



2 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
1.0 
1.3 
1.6 



P.P. 



79' 



ll"' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



168° 



Log. Sin.l d. 



28 060 
28 125 
28 189 
28 254 
28 319 



28 383 
28 448 
28 512 
28 576 
28 641 



28 705 
28 769 
28 832 
28 896 
28 960 



29 023 
29 087 
29 150 
29 213 
29 277 



29 340 
29 403 
29 466 
29 528 
29 591 



29 654 
29 716 
29 779 
29 841 
29 903 



29 965 

30 027 
30 089 
30 151 
30 213 



30 275 
30 336 
30 398 
30 459 
30 520 



30 582 
30 643 
30 704 
30 765 
30 826 



30 886 

30 947 

31 008 
31 068 
31 129 



31 189 
31 249 
31 309 
31 370 
31 429 



31489 
31 549 
31 609 
31 669 
31 728 



9 31 788 
Log. Cos. 



65 
64 
65 
64 

64 
64 
64 
64 
64 

64 
64 
63 
64 
63 

63 
63 
63 
63 
63 

63 
63 
63 
62 
63 
62 
62 
62 
62 
62 

62 
62 
62 
62 
61 
62 
61 
61 
6l 
61 

6l 
61 
61 
61 
61 
60 
61 
60 
60 
60 

60 
60 
60 
60 
59 

60 
60 
59 
60 
59 
59 



Log. Tan. c. d 



28 865 

28 932 

29 000 
29 067 
29 134 



29 201 
29 268 
29 335 
29 401 
29 468 



29 535 
29 601 
29 667 
29 734 
29 800 



29 866 
29 932 

29 998 

30 064 
30 129 



30 195 
30 260 
30 326 
30 391 
30 456 



30 522 
30 587 
30 052 
30 717 
30 781 



30 846 
30 911 

30 975 

31 040 
31 104 



31 168 
31 232 
31 297 
31 361 
31424 



31488 
31 552 
31 616 
31 679 
31 743 



31 806 
31 869 
31 933 

31 996 

32 059 



32 122 
32 ^85 
32 248 
32 310 
32 373 



32 436 
32 498 
32 560 
32 623 
32 685 



32 747 



Log. Cot 



67 
67 
67 
67 

67 
66 
67 
66 
67 

66 
66 
66 
66 
66 

66 
66 
66 
66 
65 

65 
65 
65 
65 
65 

65 
65 
65 
65 
64 

65 
64 
64 
64 
64 

64 
64 
64 
64 
63 

64 
64 
63 
65 
63 

63 
63 
63 
63 
63 

63 
63 
63 
62 
63 

62 
62 
62 
62 
62 

62 



Log. Cot. 



c.d 



71 135 
71 067 
71 000 
70 933 
70 866 



70 798 
70 732 
70 665 
70 598 
70 531 



70 465 
70 398 
70 332 
70 266 
70 200 



70 134 
70 068 
70 002 
69 936 
69 870 



69 805 
69 739 
69 674 
69 608 
69 543 



69 478 
69 413 
69 348 
69 283 
69 218 



69 153 
69 089 
69 024 
68 960 
68 896 



68 831 
68 767 
68 703 
68 639 
68 575 



68 511 
68 447 
68 384 
68 320 
68 257 



68 193 
68 130 
68 067 
68 004 
67 941 



67 878 
67 815 
67 752 
67 689 
67 626 



67 564 
67 501 
67 439 
67 377 
67 314 



Log. Cos, 



67 252 



Log. Tan, 



194 
192 
189 
187 
185 



99 



182 
180 

177 
175 
172 

170 
167 
165 
162 
160 



99 



157 
155 
152 
150 
147 



145 
142 
139 
137 
134 



132 
129 
127 
124 
122 



119 
116 
114 
111 
109 



106 
104 
101 
098 
096 



093 
091 
088 
085 
083 



080 
077 
075 
072 
069 



9-99 



101' 



Log. 
658 



067 
064 
062 
059 
056 
054 
051 
048 
046 
043 

040 
Sin. 



60 

59 
58 
57 
56 



55 
54 
53 
52 
51 
50 
49 
48 
47 
46 



45 
44 
43 
42 
_41 

40 

39 
38 
37 
36 



30 

29 
28 

27 
26 



25 
24 
23 
22 
21_ 

30 

19 
18 
17 
16 



10 

9 
8 

7 
_6 

5 
4 
3 
2 
_1 




P. P. 



67 



7 
8 
9 
10 
20 
30 
40 
50) 



6 

7 

8 

9 

10 

20 

30 

40 

50 

66 

6-6 

7 

8 
10 
11 
22 
33 
44 
55 



6 


7 


6. 


7 


9 


7. 


9 





8. 


10 


1 


10. 


11 


2 


11- 


22 


5 


22. 


33 


7 


33. 


45 





44. 


56 


.2 


55. 



66 

6.6 



64 

6.4 



64 



67 

7 
8 
9 

Q 
1 

3 
5 
6 
8 

65_ 

65 



65 

6.5 



63_ 

6-3 



63 

6.3 



63_ 

2 



63 

62 



61 



60 

6.0 

7.0 

8.0 

91 

10-1 

20.1 

30-2 

40.3 

50.4 



6 

7 

8 

9 

10 

20 

30 

41 

51 

60 

60 



1 

.21 7 
2 8 
.2 9 
.2110 
• 5i20 
.730 
.040 
.2)50 



61 

61 













300 29 
40.0139 
50.0149 



59 

5-9 



3 

6 0-3 



0.3 
0.4 
0.4 



0.5 
1.0 
1.5 
2.0 

2-5 



2_ 

0.2 
03 
0-3 
0.4 
0.4 
0.8 
1.2 
1.6 
2.1 



2 

0-2 
0.2 
0.2 
0.3 
0-3 
0.6 
1.0 
1.3 
1.^ 



P. P. 



7^ 



TABLE VII— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 167° 



Log. Sin. 



, 50 

51 

(62 

|.63 

lbs 

'^57 
68 

119 
.60 



31 788 
31 847 
31 906 

31 966 

32 025 



32 084 
32 143 
32 202 
32 260 
32 319 



32 378 
32 436 
32 495 
32 553 
32 611 



32 670 
32 728 
32 786 
32 844 
32 902 



32 960 

33 017 
33 075 
33 133 
33 190 



33 248 
33 305 
33 362 
33 419 
33 476 



33 533 
33 590 
33 647 
33 704 
33 761 



33 817 
33 874 
33 930 

33 987 

34 043 



34 099 
34 156 
34 212 
34 268 
34 324 



34 379 
34 435 
34 491 
34 547 
34 602 



34 658 
34 713 
34 768 
34 824 
34 879 



34 934 

34 989 

35 044 
35 099 
35 154 



35 209 



.og. Cos, 



d. 



59 
59 
59 
59 
59 
59 
59 
58 
59 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 
57 
58 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

55 
56 
55 
56 
55 

55 
55 
55 
55 
55 

55 
55 
55 
54 
55 
55 



d. 



Log. Tan. 



32 747 
32 809 
32 871 
32 933 
32 995 



33 057 
33 118 
33 180 
33 242 
33 303 



33 364 
33 426 
33 487 
33 548 
33 609 



33 670 
33 731 
33 792 
33 852 
33 913 



33 974 

34 034 
34 095 
34 155 
34 215 



34 275 
34 336 
34 396 
34 456 
34 515 



34 575 
34 635 
34 695 
34 754 
34 814 



34 873 
34 933 

34 99:2 

35 05l 
35 110 



35 169 
35 228 
35 287 
35 346 
35 405 



35 464 
35 522 
35 581 
35 640 
35 698 



35 756 
35 815 
35 873 
35 931 
35 989 



36 047 
36 105 
36 163 
36 221 
36 278 



9.36 336 



Log. Cot 



c.d. 

62 
62 
62 
62 

61 
61 
62 
61 
61 

61 
61 
61 
61 
61 

60 
61 
61 
60 
61 

60 
60 
60 
60 
60 

60 
60 
60 
60 
59 
60 
60 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

58 
58 
59 
58 
58 
58 
58 
58 
58 
58 

58 
58 
57 
58 
57 

58 



Log. Cot. 



67 252 
67 190 
67 128 
67 066 
67 004 



66 943 
66 881 
66 819 
66 758 
66 696 



66 635 
66 574 
66 513 
66 452 
66 390 



66 330 
66 269 
66 208 
66 147 
66 C86 



66 026 
65 965 
65 905 
65 845 
65 784 



65 724 
65 6e4 
65 604 
65 544 
65 484 



65 424 
65 364 
65 305 
65 245 
65 186 



65 126 
65 067 
65 008 
64 948 
64 889 



64 830 
64 771 
64 712 
64 653 
64 594 



64 536 
64 477 
64 418 
64 360 
64 302 



64 243 
64 185 
64 127 
64 068 
64 010 

63 952 
63 894 
63 837 
63 779 
63 721 



0.63 663 



Log. Tan. 



Log. Cos. 



99 040 
99 038 
99 035 
99 032 
99 029 



99 027 
99 024 
99 021 
99 019 
99 016 



99 013 
99 010 
99 008 
99 005 
99 002 



98 999 
98 997 
98 994 
98 991 
98 988 



98 986 
98 983 
98 t80 
98 977 
98 975 



98 972 
98 969 
98 966 
98 963 
98 961 



98 958 
98 955 
98 952 
98 949 
98 947 



98 944 
98 941 
98 938 
98 935 
98 933 



98 930 
98 927 
98 924 
98 921 
98 918 



98 915 
98 913 
98 910 
98 907 
98 904 



98 901 
98 898 
98 895 
98 892 
98 890 



98 887 
98 884 
98 881 
98 878 
98 875 



9. 98 872 



Log. Sin. 
659 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



40 

39 
38 
37 
J6 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26 



20 

19 
18 
17 
16 



P. P. 





62 


61 


6 


62 


6.1 


7 


7 


2 


72 


8 


8 


2 


82 


9 


9 


3 


9-2 


10 


10 


3 


10.2 


20 


20 


6 


20.5 


30 


31 


C 


SO 7 


4C 


41 


3 


41.0 


50 


51 


6 


51. 2 



61 

6-1 

71 

8.1 

91 

10.1 

20 3 

30-5 

40.6 

50.8 





60 


60 


59 


5fl 


6 


6.0 


6-0 


5 9 


5. 


7 


7 





7 





6 


9 


6- 


8 


8 





8 





7 


9 


7- 


9 


9 


1 


9 





8 


9 


8. 


10 


10 


1 


10 





9 


9 


9. 


20 


20 


1 


20 





19 


8 


19. 


30 


30 


2 


80 





29 


7 


29. 


40 


40 


3 


40 





S9 


6 


39. 


50 


50 


4 


50 





49 


6 


49 





58 


58 


57 


57 


6 


5.8 


5 8 


5.7 


57 


7 


6 


8 


6 


7 


6 


7 


6 


6 


8 


7 


8 


7 


7 


7 


6 


7 


6 


9 


8 


8 


8 


7 


8 


6 


8 


5 


10 


9 


7 


9 


6 


9 


6 


9 


5 


20 


19 


5 


19 


3 


19 


1 


19 





30 


29 


2 


29 





28 


7 


28 


5 


40 


39 





38 


6 


38 


3 


38 





50 


48 


7 


48 


3 


47 


9 


47 


5 



6 

7 

8 

9 

10 

20 

30 

40 

50 



56_ 

5-6 



56 

5.6 



55_ 

5.5 



55 

5-5 





54 


3 


2 


6 


5.4 


03 


0. 


7 


6 


3 





3 


0. 


8 


7 


2 





4 


0. 


9 


8 


2 





4 


0. 


10 


9 


] 





g 





20 


18 


1 


1 


C 


0. 


30 


27 


2 


1 


5 


1. 


40 


36 


1- 


2 


C 


1. 


50 


45 


4 


2 


5 


2. 



77' 



13° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



166° 



35 
36 
37 9 



Log. Sin. 



35 209 
35 263 
35 318 
35 372 
35 427 



35 481 
35 536 
35 590 
35 644 
35 698 



35 752 
35 806 
35 860 
35 914 
35 968 



36 021 
36 075 
36 128 
36 182 
36 235 



36 289 
36 342 
36 395 
36 448 
36 501 



36 554 
36 607 
36 660 
36 713 
36 766 



36 818 
36 871 
36 923 

36 976 

37 028 



37 081 
37 133 
37 185 
37 237 
37 289 



37 341 
37 393 
37 445 
37 497 
37 548 



37 600 
37 652 
37 703 
37 755 
37 806 



37 857 
37 909 

37 960 

38 011 
38 062 



38 113 
38 164 
38 215 
38 266 
38 317 



9. 38 367 



Log, Cos. 



54 
54 
54 
54 

54 
54 
54 
54 
54 
54 
54 
54 
53 
54 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
52 
53 
52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
51 
52 
5l 
52 
5l 
5l 
51 
51 

51 
5l 
51 
5l 
51 

51 
51 
50 
51 
51 

50 



Log. Tan. 



36 
9 36 
9. 36 
9.36 
9.36 



9. 36 
9.36 
9-36 
36 
9. 36 



9. 36 
9. 36 
9.37 
9.37 
9.37 



9.87 
9.37 
9. 37 
9.37 
9. 37 



9. 37 
9.37 
9. 37 
9. 37 
9.37 



9.37 
9. 37 
9.37 
9. 37 
9-37 



9. 38 
9. 38 
9-38 
9-38 
9. 38 



336 
394 
451 
509 
566 

623 
681 
738 
795 
852 

909 
966 
023 
080 
136 

193 
250 
306 
363 
419 

475 
532 
588 
644 
700 

756 
812 
868 
924 
979 

035 
091 
146 
202 
257 



9. 38 
9. 38 
9-38 
9. 38 
9-38 



313 
368 
423 
478 
533 



c.d. 



9-38 
9. 38 
9.38 
9. 38 
9. 38 



589 
644 
698 
753 
808 



9. 38 
9. 38 

9. 38 
9.39 

9. 39 



863 
918 
972 
027 
081 



9. 39 
9. 39 
9. 39 
9. 39 
9. 39 



9. 39 
9. 39 
9.39 
9.39 
9. 39 



136 
190 
244 
299 
3_5| 

407 
46l 
515 
569 
623 



9-39 
Log. 



677 
Cot. 



57 
57 
57 
57 
57 
57 
57 
57 
57 

57 
57 
56 
57 
56 

57 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
55 
56 
56 
55 

56 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
54 
55 
55 

54 
55 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

53 



Log. Cot. 



63 663 
63 606 
63 548 
63 491 
63 433 



63 376 
63 319 
63 262 
63 204 
63 147 



0.63 090 
0.63 033 
0.62 977 
0.62 920 
0.62 863 



62 806 
62 750 
62 693 
62 637 
.62 580 



0.62 524 
. 62 468 
0.62 412 
0.62 356 
0.62 299 



0.62 243 
0.62 188 
0.62 132 
0.62 076 
. 62 020 



0.61 
0.61 
0.61 
0.61 
061 



964 
909 
853 
798 
742.9 



Log. Cos. 



0.61 
0.61 
061 
061 
061 



687 
632 
576 
521 
466 



0-61 411 
061 356 
0.61 301 
0-61 246 
061 19l 



61 137 
61 082 
61 027 
60 973 
60 918 



0.60 864 
0.60 809 
0.60 755 
0-60 701 
0.60 647 



0-60 592 
0.60 538 
0-60 484 
0.60 430 
0-60 376 



n.fiO 323 



Log. Tan 



98 872 
98 869 
98 866 
98 863 
98 860 



98 858 
98 855 
98 852 
98 849 
98 846 



98 843 
98 840 
98 837 
98 834 
98 831 



98 828 
98 825 
98 822 
98 819 

98 816 



98 813 
98 810 
98 807 
98 804 
98 80l 



98 798 
98^795 
98 792 
98 789 
98 786 



98 783 
98 780 
98 777 
98 774 
98 771 



98 768 
98 765 
98 762 
98 759 
98 755 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 
42 

M. 
40 
39 
38 
37 
36 



98 752 
98 749 
98 746 
98 743 
98 740 



98 737 
98 734 
98 731 
98 728 
98 725 



98 721 
98 718 
98 715 
98 712 
98 709 



98 706 
98 703 
98 700 
98 696 
98 693 



98 690 
Log. Sin. 



30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
li 

15 
14 
13 
12 
11 



10 

9 
8 
7 
6 



P.p. 





57 


57 


56 


56 


6 


5 7 


5 7 


5.6 


5. 


7 


6 


7 


6 


6 


6 


g 


6. 


8 


7 


6 


7 


6 


7 


5 


7. 


9 


8 


6 


8 


5 


8 


5 


8. 


10 


9 


6 


9 


5 


9 


4 


9. 


20 


19 


1 


19 





18 


3 


18. 


30 


28 


7 


28 


5 


28 


2 28. 


40 


38 


3 


38 





37 


6 37. 


50 


47 


9 


47 


5 


47 


1'46. 





55 


55 


54 


54 


6 


5.5 


5.5 


5.4 


5.4 


7 


6.5 


6 


4 


6 


3 


6 


3 


8 


7.4 


7 


c 


7 


2 


7 


2 


9 


8 3 


8 


2 


8 


2 


8 


1 


30 


9.2 


9 


1 


9 


1 


9 





20 


18.5 


18 


3 


18 


1 


18 





30 


27.7 


27 


5 


27 




27 





40 


37.0 


36 


6 


36 


3 


36 





50 


46.2 


45 


8 


45 


4 


45 








53 


53 


52 


52 


6 


5.3 


5.3 


5.2 


5. 


7 


6 


2 


6 


2 


6 


1 


6. 


8 


7 


1 


7 





7 





6. 


9 


8 





7 


g 


7 


9 


7. 


10 


8 


g 


8 


3 


8 


7 


8. 


20 


17 


8 


17 


6 


17 


5 


17 


80 


26 


7 


26 


5 


26 


2 


26 


40 


35 


6 


35 


3 


35 





34 


50 


44 


6 


44 


1 


43 


7 


43 



51 

5.1 



51 



50 

5.0 





3 


3 




6 


0.3 


0.31 


7 


0.4 





3 


8 


0.4 





4 


9 


0.5 





4 


10 


0.6 





5 


20 


1.1 


1 





30 


1.7 


1 


5 


40 


2.3 


2 





50 


2.9 


2 


5 



2 

0.2 
0.3 
0.3 
0.4 
0.4 
0.8 
1.2 
1.6 
2.1 



P.P. 



103*' 



660 



761 



TABLE VII —LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



165® 



Log. Sin. 



38 367 
38 418 
38 468 
38 519 
38 569 



38 620 
38 675 
38 720 
38 771 
38 821 



38 871 
38 921 

38 971 

39 021 
39 071 



39 125 
39 170 
39 220 
39 269 
39 319 



39 368 
39 418 
39 467 
39 516 
39 566 



39 615 
39 664 
39 713 
39 762 
39 811 



39 860 
39 909 

39 957 

40 006 
40 055 



40 103 
40 152 
40 200 
40 249 
40 297 



40 345 
40 394 
40 442 
40 490 
40 538 



40 586 
40 634 
40 682 
40 730 
40 777 



40 825 
40 873 
40 920 

40 968 

41 015 



41 063 
41 110 
41 158 
41 205 
41 252 



9 .41 299 



Log. Cos. 



d. 



50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
48 
48 
49 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 

47 
48 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

T 



Log. Tan, 



39 677 
39 731 
39 784 
39 838 
39 892 



39 945 

39 999 

40 052 
40 106 
40 159 



40 212 
40 265 
40 318 
40 372 
40 425 



40 478 
40 531 
40 583 
40 636 
40 689 



40 742 
40 794 
40 847 
40 899 
40 952 



41 004 
41 057 
41 109 
41 16l 
41 213 



41 266 
41 318 
41 370 
41 422 
41 474 



41 525 
41 577 
41 629 
41 681 
41 732 



41 784 
41 836 
41 887 
41 938 
41 990 



42 04l 
42 092 
42 144 
42 195 
42 246 



42 297 
42 348 
42 399 
42 450 
42 501 



42 552 
42 602 
42 653 
42 704 
42754 

42 805 



Log. Cot. 



c. d. 

54 
53 
54 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
52 
53 
52 

53 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

51 
52 
52 
5l 
5l 

51 
52 
5l 
51 
5l 
5l 
51 
51 
51 
51 

51 
51 
51 
51 
50 

51 
50 
51 
50 
50 
50 



Log. Cot 



60 323 
60 269 
60 215 
60 16l 
60 108 



60 054 
60 001 
59 947 
59 894 
59 841 



59 787 
59 734 
59 68l 
59 628 
59 575 



59 522 
59 469 
09 416 
59 363 
59 311 



258 

20 

153 

IOC 

048 

995 
943 
891 
838 
786 



58 734 
58 682 
58 63C 
58 578 
58 526 



58 474 
58 422 
58 37C 
58 319 
58 2R7 



58 216 
58 164 
58 112 
58 061 
58 OIC 



958 
907 
856 
805 
753 
702 
65l 
600 
549 
499 



57 448 
57 397 
57 346 
57 296 
57 245 



0.57 195 



Log, Tan. 



Log. Cos, 



68 6GC 
98 687 
98 684 
98 681 
98 678 



98 674 
98 671 
98 668 
98 665 
98 662 



98 658 
98 655 
98 652 
98 649 
98 646 



98 642 
98 639 
98 636 
98 633 
98 630 



98 626 
98 623 
98 620 
98 617 
98 613 



98 610 
98 607 
98 604 
98 600 
98 597 



98 594 
98 591 
98 587 
98 584 
98 581 



98 578 
98 574 
98 571 
98 568 
98 564 



98 561 
98 558 
98 55i 
98 55l 
98 548 



98 544 
98 54l 
98 538 
98 534 
98 531 



98 528 
98 524 
98 521 
98 518 
98 514 



98 5ll 
98 508 
98 504 
98 501 
98 498 



9-98 494 



Log. Sin. d. 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 



30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 
2 
1 
O 



P. P. 



8 

7 

8 

9 

10 

20 

30 

40 

50 



54 

5.4 

63 

7.2 

8.1 

9.0 

18. 

27.0 

36.0 

C 



45 



53 

5.5 



53 

5.3 





5? 


I 


53 


5T 


51 


6 


5.2 


5.2 


5.1 


5. 


7 


6 


1 


6 





6 





5. 


8 


7 





6 


9 


6 


8 


6. 


9 


7 


9 


7 


8 


7 


7 


7. 


10 


8 


7 


8 


6 


8 


6 


8. 


20 


17 


5 


17 


3 


17 


\ 


17. 


30 


26 


^ 


26 




25 


7 


25. 


40 


35 





34 


g 


34 


3 


34. 


50 


43 


7 


43 


3 


42 


9 


42. 





^^A 


50 


49 


49 


6 


5.d 


5.0 


4-9 


4. 


7 


5 


9 


5 


8 


5 


8 


5. 


8 


6 


7 


6 


g 


6 


6 


6. 


9 


7 


6 


7 


5 


7 


4 


7. 


10 


8 


4 


8 


3 


8 


2 


8. 


20 


16 


8 


16 


6 


16 


5 


16. 


30 


25 


2 


25 





24 


7 


24. 


40 


33 


6 


33 


3 


33 





32. 


50 


42 


1 


41 


6 


41 


2 


40. 





48 


48 


47 


6 


4.8 


48 


4.7 


7 


5 


6 


5 


6 


5 


5 


8 


6 


4 


6 


4 


6 


3 


9 


7 


3 


7 


2 


7 


1 


10 


8 


1 


8 





7 


9 


20 


16 


1 


16 





15 


8 


30 


24 


2 


24 





23 


7 


40 


32 


3 


32 





31 


6 


50 


40 


4 


40 





39 


6 



47 

4.7 
5.5 
6.2 
7.0 





3 




6 


0.31 


7 





4 


8 





4 


9 





5 


10 





g 


20 


1 


1 


30 


1 


7 


40 


2 


3 


50 


2 


9 



3 

0.3 
0.3 
0.4 



20 
2.5 



h. P. 



661 



75' 



15' 



TABLE VIT— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



164' 



Log. Sin. 



O 

1 

2 

3 

_£ 

5 

6 

7 

8 

J_ 

10 

11 

12 

13 

ii 

15 

16 

17 

18 

19 



41 299 
41 346 
41 394 
9.41441 
9. 41 488 



20 

21 

22 

23 

24_ 

25 

26 

27 

28 

21 

30 

31 

32 

33 

31 

35 
36 
37 
38 
39 



d. 



Log. Tan. 



• 41 534 
.41 581 
.41 628 
.41 675 
.41 72T 



.41 768 
.41815 
.41 861 
.41908 
.41 954 



9. 42 000 
9.42 047 
9.42 093 
9.42 139 
9.42 185 



42 232 
42 278 
.42 324 
.42 369 
9.42 415 



9.42 461 
9.42 507 
9.42 553 
9. 42 598 
9-42 644 



42 690 
.42 735 
.42 781 
.42 826 
• 42 87l 



.42 917 
.42 962 
.43 007 
.43 052 
.4.S OQ*? 



40 

41 
42 
43 
44 

45 

46 

47 

48 

49_ 

50 

51 

52 

53 

51 

55 
56 
57 
58 
59, 
60 



43 143 
43 188 
9-43 233 
9. 43 278 
9.43 322 



43 367 
43 412 
9.43 457 
9.43 501 
9-43 546 



.43 591 
.43 635 
9.43 680 
9.43 724 
.43 768 



9.43 813 
9. 43 857 
9.43 90T 
9.43 945 
9. 43 989 



9.44 034 



47 
47 
47 
47 

46 
47 
47 
46 
46 
47 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 
45 
46 

46 
46 
45 
45 
46 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
44 

45 
44 
45 
44 
44 

45 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 



42 805 

42 856 

9.42 906 

9.42 956 

9.43 007 



9.43 057 
9.43 107 
9.43 157 
9-43 208 
9. 43 258 



c.d. 



9.43 308 
9.43 358 
9.43 408 
9.43 458 
9.43 508 

9.43 557 
.43 607 
.43 657 
.43 706 
.43 756 



.43 806 
43 855 

9.43 905 
9-43 954 

9.44 003 



9.44 053 
9.44 102 
9.44 151 
9.44 200 
9-44 249 



44 299 
44 348 

9.44 397 
44 446 

9. 44 494 



44 543 
9.44 592 
9-44 641 
9.44 690 
9.44738 



9.44 787 
9.44 835 
9.44 884 
9.44 932 
9.44 981 



9.45 029 
9.45 077 
45 126 
9-45 174 
9-45 222 



45 270 
45 318 
45 367 
45 415 
45 463 



Log. Cos, 



.45 510 
-45 558 
9-45 606 
9-45 654 
9-45 702 



9-45 749 



51 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
49 
49 
50 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
48 

49 
49 
48 
a.d 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

47 
48 
48 
47 
48 

47 



Log. Cot. 



Log. Cos. 



57 195 

57 144 

57 094 

.57 043 

.56 993 



56 942 
56 892 
.56 842 
.56 792 
.56 742 



0-56 
0-56 
0.56 
0.56 
0-56 



692 
642 
592 
542 
492 



0.58 
0.56 
0.56 
0.56 
0.56 



442 
392 
343 
293 

243 



98 494 
98 491 
98 487 
98 484 
98 481 



.98 477 
.98 474 
.98 470 
.98 467 
-98 464 



.98 460 
98 457 
.98 453 
.98 450 
-98 446 



0.56 194 
0.56 144 
0.56 095 
0.56 045 
0.55 996 



0.55 
0.55 
0.55 
0.55 
0.55 



947 
898 
848 
799 
750 



98 443 
98 439 
.98 436 

• 98 433 

• 98 429 



98 426 
98 422 
98 419 
98 415 
98 412 



0.55 
0.55 
0.55 
55 
0.5.'> 



701 
652 
603 
554 
605 



0.b5 456 
0.55 407 
0.55 359 
0.55 310 
0-55 261 



Log. Cot 



.98 408 
98 405 
• 98 401 
.98 398 
-98 394 



.98 391 
.98 387 
-98 384 
.98 380 
.98 377 



98 373 
98 370 
.98 366 
.98 363 
.98 359 



0.55 213 
0.55 164 
0.55 116 
0.55 067 
0.55 019 



0.54 970 
0.54 922 
0^.54 874 
0.54 825 
0-54 777 



0-54 
0.54 
0.54 
0.54 
0-54 



729 
681 
633 
585 
537 



c.d. 



0-54 
0-54 
0-54 
0-54 
0-54 



489 
441 
393 
346 
298 



054 250 



98 356 
98 352 
98 348 
98 345 
98 341 



.98 338 
.98 334 
.98 331 
.98 327 
.98 324 



.98 320 
.98 316 
.98 313 
-98 309 
.98 306 



.98 302 
.98 298 
.98 295 
.98 29! 
.98 288 



998 284 



Log. Tan. Log. Sin. 



60 

59 
58 
57 
56 



55 

54 
53 
52 
51 

50 

49 
48 
47 
46 



40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 
30 
19 
18 
17 

li 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 
2 
_1 




106' 



662 



P.P. 



50 



6 

7 
8 

9 
10 
20 
30 
40 
50 



5-0 


5. 


5.9 


5. 


6.7 


6. 


7.6 


7. 


8.4 


8. 


16.8 


16. 


25.2 


25. 


33.6 


33. 


42.1 


41. 



50 


8 
6 
5 
3 
6 

3 
6 





49 


49 


48 


48 


6 


4.9 


4.9 


4.8 


48 


7 


5 


8 


5.7 


5.6 


5 


6 


8 


6 


6 


6.5 


6.4 


6 


4 


9 


7 


4 


7.3 


7.3 


7 


2 


10 


8 


2 


8.1 


8.1 


8 





20 


16 


5 


16.3 


16.1 


16 





30 


24 


7 


24.5 


24.2 


24 





40 


33 





32.6 


32.3 


32 





50 


41 


2 


40.8 


40.4 


40 






47_ 

4-7 

5.5 

6.3 

7.1 

7.9 

15-8 

23-7 

31-0 

39.6 



47 

4.7 



46_ 

4.6 

5.4 

6-2 

7.0 

7.7 

15.5 

23.2 

31.0 

38.7 



46 

46 





45 


45 


44 


44 


6 


4-5 


4-5 


4.4 


4. 


7 


5 


3 


5-2 


5.2 


5. 


8 


6 


d 


6-0 


5.9 


5. 


9 


6 


8 


6-7 


6.7 


6. 


10 


7 


6 


7-5 


7.4 


7- 


20 


15 


1 


15-0 


14.8 


14- 


30 


22 


7 


22-5 


22.2 


22 


40 


30 


3 


30.0 


29.6 


29- 


50 


37 


9 


37.5 


37.1 


i36. 





4 


3 


a 


6 


0.4 


0.3 


0. 


7 


0.4 


0.4 


0. 


8 


0.5 


0.4 


0. 


9 


0.6 


0.5 


0. 


10 


0.6 


0.6 


0. 


20 


1 .3 


1.1 


1. 


30 


2.0 


1.7 


1. 


40 


2.6 


2.3 


2 


50 


3.3 


2.9 


2. 



P.P. 



74^1 



16^ 



TABLE VIl— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



163' 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

a 

12 
13 

M 

15 
16 
17 
18 
19 



log. Sin. 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39, 

40 

41 
42 
43 
4£ 

45 
46 
47 
48 
49^ 

50 

51 
2 
3 
4 



9.44 034 
9-44 078 
9-44 122 
9.44 166 
9-44 209 



44 253 
44 297 
44 341 
44 384 
44 428 



44 472 
44 515 
44 559 
44 602 
44 646 



44 689 
44 732 
44 776 
44 819 
44 862 



44 905 
44 948 

44 991 

45 034 
45 077 



45 120 
45 163 
45 206 
45 249 
45 291 



45 334 
45 377 
45 419 
45 462 
45 504 



45 547 
45 589 
45 631 
45 674 
45 716 



45 758 
45 800 
45 842 
45 885 
45 927 



45 969 

46 Oil 
46 052 
46 094 
46 136 



46 178 
46 220 
46 261 
46 303 
46 345 



9-46 386 
9-46 428 
9-46 469 
9.46 511 
9.46 552 



9^46 593 
Cos. 



Log. 



44 
44 
44 
43 
44 
44 
43 
43 
44 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43^ 

43 
43 

43 
42 
43 
43 
42 

42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
41 
42 
42 
4l 
42 
41 
41 
42 
41 
41 
41 
41 
41 

4l 



Log. Tan. 



9-45 749 
9-45 797 
9-45 845 
9-45 892 
9-45 940 



45 987 

46 035 
9-46 082 
9-46 129 
9-46 177 



9-46 224 
9.46 271 
46 318 
46 366 
46 413 



46 460 
46 507 
46 554 
46 601 
46 647 

46 694 
46 741 
46 788 
9-46 834 
9j_46 881 

9-46 928 
9-46 974 
9-47 021 
9 47 067 
9-47 114 



9-47 160 
9-47 207 
9-47 253 
9-47 299 
9-47 345 



47 392 
47 438 
47 484 
47 530 
47 576 



47 622 
47 668 
47 714 
47 760 
47 806 



47 851 
47 897 
47 943 

47 989 

48 034 



48 080 
48 125 
48 171 
48 216 
48 262 



48 307 
48 353 
48 398 
48 443 
48 488 



9-48 534 
Log. Cot. 



c. d. 

48 

47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
46 

47 
47 
46 
46 
47 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

46 
46 
45 
46 
46 

45 
46 
45 
46 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 



Log. Cot. Log. Cos 



0-54 250 
0-54 202 
0-54 155 
0-54 107 
0-54 060 



0-54012 
0-53 965 
0-53 917 
0.53 870 
0-53 823 



53 776 
53 728 
53 681 
53 634 
53 587 



0.53 540 
0-53 493 
0.53 446 
53 399 
0-53 352 



0-53 305 
0.53 258 
0.53 212 
0-53 165 
0-53 118 



53 072 
53 025 
52 979 
52 932 
52 886 



0-52 839 
0-52 793 
0-52 747 
0-52 700 
0-52 654 



0-52 608 
0-52 562 
0-52 516 
0-52 469 
0.52 423 



9-98 284 
9-98 280 
9-98 277 
9.98 273 
9.98 269 



9.98 266 
9.98 262 
9.98 258 
9-98 255 
9-98 25l 



98 247 
98 244 
98 240 
98 236 
98 233 



9-98 229 
9-98 225 
9.98 222 
9-98 218 
9-98 214 



9.98 211 
9.98 207 
9.98 203 
9-98 200 
9-98 196 



98 192 
98 188 
98 185 
98 181 
98 177 



98 173 
98 170 
9.98 166 
9.98 162 
9.98 158 



52 377 
52 331 
52 286 
52 240 
52 194 



0.52 148 
0.52 102 
0.52 057 
0-52 Oil 
0-5] 965 



0-5] 920 
05] 874 
0-51 829 
0-51 783 
0-51 738 



0-51 692 
0-51 647 
0-51 602 
0-51 556 
0-51 5ll 



0-51 466 



Log. Tan, 



98 155 
98 151 
98 147 
98 143 
98 140 



9-98 ],36 
9-98 132 
9-98 128 
9.98 124 
9-98 121 



98 117 
98 113 
98 109 
98 105 
98 102 



9-98 098 
9-98 094 
98 090 
9-98 086 
9. 98 082 



9-98 079 
9-98 075 
9-98 071 
9-98 067 
9-98 063 



9-98 059 
Log. Sin. 



60 

59 
58 
57 
56 



55 
54 
53 
52 
-51 
50 
49 
48 
47 
-46 
45 
44 
43 
42 

11 
40 

39 

38 

37 

J6_ 

35 
34 
33 
32 
31 

30 

29 
28 

27 
26 

25 
24 
23 
22 

21 
20 

19 
18 

17 
jL6 

15 
'14 
13 
12 
1] 

10 

9 

8 

7 

_6^ 

5 
4 
3 
2 

1 



P. P. 





48 


47 


6 


4.8 


4-71 


7 


5 


6 


5 


5 


8 


6 


4 


6 


3 


9 


7 


2 


7 


1 


10 


8 





7 


9 


20 


16 





15 


3 


30 


24 





23 


7 


40 


32 





31 


6 


50 


40 





39 


6 



47 

4.7 

5.5 

6-2 

7.0 

7 8 

15-6 

23.5 

31.3 

39.1 





46 


46 


45 


6 


4.6 


4.6 


4-51 


7 


5 


4 


5 


3 


5 


3 


8 


6 


2 


6 


1 


6 





9 


7 





6 


g 


6 


8 


10 


7 


7 


7 


g 


7 


6 


20 


15 


5 


15 


3 


15 




30 


23 


2 


23 





22 


7 


40 


31 





30 


6 


30 


3 


50 


38 


7 


38 


3 


37 


9 



45 

4.5 

5-2 

6-0 

6.7 

7-5 

15-0 

22-5 

30-0 

37-5 



6 

7 

8 

9 

10 

20 

30 

40 

50 



44 

4.4 



43 

4-3 



43 

4-3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



43_ 

4-2 



42 



4-9 

5-6 

6.4 

7-1 

14-1 

21-2 

28-3 

35-4 



41_ 

4-1 

4-8 

5-5 

6-2 

6-9 

13-8 

20-7 

27-6 

34-6 



41 

4-1 



4 

5 

6 

6 

13 

20 

27 

34. 





4 


6 


0-4 


7 


0-4 


8 


0-5 


9 


0-6 


10 


0-6 


20 


1-3 


30 


2-0 


40 


2-6 


50 


3-3 



3 

0-3 
0-4 
0-4 
0-5 
0-6 

11 
1-7 
2-3 
2.9 



P.P. 



jioe' 



663 



73*" 



17** 



TABLE VII -LOGARITHMIC SINES, COSINES. TANGENTS 
AND COTANGENTS. 



163® 



O 

1 

2 

3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19 
30 9 

21 9 

22 9 



9.46 593 
9.46 635 
9-46 676 
46 717 
9.46 758 



Log. Sin. 



.46 799 
.46 840 
.46 881 
.46 922 
.46 963 



.47 004 
.47 045 
.47 086 
.47 127 
.47 168 



.47 208 
.47 249 
.47 290 
.47 330 
.47 371 



23 
21 
25 
26 
27 
28 
29 

30 

31 

32 

33 

3i 

35 

36 

37 

38 9 

319 

40 9 

41 

42 

43 

4i 

45 

46 

47 

48 

49 

60 

51 
52 
53 
5i 
55 
56 
57 
58 
59 

22 



.47 411 
.47 452 
.47 492 
.47 532 
.47 573 



.47 613 
.47 653 
• 47 694 
.47 734 
.47 774 



4l 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
40 
41 

40 
40 
41 
40 
40 
40 
40 
40 
40 
40 

40 
40 
40 
40 
40 



9. 48 534 
9.48 579 
9-48 624 
9. 48 669 
48 714 



Log. Tan, 



c.d. 



9.48 759 
9.48 804 
9-48 849 
9. 48 894 
9. 48 939 



9. 48 984 

9. 49 028 
9.49 073 
9.49 118 
9-49 162 



Log. Cot 



9-49 207 
49 252 
49 296 
9.49 341 
9.49 385 



.47 814 
.47 854 
.47 894 
.47 934 
.47 974 



.48 014 
.48 054 
.48 093 
.48 133 
■ 48 173 



9.49 430 
9.49 474 
9-49 518 
9.49 563 
9.49 607 



.48 213 
.48 252 
.48 292 
.48 331 
.48 371 



.48 410 
.48 450 
.48 489 
.48 529 
.48 568 



9-48 607 
48 646 
9-48 686 
9.48 725 
9.48 764 



9-48 803 
9.48 842 
9.48 881 
9.48 920 
48 959 



107^ 



9. 48 998 
Loc:. Cos. 



40 
40 
40 
40 
40 

40 
40 
39 
4C 
39 
40 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 

39 
39 
39 
39 
39 
38 

T 



9. 49 651 
9.49 695 
9.49 740 
9.49 784 
9. 49 828 



.49 872 
.49 916 
.49 960 
.50 004 
.50 048 



50 092 
50 136 
50 179 
9-50 223 
9-50 267 



9.50 311 
9-50 354 
50 398 
9 . 50 442 
9.50 485 



45 
45 
45 
45 

45 
45 
44 
45 
45 

45 
44 
45 
44 
44 

45 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
43 
44 

44 
44 
43 
44 
43 

44 
43 
43 
44 
43 



51 466 

.51421 

.51376 

51 330 

51 285 



.51 240 
.51 195 
.51 151 
.51 106 
51 061 



0.51 016 
0.50 971 
0.50 926 
0.50 882 
0.50 837 



Log. Cos 



98 059 

98 056 

9.98 052 

9. 98 048 

9-98 044 



9-98 040 
9.98 036 
9. 98 032 
9-98 028 
9.98 024 



50 792 

50 748 

.50 703 

.50 659 

.50 614 



50 570 
50 525 
0. 50 481 
0.50 437 
0. 50 392 



98 021 
.98 017 
.98 013 
.98 009 
9,98 0^ 

9 



98 001 
97 997 
97 993 
97 989 
97 985 



9.97 981 
9-97 977 
9.97 973 
97 969 
9. 97 966 



0.50 348 
0-50 304 
0-50 260 
0-50 216 
0.50 172 



50 128 
50 083 
. 50 039 
.49 996 
.49 952 



.97 962 
.97 958 
.97 954 
.97 950 
.97 946 



9-97 942 
9.97 938 
9-97 934 
9-97 930 
9-97 926 



0.49 908 
. 49 864 
0.49 820 
0.49 776 
049 733 



.97 922 
.97 918 
.97 914 
.97 910 
.97 906 



9.50 529 
9.50 572 
9.50 616 
9.50 659 
9-50 702 



50 746 
50 789 

9.50 832 
50 876 

9.50 919 



9.50 962 

9.51 005 
9.51 048 
9. 51 091 
9.51 134 



9-51 177 
Log. Cot 



43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 



. 49 .689 
. 49 645 
. 49 602 
049 558 
0.49 514 



9.97 902 
9.97 898 
9.97 894 
9.97 890 
9.97 886 



0.49 471 
0.49 427 
0.49 384 
0.49 340 
0-49 297 



0.49 254 
0.49 210 
0.49 167 
0.49 124 
0.49 081 



97 881 
9.97 877 
9.97 873 
9.97 869 
9.97 865 



9.97 861 
9.97 857 
9-97 853 
97 849 
9. 97 845 



0.49 038 
0.48 994 
0.48 951 
0..48 908 
0-48 865 



0-48 822 
Log. Tan 



9.97 841 
9.97 837 
97 833 
9.97 829 
9-97 824 



9.97 820 



3 
4 
4 
4 

3 
4 
4 
4 
4 

3 

4 

4 

4. 

4 

4 

3 

4 

4 

4 

4 

4 

4 

4 

3 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 
4 
4 
4 

4 

4 
4 
4 
4 
4 
4 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



P.P. 



10 
20 
30 
40 
50 



45. 

4-5 



45 

4-5 



44 

4.4 

5-2 

5-9 

6-7 

7.4 

14.8 

22.2 

29.6 

37.1 



44 

44 



Log. Sin 
664 



45 
44 
43 
42 
41 

40 

39 
38 

37 

35 
34 
33 
32 
II 
30 
29 
28 
27 
1^ 
25 
24 
23 
22 
-21 
30 
19 
18 
17 

li 

15 
14 
13 
12 
il 

10 

9 

8 

7 

_6 

5 
4 
3 
2 

J, 
O 



6 

7 

8 

9 

10 

20 

30 

40 

50 



41 

4.1 
4.8 
5-5 
6.2 
69 



43 43 



4.3 


4. 


3 


5.1 


5 





58 


5 


7 


6.5 


6 


4 


7-2 


7 


1 


14-5 


14 


3 


21-7 


21 


5 


29.0 


28 


. 6 


136.2 


35 


8 



20 13.8 
30 20.7 



41 

4.1 



■ 6 
•5 
40:27. § 27-3 
50 34. 6 34. 1 



40 

4.0 
4-7 
5-4 



40 

4.0 



d. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



39 


39 


38 


3 9 


39 


38 


4 


(\ 


4 


5 


4 


5 


5 


2 


5 


2 


5 


1 


5 


9 


5 


8 


5 


8 


6 


6 


6 


5 


6 


4 


13 


T 


13 





12 


8 


19 


7 


19 


5 


19 


2 


26 


• 3 


26 


.0 


25 


.6 


32 


9 


32 


5 


32 


•1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



4 3_ 

0-40.3 
40.4 
0-5 0.4 
0-60 

60 

31 

1 

6 2 



3.3 29 



K. P. 



7»^ 



TABLE VII.— LOGARIT?IMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 161* 



Log. Sin. 



9. 48 
49 



998 
037 
076 
114 
153 



192 
231 
269 
308 
346 

385 
423 
462 
500 
539 

577 
615 
653 
692 
730 



768 
806 
844 
882 
920 



958 
996 
034 
072 
110 



147 
185 
223 
260 
298 



336 
373 
411 
448 
486 



523 
561 
598 
635 
672 



710 
747 
784 
821 
858 



895 
932 
969 
006 
043 



080 
117 
154 
190 

227 



51 264 



Log. Cos 



39 
39 
38 
39 

38 
39 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 
38 
38 
37 
38 
38 

37 
38 
37 
37 
38 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 

37 
36 
37 
36 
37 

36 



Log. Tan. 



51 177 
51 220 
51 263 
51 306 
51 349 



51 392 
51 435 
51477 
51 520 
51 563 



51 605 
51 648 
51 691 
51 733 
51 776 



51 818 
51 861 
51 903 
51 946 
51 988 



52 030 
52 073 
52 115 
52 157 
52 199 



52 241 
52 284 
52 326 
52 368 
52 410 



52 453 
52 494 
52 536 
52 578 
52 619 



52 661 
52 703 
52 745 
52 787 
52 828 



52 870 
52 912 
52 953 

52 995 

53 036 



53 078 
53 119 
53 161 
53 202 
53 244 



53 285 
53 326 
53 368 
53 409 
53 450 



53 491 
53 533 
53 574 
53 615 
53 656 



9-53 697 
Log. Cot. 



c, d. 



43 
43 
43 
43 
42 
43 
42 
43 
42 

42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
41 

42 
42 
41 
42 
41 

4l 
42 
4l 
4l 
4l 

4l 
41 
41 
41 
41 

4l 
41 
41 
41 
4l 

41 
41 
41 
41 
41 

41 
c. d, 



Log. Cot. 



48 822 
48 779 
48 736 
48 693 
48 650 



48 608 
48 565 
48 522 
48 479 
48 437 



48 394 
48 351 
48 309 
48 266 
48 224 



48 181 
48 139 
48 096 
48 054 
48 012 



47 969 
47 927 
47 885 
47 842 
47 800 



47 758 
47 716 
47 674 
47 632 
47 59C 



47 548 
47 506 
47 464 
47 422 
47 380 



47 338 
47 296 
47 255 
47 213 
47 171 



47 130 
47 088 
47 046 
47 005 
46 963 



46 922 
46 880 
46 839 
46 797 
46 756 



46 714 
46 673 
46 632 
46 591 
46 549 



46 508 
46 467 
46 426 
46 385 
46 344 



46 303 
Log, Tan. 



Log. Cos, 



97 820 
97 816 
97 812 
97 808 
97 804 



97 800 
97 796 
97 792 
97 787 
97 783 



97 779 
97 775 
97 771 
97 767 
97 763 



97 758 
97 75i 
97 750 
97 746 
97 742 



97 737 
97 733 
97 729 
97 725 
97 721 



97 716 
97 712 
97 708 
97 704 
97 700 



97 695 
97 691 
97 687 
97 683 
97 678 



97 674 
97 670 
97 666 
97 661 
97 657 



97 653 
97 649 
97 644 
97 640 
97 636 



97 632 
97 627 
97 623 
97 619 
97 614 



97 610 
97 606 
97 601 
97 597 
97 593 



97 588 
97 584 
97 580 
97 575 
97 571 



9-97 567 
Log. Sin. 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



35 
34 
33 
32 
_3I 
30 
29 
28 
27 
26 



25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 



10 

9 
8 

7 
_i 

5 
4 
3 
2 
_1 

O 



P. P. 





43 


43 


6 


4.3 


4-2 


7 


5 





4-9 


8 


5 


7 


5.6 


9 


6 


4 


6.4 


10 


7 


1 


7.1 


20 


14 


3 


14.1 


30 


21 


5 


21.2 


40 


28 


6 


28.3 


50 


35 


8 


35.4 



43 

4.2 
4.9 
5.6 
6.3 



4 


I 


4 


3 


5 


5 


6 


2 


6 




13 


3 


20 


7 


27 


6 


34 


6 



41 

4.1 

48 

5.4 

6.1 

68 

13.6 

20.5 

27.3 

34.1 





39 


38 


3? 


6 


3.9 


3.8 


3. 


7 


4 


5 


4.5 


4. 


8 


5 


2 


5.1 


5. 


9 


5 


3 


5.8 


5. 


10 


6 


5 


6.4 


6. 


20 


13 





12.8 


12. 


30 


19 


5 


19.2 


19. 


40 


26 





25.6 


25. 


50 


32 


5 


32.1 


31. 





37 


37 


36 


6 


3.7 


3.7 


3.6 


7 


4 


4 


4 


3 


4 


2 


8 


5 





4 


9 


4 


8 


9 


5 


6 


5 


5 


5 


5 


10 


6 


2 


6 


1 


6 


1 


20 


12 


5 


12 


3 


12 


1 


30 


18 


7 


18 


5 


18 


2 


40 


25 




24 


(3 


24 


3 


50 


31 


2 


30 


8 


30 


4 





^ 




6 


0.41 


7 





5 


8 





6 


9 





7 


10 





7 


20 


1 


5 


30 


2 


2 


40 


3 





50 


3 


7 



2.6 
3.3 



P. P. 



665 



71* 



19° 



TABLE Vll— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



160' 



Log. Sin. 



51 264 
51 301 
51 337 
51 374 
51 410 



51447 
51 483 
51 520 
51 556 
51 593 



51 629 
51 665 
51 702 
51 738 
51 774 



51 810 
51 847 
51 883 
51 919 
51 955 



51 991 

52 027 
52 063 
52 099 
52 135 



52 170 
52 206 
52 242 
52 278 
52 314 



52 349 
52 385 
52 421 
52 456 
52 492 



52 527 
52 563 
52 598 
52 634 
52 669 



52 704 
52 740 
52 775 
52 810 
52 848 



52 881 
52 916 
52 951 

52 986 

53 021 



53 056 
53 091 
53 126 
53 161 
53 196 



53 231 
53 266 
53 301 
53 335 
53 370 



9-53 405 



Log.' Cos. 



d. Log. Tan 



37 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

35 
36 
36 
35 
36 

35 
35 
36 
35 
35 

35 
35 
35 
35 
35 

35 
35 

35 
35 
35 

35 

35 
35 
35 
35 

35 
35 
35 
35 
35 

34 
35 
35 
34 
35 

34 



53 697 
53 738 
53 779 
53 820 
53 861 



9 
9^ 

9.53 902 
9-53 943 
9-53 983 
9-54 024 
9-54 065 



9-54 106 
9-54 147 
9-54 187 
9-54 228 
9-54 269 

9-54 309 
9-54 350 
9-54 390 
9-54 431 
9-54 471 



54 714 
54 754 
9-54 794 
9-54 834 
9-54 874 



54 512 
54 552 
54 593 
54 633 
54 673 



9-54915 
54 955 

54 995 

55 035 
55 075 



55 115 
55 155 
55 195 
55 235 

55 275 



55 315 
55 355 
55 394 
9-55 434 

55 474 



9-55 514 
9-55 553 
55 593 
9-55 633 
9-55 672 



55 712 
55 751 
55 791 
55 831 
55 870 



55 909 
55 949 

55 988 

56 028 
56 067 



9.56 106 



Log. Cot. 



c.d. 



41 
41 
41 
41 

41 
41 
40 
41 
41 

40 
41 
40 
40 
41 

40 
40 
40 
40 
40 
40 

^:o 

40 
40 
40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 

40 
39 
40 
40 
40 

40 
40 
39 
40 
39 

40 
39 
40 
39 
39 

39 
39 
40 
39 
39 

39 

39 
39 
39 
39 

39 



Log. Cot. 



46 303 
46 262 
46 221 
46 180 
46 139 



0-46 098 
0-46 057 
0-46 016 
0-45 975 
0-45 934 



45 894 
45 853 
45 812 
45 772 
45 731 



0-45 690 
0-45 650 
0-45 609 
0-45 569 
0-45 528 



0-45 488 
0-45 447 
0-45 407 
0-45 367 
0.45 326 



0-45 286 
0-45 246 
0-45 205 
0-45 165 
0-45 125 



45 085 
45 045 
0-45 005 
0-44 965 
0-44 925 



Log. Cos. d 



97 567 
97 562 
97 558 
97 554 
97 549 



97 545 
97 541 
97 536 
97 532 
97 527 



97 523 
97 519 
97 514 
97 510 
97 505 



97 501 
97 497 
97 492 
97 488 
97 483 



97 479 
97 475 
97 470 
97 466 
97 461 



97 457 
97 452 
97 448 
97 443 
97 439 



97 434 
97 430 
97 425 
97 421 
97 416 



44 884 
44 845 
44 805 
44 765 
44 725 



44 685 
44 645 
44 605 
44 565 
44 526 



44 486 
44 446 
44 406 
44 367 
44 327 



0-44 288 
0-44 248 
0.44 208 
0-44 169 
0.44 129 



44 090 
44 051 
0-44 011 
0-43 972 
0.43 93? 



0.43 893 



Log. Tan, 



97 412 
97 407 
97 403 
97 398 

97 394 



97 389 
97 385 
97 380 
97 376 
97 371 



97 367 
97 362 
97 358 
97 353 
97 349 



97 344 
97 340 
97 335 
97 330 
97 326 



97 321 
97 317 
9.97 312 
9.97 308 
9.97 303 



9.97 298 



Log. Sin. 



d. 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 



35 
34 
33 
32 
11 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 

n 

10 

9 
8 

7 
_6 

5 
4 
3 
2 
_1 





P. P. 



41 

4-1 

4 

5 

6 

6 
13 
20 
27 
34 



40_ 

4.0 



40 

4.0 



39 



3 


9 


3- 


4 


g 


4. 


5 


2 


5. 


5 


9 


5- 


6 


6 


6- 


13 


1 


13- 


19 


7 


19- 


26 


3 


26- 


32 


9 


32. 



39 

9 
5 
2 
8 
5 

5 

5 





37 


36 


36 


6 


3-7 


3-6 


3- 


7 


4 


3 


4 


2 


4- 


8 


4 


9 


4 


8 


4- 


9 


5 


5 


5 


5 


5- 


10 


6 


1 


6 


1 


6- 


20 


12 


3 


12 


1 


12- 


30 


18 


5 


18 


2 


18- 


40 


24 


6 


24 


3 


24- 


50 


30 


8 


30 


4 


30- 



35 35 34 1 


6 


3-5 


3-51 3-4 1 


7 


4 


1 


4-1 


4 





8 


4 


7 


4-6 


4 


6 


9 


5 


3 


5-2 


5 


2 


10 


5 


9 


5-8 5 


7 


20 


11 


8 


11-6 11 


5 


30 


17 


7 


17-5 17 


2 1 
1 


40 


23 


5 


23-3 23 


50 


29-6 


29.128-7 J 






5 


? 


4 


i 






5 





4 





g 





5 





6 





6 





7 





7 





8 





7 


1 


6 


1 


5 


2 


5 


2 


2 


3 


3 


3 





4 


1 


3 


7 



0-4 
0-5 
0-6 
0-6 
1-3 
2.0 
2.6 
3-3 



P.P. 



109° 



666 



70' 



20° 



TABLE VIl— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



159'' 



Log. Sin. 



53 405 
53 440 
53 474 
53 509 
53_544 

53 578 
53 613 
53 647 
53 682 
53 716 



53 750 
53 785 
53 819 
53 854 
53 888 
53 922 
53 956 

53 990 

54 025 
54 059 



54 093 
54 127 
54 161 
54 195 
54 229 



54 263 
54 297 
54 331 
54 365 
54 398 



54 432 
54 466 
54 500 
54 534 
54 567 



54 601 
54 634 
54 668 
54 702 
54 735 



54 769 
54 802 
54 836 
54 869 
54 902 



54 936 

54 969 

55 002 
55 036 
55 069 



55 102 
55 135 
55 168 
55 202 
55 235 



55 268 
55 301 
55 334 
55 367 
55 400 



9 • 55 433 



Log. Cos, 



35 
34 
34 
35 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

33 
34 
34 
34 
33 
34 
34 
33 
34 
33 

33 
33 
34 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 
33 



Log. Tan, 



9-56 
56 



106 
146 
185 
224 
263 

303 
342 
381 
420 
459 
498 
537 
576 
615 
654 



693 
732 
771 
810 
848 



887 

926 
965 
003 
042 

081 
119 
158 
196 
235 



57 274 
57 312 
57 350 
57 38? 
57 427 



466 
504 
542 
581 
619 



657 
696 
734 
772 
810 

848 
886 
925 
963 
001 




Log. Cot. 



c.d. 



39 
39 
39 
39 

39 
39 
39 
3? 
39 

39 
39 
39 
39 
38 

39 
39 
39 
39 
38 
39 
38 
3? 
38 
38 

39 
38 
38 
38 
38 

39 
38 
38 
38 
38 

38 
38 
38 
38 
38 
38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
37 

38 
38 
38 
37 
38 

37 
c.d. 



Log. Cot. 



43 893 
43 854 
43 815 
43 775 
43 736 



43 697 
43 658 
43 619 
43 580 
43 540 



43 501 
43 462 
43 423 
43 384 
43 346 



43 307 
43 268 
43 229 
43 19C 
43 15l 



43 112 
43 074 
43 035 
42 996 
42 958 



42 919 
42 88C 
42 842 
42 803 
42 765 



42 726 
42 687 
42 648 
42 611 
42 572 



42 534 
42 495 
42 457 
42 419 
42 38C 



42 34 
42 304 
42 266 
42 227 
42 189 



42 151 
42 113 
42 075 
42 037 
41 999 

41 961 
41 923 
41 885 
41 847 
41 809 



41 771 
41 733 
41 695 
41 658 
41 620 



0.41 582 



Log. Tan. 



Log. Cos, 



97 298 
97 294 
97 289 
97 285 
97 280 



97 275 
97 271 
97 266 
97 261 
97 257 
97 252 
97 248 
97 243 
97 238 
97 234 



97 229 
97 224 
97 220 
97 215 
97 210 



97 206 
97 201 
97 196 
97 191 
97 187 



97 182 
97 177 
97 173 
97 168 
97 163 



97 159 
97 154 
97 149 
97 144 
97 140 



97 135 
97 130 
97 125 
97 121 
97 116 



97 111 
97 106 
97 102 
97 097 
97 092 



97 087 
97 082 
97 078 
97 073 
97 068 



97 063 
97 058 
97 054 
97 049 
97 044 



97 03? 
97 034 
97 029 
97 025 
97 020 



9.97 015 



110' 



Log. Sin 
667 



d. 



60 

59 
58 
57 
-56_ 

55 
54 
53 
52 
51 

50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
16 

35 
34 
33 
32 
11 
30 
29 
28 
27 
-26 

25 
24 
23 
22 
21 

20 

19 
18 
17 

JA 
15 
14 
13 
12 

JJ, 

10 

9 
8 

7 
_6 

5 
4 
3 
2 
1 





P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



39 

3-9 



39 

3 





38 


38 


37 


6 


3.8 


38 


3. 


7 


4 


5 


4 


4 


4. 


8 


5 


1 


5 





5. 


9 


5 


8 


5 


7 


5. 


10 


6 


4 


6 


3 


6. 


20 


12 


8 


12 


g 


12. 


30 


19 


2 


19 





18. 


40 


25 


6 


25 


3 


25. 


50 


32 


1 


31 


6 


31. 





35 


34 


34 


6 


3-5 


3.4 


3.4 


7 


4 


1 


4 





3 


9 


8 


4 


6 


4 


6 


4 


5 


9 


5 


2 


5 


2 


5 


1 


10 


5 


3 


5 


7 


5 


6 


20 


11 


6 


11 


5 


11 


3 


30 


17 


5 


17 


2 


17 





40 


23 


3 


23 





22 


g 


50 


29 


1 


28 


7 


28 


3 



33_ 

33 



33 

3.3 



6 

7 

8 

9 

10 

20 

30 

40 

50 






5 


0. 





6 


0. 





6 


0. 





7 


0. 





3 


0. 


1 


6 


1. 


2 


5 


2. 


3 


3 


3 


4 


1 


3. 



P. p. 



69' 



21* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 158* 



Log. Sin. I d, iLog. Tan. c, d. Log. Cot, Log. Cos. 



55 433 
55 466 
55 498 
55 531 
55 564 



55 597 
55 630 
55 662 
55 695 
55 728 



55 760 
55 793 
55 826 
55 858 
55 891 



55 923i 
55 956 

55 988 

56 020 
56 0531 



56 085 
56 118 
56 150 
56 182 
56 214 



56 247 
56 279 
56 311 
56 343 
56 375 



56 407 
56 439 
56 471 
56 503 
56 535 



56 567 
56 599 
56 631 
56 663 
56 695 



56 727 
56 758 
56 790 
56 822 
56 854 



56 885 
56 917 
56 949 

56 980 

57 012 



57 043 
57 075 
57 106 
57 138 
57 169 



57 201 
57 232 
57 263 
57 295 
57 326 



57 357 



Log. Cos. 



33 
32 
33 
33 

32 
33 
32 
33 
32 

32 
32 
33 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
3l 
32 

32 
3l 
32 
3l 

32 

31 
31 
32 
31 
31 

3l 
31 
31 
31 
3l 

3l 
3l 
31 
31 
31 

31 



58 417 
58 455 
58 493 
58 531 
58 58R 



58 606 
58 644 
58 681 
58 719 
58 756 



58 794 
58 831 
58 869 
58 906 
58 944 



58 981 

59 019 
59 056 
59 093 

59 T3T 



59 168 
59 205 
59 242 
59 280 
59 317 



59 354 
59 391 
59 428 
59 465 
59 502 



59 540 
59 577 
59 614 
59 651 
59 688 



59 724 
59 761 
59 798 
59 835 
59 872 



59 909 

59 946 
5a 982 

60 019 
60 056 



60 093 
60 129 
60 166 
60 203 
60 239 



60 276 
60 312 
60 349 
60 386 
60 422 



60 459 
60 495 
60 531 
60 568 
60 604 



9 

9 . 60 641 



Log. Cot. 



38 
37 
38 
37 

37 
38 
37 
37 
37 

37 
37 
37 

37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

36 
37 
37 
37 
36 

37 
37 
36 
37 
36 

37 
36 
37 
36 
36 
36 
36 
37 
36 
36 

36 
36 
36 
36 
36 
36 



c.d. 



41 582 
41 544 
41 507 
41 469 
41 431 



41 394 
41 356 
41 318 
41 281 
41 243 



41 206 
41 168 
41 131 
41 093 
41 056 



41 018 
40 981 
40 944 
40 906 
40 8RP 



40 832 
40 794 
40 757 
40 720 
40 683 



40 646 
40 608 
40 571 
40 534 
40 497 



40 460 
40 423 
40 386 
40 349 
40 312 



40 275 
40 238 
40 201 
40 164 
40 12R 



40 091 
40 054 
40 017 
39 980 
39 944 



39 907 
39 870 
39 833 
39 797 
39 760 



39 724 
39 687 
39 650 
39 614 
39 577 



39 541 
39 504 
39 468 
39 432 
39 395 



0.39 359 



Log, Tan, 



97 015 
97 010 
97 005 
97 000 
96 995 



96 991 
96 986 
96 981 
96 976 
96 971 



96 966 
96 961 
96 956 
96 952 
96 947 



96 942 
96 937 
96 a32 
96 927 

96 922 



96 917 
96 912 
96 907 
96 902 
96 897 



96 892 
96 887 
96 882 
96 877 
96 873 



96 868 
96 863 
96 858 
96 853 
96 848 



96 843 
96 838 
96 833 
96 828 
96 823 



96 818 
96 813 
96 808 
96 802 
96 797 



96 792 
96 787 
96 782 
96 777 
96 772 



96 767 
96 762 
96 757 
96 752 
96 747 



96 742 
96 737 
96 732 
96 727 
96 721 



9.96 716 



Log. Sin, 



60 

59 
58 
57 
_56 

55 
54 
53 
52 
51 

50 

49 
48 

47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
3i 
35 
34 
33 
32 
31 

30 

29 
28 
27 
26. 
25 
24 
23 
22 
21 
30 
19 
18 
17 
16_ 
15 
14 
13 
12 
U 
10 
9 
8 
7 
_6 

5 
4 
3 

2 

O 



P. P. 





38 


37 


6 


38 


3-71 


7 


4-4 


4 


4 


8 


5.0 


5 





9 


5.7 


5 


g 


10 


6.3 


6 


^ 


20 


12.6 


12 


5 


30 


19.0 


18 


7 


40 


25.3 


25 





50 


31-6 


31 


2 



37 

37 



4. 

4. 

5 

6. 
12. 
18. 
24.6 
30.8 





36 


36 


6 


3.6 


3.6 


7 


4 


2 


4 


2 


8 


4 


3 


4 


8 


9 


5 


5 


5 


4 


10 


6 


1 


6 





20 


12 


1 


12 





30 


18 


2 


18 





40 


24 


3 


24 





50 


30 


4 


30 






33 


33 


3.3 


3.2| 


3.8 


3 


8 


4.4 


4 


3 


4.9 


4 


9 


5.5 


5 


4 


11.0 


10 


g 


16.5 


16 


2 


22.0 


21 


6 


27.5 


27 


1 



3JR 

3.2 

3.7 

4.2 

48 

5.3 

10-6 

16.0 

21.3 

26.6 



31 

3.1 

3.7 

4.2 

4.7 

5.2 

10.5 

15.7 

21.0 

26.2 



31 

31 

3 

4 

4 

5 
10 
15 
20 
25 



7 
8 
9 

10 
20 
30 
40 
50 



5 


5 


^ 


0.5 


0.5 


0. 


0.6 


0-6 


0. 


0.7 


0.6 


0. 


0.8 


0-7 


0. 


0.9 


0.8 


0. 


1.8 


1-6 


1. 


2.7 


2.5 


2. 


3.6 


3-3 


3. 


4.6 


41 


3. 



P. p. 



tOc 



es'' 



23' 



TABLE VII —LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



157' 





1 
2 
3 

5 
6 
7 
8 
J^ 

10 

11 
12 
13 
11 
15 
16 
17 
18 

IL 

20 

21 

22 

23 

24. 

25 

26 

27 

28 

29, 

30 

31 

32 

33 

34 

35 

36 

37 

38 

3P. 

40 

41 

42 

43 

44 



Log. Sin. 

9. 57 357 
9-57 389 
9-57 420 
9.57 451 
9. 57 482 



57 513 
57 544 
57 576 
57 607 
57 638 



57 669 
57 700 
57 731 
57 762 
57 792 



57 823 
57 854 
57 885 
57 916 
57 947 



57 977 

58 ooe 

58 039 
58 070 
58 100 



58 131 
58 162 
58 192 
58 223 
58 253 



45 
46 
47 
48 

50 

51 
52 
53 
51 
55 
56 
57 
58 
59_ 

60 



58 284 
58 314 
58 345 
9-58 375 
9. 58 406 



9.58 436 
9.58 466 
9-58 497 
9.58 527 
9-58 557 



58 587 

58 618 

9-58 648 

9-58 678 

9-58 708 



9-58 738 
9-58 769 
9-58 799 
9.58 829 
9. 58 859 

9.58 889 
58 919 
58 949 

58 979 

59 009 



59 038 
59 068 
59 098 
9. 59 128 
9.59 158 



9-59 188 



Log. Cos. 



31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
30 

31 
31 
31 
30 
31 

30 
31 
30 
31 
30 

30 
31 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

29 
30 
30 
29 
30 
30 



Log. Tan. 



60 641 
60 677 
60 713 
60 750 
60 786 



9.60 822 
9. 60 859 
9.60 895 
9. 60 931 
9- 60 967 



61 003 
61 039 
61 076 
61 112 
61 148 



61 184 
61 220 
9.61 256 
9-61 292 
Q.61 328 



61 364 
9-61 400 
9-61 436 
9-61 472 
9-61 507 



61 543 
61 579 
61 615 
61 651 
61 686 



9-61 722 
9.61 758 
9.61 794 
9.61 829 
9. 61 865 



61 901 
61 936 

61 972 

62 007 

9-62 043 



9.62 078 
62 114 
9.62 149 
9.62 185 
9.62 220 



9.62 256 
62 29l 
9.62 327 
9- 62 362 
9.62 397 



62 433 
62 468 
62 503 
62 539 
62 574 



9-62 609 
9 . 62 644 
9.62 679 
62 715 
62 750 



c.d. 



9. 62 785 
Log. Cot. 



36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
35 

36 
36 
35 
36 
35 

36 
35 
36 
35 
35 

36 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 



Log. Cot. 



0-39 359 
0-39 322 
0-39 286 
0-39 250 
0-39 213 



0-39 177 
0-39 141 
0-39 105 
0-39 068 
0-39 C32 



38 996 
38 960 
38 924 
38 888 
38 852 9 



Log. Cos. 



96 716 
96 711 
96 706 
96 701 
96 696 



96 691 
96 686 
96 681 
96 675 
96 670 



0-38 816 
038 780 
0.38 744 
038 708 
0.38 672 



0.38 636 
0.38 600 
0-38 564 
038 528 
0-38 492 



38 456 
38 420 
38 385 
38 349 
38 313 



0.38 277 
0.38 242 
0.38 206 
038 170 
0-38 135 



96 665 
96 660 
96 655 
96 650 
96 644 

96 639 
96 635 
96 629 
96 624 
96 619 



■ 96 613 
96 608 
96 603 
96 598 
96 593 



96 587 
96 582 
96 577 
96 572 
96 567 



0-38 099 
0-38 063 
0.38 028 
0.37 992 
0-37 957 



37 921 
37 886 
0.37 850 
0.37 815 
0-37 779 



0.37 744 
0.37 708 
037 673 
0.37 637 
0.37 602 



37 567 
37 531 
37 496 
37 461 
37 426 



0-37 390 
0-37 355 
0.37 320 
037 285 
0.37 250 



96 561 
96 556 
96 551 
96 546 
96 540 



96 535 
96 530 
96 525 
96 519 
96 514 



96 509 
96 503 
96 498 
96 493 
96 488 



9.96 482 
9.96 477 
9.96 472 
996 466 
9-96 461 



9-96 456 
9-96 450 
9-96 445 
9-96 440 
P-96 434 



0-37 215 



Log. Tan, 



9-96 429 
9-96 424 
9-96 418 
9-96 413 
9-96 408 



9-96 402 
Log. Sin. 



d. 

5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

5 



60 

59 
58 
57 

55 
54 
53 
52 

50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
31 
35 
34 
33 
32 
IL 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

30 

19 
18 
17 
16 



10 

9 
8 
7 
6 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



36 

3.6 



36 

3 





35 


35 


6 


3.5 


3.5 


7 


4 


•1 


4-1 


8 


4 


7 


4.6 


9 


5 


3 


5.2 


10 


5 


9 


5.8 


20 


11 


g 


11.6 


30 17 


7 


17-5 


40 23 


6 


23-3 


5029 


6 


29.1 






30 


30 


6 


3.0 


3-0 


7 


3 


• 5 


3.5 


8 


4 





4.0 


9 


4 


6 


4.5 


10 


5 


1 


5.0 


20 


10 


1 


10.0 


30 


15 


2 


15.0 


40 


20 


3 


20.0 


50 


25 


4 


25.0 



39 

2.9 

3-4 

3.9 

4.4 

4.9 

9.8 

14.7 

19.6 

24. i 



6 


0. 


7 


0. 


8 


0. 


9 


0. 


10 


0. 


20 


1. 


30 


2- 


40 


3. 


50 


4. 



5 5 

5(0.5 
60.6 
7;0.6 
8,0.7 
9 0.8 
8a. 6 
7,2.5 
63.3 
6i4.1 



P.P. 



113^ 



669 



Of 



y^ 



33' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 156« 



Log. Sin. 



59 188 
59 217 
59 247 
59 277 
59 306 



59 336 
59 366 
59 395 
59 425 
59 454 



59 484 
59 513 
59 543 
59 572 
59 602 



59 631 
59 661 
59 690 
59 719 
59 749 



59 778 
59 807 
59 837 
59 866 
59 895 



59 924 
59 953 

59 982 

60 012 
60 041 



60 070 
60 099 
60 128 
60 157 
60 186 



60 215 
60 244 
60 273 
60 301 
60 330 



60 359 
60 388 
60 417 
60 445 
60 474 



60 503 
60 532 
60 560 
60 589 
60 618 



60 646 
60 675 
60 703 
60 732 
60 760 



60 789 
60 817 
60 846 
60 874 
60 903 



60 93l 



Log. Cos 



29 
30 
29 
29 
30 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29. 
29 
28 
29 

29 
28 
29 
28 
29 

28 
29 
28 
28 
29 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 



Log. Tan. c. d 



62 785 
62 820 
62 855 
62 890 
62 925 



62 960 

62 995 

63 030 
63 065 
63 100 



63 135 
63 170 
63 205 
63 240 
63 275 



63 310 
63 344 
63 379 
63 414 
63 449 



63 484 
63 518 
63 553 
63 588 
63 622 



63 657 
63 692 
63 726 
63 761 
63 795 



63 830 
63 864 
63 899 
63 933 
63 968 



64 002 
64 037 
64 071 
64 106 
64 140 



64174 
64 209 
64 243 
64 277 
64 312 



64 346 
64 380 
64 415 
64 449 
64 483 



64 517 
64 551 
64 585 
64 620 
64 654 



64 688 
64 722 
64 756 
64 790 
64 824 



64 858 



Log. Cot. 



35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
34 
35 

35 
34 
35 
35 

34 

35 

34 
34 
35 
34 
34 
35 
34 
34 
34 

34 
34 
34 
34 
34 
34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 



Log. Cot. 



37 215 
37 179 
37 144 
37 109 
37 074 



37 039 
37 004 
36 969 
36 934 
36 899 



36 864 
36 829 
36 794 
36 76C 
36 725 



36 69C 
36 655 
36 620 
36 585 
36 551 



36 516 
36 481 
36 447 
36 412 
36 377 



36 343 
36 308 
36 273 
36 230 
36 204 



36 170 
36 135 
36 101 
36 066 
36 032 



35 997 
35 963 
35 928 
35 894 
35 8 



35 825 
35 791 
35 756 
35 722 
35 688 



35 658 
35 619 
35 585 
35 551 
35 517 



35 482 
35 44F. 
35 414 
35 38C 
35 346 



35 312 
35 278 
35 244 
35 209 
35 175 



0.35 141 



Log. Tan 



Log. Cos. 



96 402 
96 397 
96 392 
96 386 
96 381 



96 375 
96 370 
96 365 
96 359 
96 354 

96 349 
96 343 
96 338 
96 332 
96 327 



96 321 
96 316 
96 311 
96 305 
96 300 



96 294 
96 289 
96 283 
96 278 
96 272 



96 267 
96 261 
96 256 
96 251 
96 245 



96 240 
96 234 
96 229 
96 223 
96 218 



96 212 
96 206 
96 201 
96 195 
96 190 



96 184 
96 179 
96 173 
96 168 
96 162 



96 157 
96 151 
96 146 
96 140 
96 134 



96 129 
96 123 
96 118 
96 112 
96 106 



96 101 
96 095 
96 090 
96 084 
96 078 



9.96 073 



log. Sin 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 

35 
34 
33 
32 
_31 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

30 

19 
18 
17 
16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 
4 
3 
2 
1 





P.P. 



35 



10 



6 

7 

8 

9 

10 

20 

30 

40 

50 



3 


5 


3. 


4 


1 


4. 


4 


7 


4. 


5 


3 


5. 


5 


9 


5. 


11 


3 


11. 


17 


7 


17. 


23 


6 


23. 


29 


6 


29. 



34 



3 


4 


3. 


4 





3. 


4 


6 


4. 


5 


2 


5. 


5 


7 


5 


11 


5 


11. 


17 


2 


17. 


23 





22. 


28 


7 


28. 



35 

5 
1 
6 
2 
8 
6 
5 
3 
I 



34 

4 
9 
5 
1 
6 
3 

6 
3 



6 


3. 


7 


3. 


8 


4. 


9 


4. 


10 


5. 


20 


10. 


30 


15. 


4C 


20. 


50 


25. 



30 


5 

5 








I _ 





39 


39 


6 


2.9 


2.9 


7 


3 


4 


3.4 


8 


3 


9 


38 


9 


4 


4 


4.3 


10 


4 


9 


4.8 


20 


9 


8 


9.6 


30 


14 


7 


14.5 


40 


19 


6 


19.3 


50 


24 


6 


24.1 



6 


5 


60.610. 51 


70 


70 


6 


80 


80 


7 


90 


90 


8 


10 1 


olo 


9 


20 2 





1 


8 


30 3 





2 


7 


40 4 


0i3 


6 


50 5 





4 


6 



38. 

2.8 

3.3 

38 

4.3 

4.7 

9.5 

14.2 

19.0 

23.7 



5 

0.5 
0.6 
0.6 
0.7 
0.8 
1.6 
2.5 
3.3 
4-1 



P.P. 



113* 



670 



66? 



1ABLE VII.— LOGARITHMIC SINES, COSINES, 
AND COTANGENTS. 



TANGENTS, 



155° 



' Log. Sip. d. Log. Tan. c.d. Log. Cot. Log. Cos 



60 931 
60 959 

60 988 

61 016 
61 044 



61 073 
61 101 
61 129 
61 157 
61 186 



61 214 
61 242 
61 270 
61 298 
61 326 



61 354 
61 382 
61 410 
61438 
61 466 



61 494 
61 522 
61 550 
61 578 
61 606 



61 634 
61 661 
61 689 
61 717 
61 745 



61 772 
61 800 
61 828 
61 856 
61 883 

61 911 
61 938 
61 966 

61 994 

62 021 



62 049 
62 076 
62 104 
62 131 
62 158 



62 186 
62 213 
62 241 
62 268 
62 295 



62 323 
62 350 
62 377 
62 404 
62 432 



62 459 
62 486 
62 513 
62 540 
62 567 



962 595 



Log. Cos. 



28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
27 
28 
28 
28 
27 
28 
27 
28 

27 
28 
27 
28 
27 

27 
27 
27 
28 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 



64 858 
64 892 
64 926 
64 960 
64 994 



65 028 
65 062 
65 096 
65 129 
65 163 



65 197 
65 231 
65 265 
65 299 
65 332 



65 366 
65 400 
65 433 
65 467 
65 501 



65 535 
65 568 
65 602 
65 635 
65 669 



65 703 
65 736 
65 770 
65 803 
65 837 



65 870 
65 904 
65 937 

65 971 

66 004 



66 037 
66 071 
66 104 
66 137 
66 171 



66 204 
66 237 
66 271 
66 304 
66 337 



66 370 
66 404 
66 437 
66 470 
66 503 



66 536 
66 570 
66 603 
66 636 
66 669 



66 702 
66 735 
66 768 
66 801 
66 834 



JLog. 



66 867 
Cot. 



34 
34 
33 
34 

34 
34 
34 
33 
34 

34 
33 
34 
34 
33 

34 
33 
33 
34 
33 
34 
33 
33 
33 
33 

34 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 
33 



0-35 
35 



141 
107 
073 
040 
006 



972 
938 
904 
870 
836 
802 
769 
735 
701 
667 



633 
600 
566 
532 
499 



465 
431 
398 
364 
331 



297 
263 
230 
196 
163 



129 
096 
062 
029 
996 

962 
929 
895 
862 

829 



795 
762 
729 
696 
662 

629 
596 
563 
529 
496 



Log. 



463 
430 
397 
364 
331 

298 
265 
232 
198 
165 
132 
Tan. 



96 073 
96 067 
96 062 
96 056 
96_050 

96 045 
96 039 
96 033 
96 028 
96 022 



96 016 
96 011 
96 005 
95 999 
95^94 

95 988 
95 982 
95 977 
95 971 
95 965 



95 959 
95 954 
95 948 
95 942 
95 937 



95 931 
95 925 
95 919 
95 914 
95 908 



95 902 
95 896 
95 891 
95 885 
95 879 



95 873 
95 867 
95 862 
95 856 
95 850 



95 844 
95 838 
95 833 
95 827 
95 821 



95 815 
95 809 
95 804 
95 798 
95 792 



95 786 
95 780 
95 774 
95 768 
95 763 



95 757 
95 751 
95 745 
95 739 
95 733 



9-95 727 
Log. Sin. 



d. 

5 
5 
6 
5 

5 
6 
5 
5 
6 

5 
5 
6 
5 
5 
6 
5 
5 
6 
5 
6 
5 
6 
5 
5 
6 
5 
6 
5 
6 

5 
6 
5 
6 
6 

5 
6 

5 
6 

. 5 

6 
6 
5 
6 
5 
6 
6 
5 
6 
6 

5 
6 
6 
6 
5 

6 
6 
6 
5 
6 



60 

59 
58 
57 
56_ 
55 
54 
53 
52 
II 
50 
49 
48 
47 
j46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 

32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

30 

19 
18 

17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 

2 

1 



P. P. 





34 


33 


3( 


6 


3.4 


3.3 


3. 


7 


3.9 


3 


9 


3. 


8 


4.5 


4 


4 


4. 


9 


5.1 


5 





4. 


10 


5.6 


5 


6 


5. 


20 


11.3 


11 


1 


11. 


30 


17.0 


16 


7 


16. 


40 


22.6 


22 


3 


22. 


50 


28.3 


27 


9 


27. 



38 



6 

7 

8 

9 

10 

20 

30 

40 

50 



2 


8 


2. 


3 


3 


3- 


3 


8 


3. 


4 


3 


4. 


4 


7 


4. 


9 


5 


9. 


14 


2 


14. 


19 





18. 


23 


7 


23. 



21 



2-7 


2. 


3 


2 


3. 


3 


6 


3. 


4 


1 


4. 


4 


6 


4- 


9 


1 


9. 


13 




13. 


18 


3 


18. 


22 


9 


22. 



28 
8 
2 
7 
2 
6 
3 

6 
3 



27 
7 
I 
6 

5 

5 

5 



6 5 






60. 





7,0. 





8 


0. 





9 


0. 


1 





0. 


2 





1. 


3 





2. 


4 





3. 


5 





4. 



P. p. 



14= 



671 



65^ 



26*^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



154' 



Log. Sin.l d. 



62 595 
62 622 
62 649 
62 676 
62 703 



62 730 
62 757 
62 784 
62 811 
62 838 



62 864 
62 891 
62 918 
62 945 
62 972 



62 999 

63 025 
63 052 
63 079 
63 106 



63 132 
63 159 
63 186 
63 212 
63 239 



63 266 
63 292 
63 319 
63 345 
63 372 



63 398 
63 425 
63 451 
63 478 
63 504 



63 530 
63 557 
63 583 
63 609 
63 636 



662 
688 
715 
741 
767 
793 
819 
846 
872 
898 



924 
950 
976 
002 
028 



054 
080 
106 
132 
158 



64 184 



Log. Cos, 



Log. Tan. c. d. Log. Cot 



867 
900 
933 
966 



68 



Log. 



032 
065 
097 
130 
163 

196 
229 
262 
294 
327 
360 
393 
425 
458 
491 

523 
556 
589 
621 
654 

687 
719 
752 
784 
817 
849 
882 
914 
947 
979 

012 
044 
077 
109 
141 

174 
206 
238 
271 
303 

335 
368 
400 
432 
464 

497 
529 
561 
593 
625 

657 
690 
722 
754 
786 

818 
Cot. 



32 
33 
33 
33 

33 
33 
32 
33 
33 

33 
32 
33 
32 
33 
32 
33 
32 
33 
32 

32 
33 
32 
32 
33 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 

32 
32 
32 
32 

32 
32 
32 
32 
32 

32 

32 
32 
32 
32 
32 
32 
32 
32 
32 

32 
c. d. 



0.33 132 
0.33 100 
0.33 067 
. 33 034 
0^_83_P01 

0-32 968 
0-32 935 
0.S2 902 
0-32 869 
0^32^836 
. 32 803 
0.32 771 
0.32 738 
0.32 705 
0.32 672 



. 32 640 
0.32 607 
0.32 574 
0-32 541 
0.32 509 



0.32 476 
0.32 443 
0.32 411 
0.32 378 
0.32 345 



0.32 313 
0.32 280 
0.32 248 
0.32 215 
0.32 183 



0.32 150 
0.32 118 
0.32 085 
0.32 053 
0.32 020 



0.31 
0.31 
0.31 
0..31 
0.31 



0-31 
0.31 
0.31 
0.31 
0.31 



988 

955 
923 
891 

82e 
79£ 
761 
729 
696 



0.31 
0.31 
0.31 
0.31 
0.31 



664 
632 
600 
567 
535 

503 
471 
439 
406 
374 



0.31 342 
0.31 310 
0.31 278 
0.31 246 
0.31 214 



0-31 182 



Log. Tan, 



Log. Cos, 



95 727 
95 721 
95 716 
hd 710 
m 704 



95 698 
95 692 
95 686 
95 680 
95 674 



95 668 
95 662 
95 656 
95 650 
95 644 



95 638 
95 632 
95 627 
95 621 
95 615 



95 609 
95 603 
95 597 
95 591 
95 585 



95 579 
95 573 
95 567 
95 561 
95 555 



95 549 
95 543 
95 537 
95 530 
95 524 



95 518 
95 512 
95 506 
95 500 
95 494 



95 488 
95 482 
95 476 
95 470 
95 464 



95 458 
95 452 
95 445 
95 439 
95 433 



95 427 
95 421 
95 415 
95 409 
95 403 



95 397 
95 390 
95 384 
95 "378 
95 372 
9-95 366 



Log. Sin, 



d. 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



35 
34 
33 
32 
31 
30 
29 
28 
27 
26 



30 

19 
18 
17 
16 



10 

9 

8 

7 

_6 

5 
4 
3 
2 
_1^ 




P. P. 





33 


33 


32 


6 


33 


3.2 


3. 


7 


3 


8 


38 


3. 


8 


4 


4 


4.3 


4. 


9 


4 


9 


4.9 


4. 


IC 


5 


5 


5.4 


5. 


2C 


11 





10.8 


10. 


3C 


16 


5 


16.2 


16. 


4C 


22 





21.6 


21. 


50 


27 


5 


27-1 


26. 





27 


6 


2.7 


7 


3.1 


8 


3.6 


9 


4.0 


10 


4.5 


20 


9.0 


30 


13.5 


40 


18.0 


50 


22.5 





2B 


26 


2, 


6 


2.6 


2.6 


2. 


7 


3 


1 


30 


3. 


8 


3 


5 


3.4 


3 


9 


4 





3.9 


3 


10 


4 


4 


4.3 


4 


20 


8 


g 


86 


8. 


3C 


]3 


2 


13 


12 


40 


17 


6 


17-3 


17- 


50 


22 


1 


21.6 


21. 





6 


6 


6 


0-6 


06 


7 





7 


0.7 


8 





8 


0.8 


9 


1 





0.9 


IG 


1 


1 


1.0 


20 


2 


1 


2.0 


30 


3 


2 


30 


40 


4 


3 


4.0 


50 


5 


4 


5.0 



P. p. 



iis** 



a72 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



153' 



Log. Sin. 



64 184 
64 210 
64 236 
64 262 
64 287 



64 313 
64 339 
64 365 
64 391 
64 416 



64 442 
64 468 
64 493 
64 519 
64 545 



64 570 
64 596 
64 622 
64 647 
64 673 



64 698 
64 724 
64 749 
64 775 
64 800 



64 826 
64 851 
64 876 
64 902 
64 927 
64 952 

64 978 

65 003 
65 028 
65 054 



65 079 
65 104 
65 129 
65 155 
65 180 



65 205 
65 230 
65 255 
65 280 
65 305 

65 331 
65 356 
65 381 
65 406 
65 431 



65 456 
65 481 
65 506 
65 53g 
65 555 



65 580 
65 605 
65 630 
65 655 
65 680 



65 704 



Log. Cos, 



26 
26 
26 
25 
26 
26 
25 
26 
25 

26 
25 
25 
26 

25 

25 
25 
26 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
24 
25 

25 
25 
24 
25 
25 
24 



Log. Tan 



68 818 
68 850 
68 882 
68 914 
68 946 



68 978 

69 010 
69 042 
69 074 
69 106 



9-69 138 
9-69 170 
9-69 202 
9-69 234 
9.69 265 



9-69 297 
9.69 329 
9.69 361 
9.69 393 
9.69 425 



9-69 456 
9. 69 488 
9.69 520 
9.69 552 
9.69 583 



69 615 
69 647 
9.69 678 
9.69 710 
9.69 742 



69 773 
69 805 
9.69 837 
9.69 868 
9.69 900 



69 931 
69 963 

9.69 994 

9.70 026 
9.70 058 



70 089 
70 121 
9.70 152 
9.70 183 
9.70 215 



9.70 246 
9.70 278 
9.70 309 
9.70 341 
9.70 372 



9 . 70 403 
9.70 435 
70 466 
9.70 497 
9.70 529 



70 560 
70 591 
70 623 
70 654 
70 685 



9.70 716 



Log. Cot. 



32 
32 
32 
32 

32 
32 
32 
31 
32 

32 

32 
32 
32 
31 

32 
32 
31 

32 
32 

3l 

32 
3l 
32 
3l 

32 
31 
31 
32 
3l 

3l 
32 
31 
31 
31 

31 
31 
3l 

32 
3l 

31 
II 

l\ 

31 
31 
31 
3l 
31 

3l 
31 

3l 
31 
3l 
3l 
31 
31 
31 
31 
31 



c.d 



Log. Cot, 



0.31 
0-31 
0-31 
0.31 
0.31 



0.31 
0.30 
0.30 
0.30 
0.30 



182 
150 
117 
085 
G58 
021 
989 
957 
926 
894 



0.30 
0.30 
0.30 
0.30 
0.30 



862 
830 
798 
766 
734 



0.30 
0.30 
0.30 
0-30 
0.30 



702 
670 
639 
607 
575 



Log. Cos 



95 366 
95 360 
95 353 
95 347 
95 341 



95 335 
95 329 
95 323 
95 316 
95 310 



95 304 
95 298 
95 292 
95 285 
95 279 



0.30 543 
0-30 511 
0.30 480 
0.30 448 
0.30 416 



0.30 384 
0.30 353 
0.30 321 
0.30 289 
0.30 258 



0.30 226 
0.30 194 
0.30 163 
-30 13l 
0.30 100 



0.30 068 
0.30 037 
0.30 005 
0.29 973 
0.29 942 



0.29 910 
0.29 879 
0.29 847 
0.29 816 
0.29 785 



0.29 
0.29 
0.29 
0.29 
0.29 



753 
722 
690 
659 
628 



0.29 
0.29 
0.29 
0.29 
0.29 



596 
565 
533 
502 
471 



29 439 
29 408 
29 377 
29 346 
29 314 



0.29 283 



Log. Tan 



95 273 
95 267 
95 260 
95 254 
95 248 



9-95 242 
9.95 235 
9.95 229 
9.95 223 
9.95 217 



95 210 
95 204 
95 198 
95 191 
95 185 



95 179 
95 173 
95 166 
95 160 
95 154 



9.95 147 
9.95 141 
9.95 135 
9.95 128 
9.95 19.9 



95 116 
95 109 
95 103 
95 097 
95 090 



95 084 
95 078 
95 071 
95 065 
95 058 



95 052 
95 046 
95 039 
95 033 
95 026 



9.95 020 
9.95 014 
9-95 007 
9-95 001 
9-94 994 



9-94 988 



Log. Sin, 



60 

59 

58 

57 

_56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
J6 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
3]^ 
30 
29 
28 
27 
21 
25 
24 
23 
22 
21_ 

30 

19 

18 

17 

16. 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 
4 
3 
2 
_]_ 
O 



P.P. 



33 



3 


2 


3 


8 


4 


3 


4 


9 


5 


4 


10 


8 


16 


2 


21 


6 


27 


1 



32 

3.2 
3.7 

4.2 
48 
5.3 
10.6 
16.0 
21.3 
26.6 





31 


31 


6 


3.1 


3.1 


7 


3 


7 


3.6 
4.1 


8 


4 


2 


9 


4 


7 


4.6 


10 


5 


2 


5.1 


20 


10 


5 


10.3 


30 


15 


7 


15.5 


40 


21 





20.6 


50 


26 


2 


25.6 



36 


25 


2.6 


2.51 


3.0 


3 





3.4 


3 


4 


3.9 


3 




4.3 


4 


2 


8.6 


8 


5 


13. 


12 


7 


17.3 


17 





21.6 


21 


2 





23 


6 


6 


2.4 


0-6 


7 


2 


8 


0-7 


8 


3 


2 


0.8 


9 


3 


7 


1-0 


10 


4 


1 


1-1 


20 


8 


1 


2. . 


30 


12 


2 


3.2 


40 


16 


3 


4.3 


50 


20 


4 


5.4 



35 

2.L 

2.9 

3.3 

3.7 

4.1 

8.3 

12.5 

16-6 

20.5 



6 

0.6 
0.7 
0.8 
0.9 



5.0 



P.P. 



673 



6^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
37* AND COTANGENTS. 152^^ 



65 704 
65 729 
9.65 754 
9.65 779 
9.65 803 



45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



60 



Log. Sin. 



65 828 
65 853 
65 878 
65 902 
65 927 



65 951 

65 976 

66 001 
66 025 
66 050 



66 074 
66 099 
66 123 
66 148 
66 172 



9.66 197 
66 221 
9.66 246 
9.66 270 
9.66 294 



66 319 
66 343 
66 367 
66 392 
66 416 



66 440 
66 465 
66 489 
66 513 
66 537 



66 561 
66 586 
66 610 
66 634 
66 658 



66 682 
66 708 

9.66 730 
66 754 

9.66 778 



9.66 802 
9.66 826 
9.66 850 
9.66 874 
9.66 898 



9.66 922 
9.66 946 
9.66 970 
9.66 994 
9-67 018 



67 042 
67 066 
67 089 
67 113 
67 137 



9.67 161 



Log. Cos, 



d. 

25 
24 
25 
24 

25 
24 
25 
24 
24 

24 
25 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
24 
24 

24 
24 
24 
23 
24 

24 
24 
23 
24 
23 
24 



Log. Tan 



9.70 716 
9.70 748 
9.70 779 
9.70 810 
9.70 841 



70 872 
70 903 
70 935 
70 966 
70 997 



9.71028 
9.71059 
9.71090 
9.71 121 
71 152 



9.71 183 
9.71 214 
9.71 245 
9.71 276 
9.71 307 



9.71 338 
9.71369 
9.71400 
9.71431 
9.71462 



71493 
71 524 
71 555 
71 586 
71 617 



9.71 647 
9.71 678 
9.71 709 
9.71 740 
71 771 



9.71 801 
9.71 832 
9.71 863 
9.71 894 
9.71 925 



9.71 955 

9.71 983 

9.72 017 
9.72 047 
9.72 078 



9.72 109 
72 139 
9.72 170 
9.72 201 
72 231 



72 262 
72 292 
72 323 
72 354 
72 384 



c.d. 



9.72 415 
9 . 72 445 
9.72 476 
9.72 506 
9.72 537 



9-72 ?iR7 



d. Log. Cot 



31 
31 
31 
31 

31 
31 
3l 
31 
31 
31 
3l 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
30 
31 
31 
31 

30 
31 
31 
30 
31 

30 

31 
31 
30 
31 

30 

30 
31 
30 
31 

30 
30 
30 
31 
30 

30 
30 
30 
31 
30 

30 
30 
30 
30 
30 

30 



Log. Cot 



0.29 283 
0.29 252 
0.29 221 
0.29 190 
0.29 158 



29 127 
29 096 
29 065 
29 034 
29 003 



28 972 
28 940 
28 909 
28 878 
28 847 



0.28 816 
0.23 785 
0.28 754 
0.28 723 
0.28 692 



0.28 661 
0.28 630 
0.28 599 
0.28 568 
0.28 537 



0.28 506 
0.28 476 
0.28 445 
0.28 414 
0.28 383 



0.28 352 
0.28 321 
0.28 290 
0.28 260 
0.28 229 



0.28 198 
0.28 167 
0.28 136 
0.28 106 
0.28 075 



0.28 044 
0.28 014 
0.27 983 
0.27 952 
0.27 921 



0.27 891 
0.27 860 
0.27 830 
0.27 799 
0.27 768 



0.27 738 
0.27 707 
0.27 677 
0.27 646 
0.27 615 



c.d, 



0.27 585 
0.27 554 
0.27 52^ 
0.27 493 
0-27 463 



Log. Cos. 



9.94 988 
9.94 981 
9.94 975 
9.94 969 
9.94 962 



9.94 956 
9.94 949 
9.94 943 
9.94 936 
9.94 930 



94 923 
94 917 
94 910 
94 904 
94 897 



9.94 891 
9.94 884 
9.94 878 
9.94 871 
9.94 865 



9.94 858 
9.94 852 
9.94 845 
9.94 839 
9.94 832 



9.94 825 
9.94 819 
9.94 812 
9.94 806 
94 799 



94 793 
9.94 786 
9.94 779. 
9.94 773 
9-94 766 

9.94 760 
9.94 753 
9.94 746 
9.94 740 
94 733 



94 727 
94 720 
94 713 
94 707 
94 700 



9.94 693 
9.94 687 
9.94 680 
9.94 674 
9.94 667 



0.27 A39. 



Log. Tan, 



9.94 660 
9-94 654 
9 - 94 647 
9-94 640 
9-94 633 

9-94 627 
9-94 620 
9-94 613 
9-94 607 
9-94 600 



9-94 503 
Log. Sin. 



60 

59 

58 

57 

_56 

55 
54 
53 
52 

11. 

50 
49 
48 
47 

j46 

45 
44 
43 
42 
41^ 

40 

39 
38 
37 

li 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26_ 

25 
24 
23 
22 

21 

30 

19 

18 

17 

11 

15 

14 

13 

12 

JJ, 

10 

9 

8 

7 

_6 

5 
4 
3 
2 
_1 




P.P. 



6 
7 
8 
9 

10 
20 
30 
40 
50 



31 31 



( 3 


I 


3.1 


3 


7 


3.6 


4 


2 


4-1 


4 


7 


4-6 


5 


2 


5.1 


10 


5 


10.3 


15 


7 


15-5 


21 





20.6 


26 


2 


25. 8j 



30 

3.0 
3-5 

4.0 
4.6 
5.1 
10.1 
15.2 
20.3 
25.4 



25 

2.5 
2.9 
3.3 
3.7 
4.1 
8.3 



30 12.5 
40 16.6 
5020.8 



22 

2.41 



24 

2.4 
2.8 
3.2 
3.6 
4.0 
8.0 
12.0 



2. 

3. 

3. 

4. 

8 
12. _ 
16.3 16.0115.6 
2O.42O.OJ19.6 



23 

2.3 
2.Z 
3.1 
3.5 
3.9 
7-8 
11.7 





7 


6 


6 


0.7 


61 


7 


0-8 





7 


8 


0-9 





3 


9 


1-0 


1 





10 


1-1 


1 


1 


20 


2.3 


2 


1 


303.5 


3 


2 


40 4.6 


4 


3 


50 


5.8 


5 


4 



6 

0.6 
0.7 
0.8 
0.9 
1.0 
2.0 
3.0 
4.0 
5.0 



P.P. 



117' 



674 



63' 



38* 



TABLE Vir.— LOGARITHMIC SINES. COSINES. TANGENTS, 
AND COTANGENTS. 



151* 



Log. Sin 



9.67 161 
9.67 184 



67 208 
67 232 
67 256 



67 279 
67 303 
67 327 
67 350 
67 374 



67 397 
67 421 
67 445 
67 468 
67 492 



67 515 
67 539 
67 562 
67 586 
67 609 



67 633 
67 656 
67 679 
67 703 
67 726 



67 750 
67 773 
67 796 
67 819 
67 843 



67 866 
67 889 
67 913 
67 936 
67 959 



67 982 

68 005 
68 029 
68 052 
68 075 



68 098 
68 121 
68 144 
63 167 
68 190 



68 213 
68 236 
68 259 
68 282 
68 305 



68 328 
68 35T 
68 374 
68 397 
6 8 420 

68 443 
68 4BR 
68 488 
68 5lT 
68 534 



9-68 557 



Log. C( 



23 
24 
23 
24 

23 
23 
24 
23 
23 

23 
24 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
22 
23 

23 
23 
22 
23 
23 
22 



Log. Tan. c. d. Log. Cot. Log. Cos 



72 567 
72 598 
72 628 
72 659 
72 689 



72 719 
72 750 
72 780 
72 811 
72 841 



72 871 
72 902 
72 932 
72 962 
72 993 



73 023 
73 053 
7S 084 
73 114 
73 144 



73 174 
73 205 
73 235 
73 265 
73 295 



73 325 
73 356 
73 386 
73 416 
73 446 



73 476 
73 506 
73 536 
73 567 
73 597 



73 627 
73 657 
73 687 
73 717 
73 747 



73 777 
73 807 
73 837 
73 867 
73 897 



73 927 
73 957 

73 987 

74 017 
74 047 



74 076 
74 106 
74 136 
74 166 
74 196 



74 226 
74 256 
74 286 
74 315 
74 345 



9-74 375 
Log. Cot. 



30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

29 
30 
30 
30 
29 

30 
30 
30 
29 
30 

29 



.27 432 
.27 402 
.27 371 
.27 341 
0.27 311 



0.27 280 



0.27 250 9.94 553 



0.27 128 
0-27 098 
0.27 067 
0.27 03? 
27 007 



9.94 593 
9.94 587 
9.94 580 
9.94 573 
9.94 566 



9.94 560 



27 219 9 
27 189 9 
27 159 9 



0.26 976 
0.26 946 
26 916 
26 886 

26 855 



26 82£ 
0.26 795 
0.26 76£ 
0.26 734 
0.26 7C4 



0.26 674 
0.26 644 
0.26 614 
0.26 584 
0.26 55S 



0.26 52S 
0-26 49£ 
0.26 46£ 
0.26 43S 
0.26 4CS 



0.26 372 
0.26 S4S 
0.26 31£ 
0.26 28£ 
0-26 25£ 



0.26 22£ 
0.26 19S 

0.26 les 

0.26 13c 
0.26 ICS 



0.26 07S 
0.26 043 
0.26 013 
0.25 98£ 
P-.25 95£ 

0.25 925 
0.25 895 
0.25 868 
0.25 833 
0.25 804 



94 546 
94 539 
94 533 

94 526 
94 519 
94 512 
9.94 506 
9.94 499 



9.94 492 
9.94 485 
9.94 478 
9-94 472 
9.94 465 



9.94 458 
9.94 451 

94 444 
9.94 437 

94 431 



9.94 424 
94 417 
94 410 
9 . 94 403 
9. 94 396 
9.94 390 
9.94 383 
9.94 376 
9.94 369 
9-94 362 

9.94 355 
9 . 94 348 
9-94 341 
9-94 335 
9- 94 328 



■ 94 321 

• 94 314 

94 307 

94 300 

94 293 



9-94 286 
9-94 279 
9-94 272 
9-94 265 
9-94 258 



9-94 251 
94 245 
9-94 238 
9-94 231 
94 224 



c. d. 



0-25 774 
0-25 744 
0-25 7U 
0-25 684 
0-25 654 



n.25 625 



-94 217 
-94 210 
.94 203 
94 196 
94 189 



9.94 182 



Log. Tan.JLog. Sin 



60 

59 
58 
57 
56 



50 

49 
48 
47 
46 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 



35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 



10 

9 

8 

7 

_6 

5 
4 
3 

2 

1 



^1 



P. P. 





30 


.^0 


6 


3.0 


5.0 


7 


3.5 


3.5 


8 


4.0 


4.0 


9 


4.6 


4-5 


10 


5.11 5-0 


20 


10.1110. 


30 


15.2115.0 


4C 


20.3i20.0 


50 


25.4 


25-01 



29_ 

2.9 

3.4 

3.9 

4.4 

4.9 

9-8 

14-7 

19.6 

24.6 





24 


6 


2.4 


7 


2.8 


8 


3.2 


9 


3.6 


10 


4.0 


20 


8.0 


30 


12.0 


40 


16.0 


50 


20.0 





23 


23 


6 


2-3 


2.3 


7 


2.7 


2-7 


8 


3.1 


3-0 


9 


3-5 


3-4 


10 


3.9 


3-8 


20 


7.8 


7-6 


30 


11-7 


11-5 


40 


15.6 


15-3 


50 


19.6 


19.1 



22 

2.2 
6 

4 
7 
5 
2 

7 





4 


f 


(R 


6 


0.7 


0.6 


7 





8 


0-7 


8 





g 


0.8 


9 


1 





i.Q 


10 


1 


1 


1.1 


20 


2 


3 


2.1 


30 


3 


5 


3-2 


40 


4 


6 


4.3 


50 


5 


8 


5.4 



P. p. 



iia' 



675 



61^ 



29' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 

AND COTANGENTS. IfiOP 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19 

30 

21 

22 

23 

21 

25 

26 

27 

28 

29_ 

30 

31 

32 

33 

31 

35 

36 

37 

38 

39_ 

40 

41 
42 
43 
44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



Log. Sin. 



68 557 
68 580 
68 602 
68 625 
68 648 

68 671 
68 693 
68 716 
68 739 
68 761 



68 784 
68 807 
68 829 
68 852 
68 874 



68 897 

68 920 
68 942 
68 965 
68 987 



69 010 
69 032 
69 055 
69 077 
69 099 



69 122 
69 144 
69 167 
69 189 
69 211 



69 234 
69 256 
69 278 
69 301 
69 323 



69 345 
69 367 
69 390 
69 412 
69 434 



69 456 
69 478 
69 500 
69 523 
69 545 



69 567 
69 589 
69 611 
69 633 
69 655 



69 677 
69 699 
69 72l 
69 743 
69 765 



69 787 
69 809 
9-69 83l 
9.69 853 
9.69 875 



9.69 897 



d. Log. Tan, c.d. Log. Cot. Log. Cos. d. 



Log. Cos. 



23 
22 
23 
22 

23 
22 
23 
22 
22 

23 
22 
22 
22 
22 

22 
23 
22 
22 
22 

22 

22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
21 
22 
22 



9.74 370 
9.74 405 
9.74 435 
9.74 464 
9.74 494 



74 524 
74 554 
9.74 583 
9.74 613 
9.74 643 



9.74 672 
9.74 702 
9.74 732 
9.74 761 
9.74 791 



9.74 821 
9.74 850 
9.74 880 
74 909 
9.74 919 



74 969 
74 998 
9.75 028 
9.75 057 
9.75 087 



75 116 
9.75 146 

75 175 
9.75 205 
9.75 234 



9.75 264 
9.75 293 
9.75 323 
9.75 352 
9.75 382 



9.75 411 
75 441 
75 470 
9.75 499 
9.75 529 



9.75 558 
9.75 588 
9.75 617 
9.75 646 
9.75 676 



9.75 705 
75 734 
9.75 764 
9.75 793 
9.75 822 



9.75 851 
9.75 881 
9.75 910 
9.75 939 
9.75 968 



9.75 998 

9.76 027 
9.76 056 
9.76 085 
9.76 115 



9.76 144 



Leg, 



Cot. 



30 
30 
29 
30 

29 
30 
29 
29 
30 

29 
30 
29 
29 
30 
2§ 
29 
29 
29 
30 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 
29 
29 
29 
29 
29 

29 

29 
29 
29 
29 

29 
29 

29 
29 
29 
29 

c.d 



0.25 625 
0.25 595 
0.25 565 
0.25 535 
0^25_505 

0.25 476 
0.25 446 
0.25 416 
0.25 387 
0-25 357 



0.25 327 
0.25 297 
0.25 268 
0.25 238 



9.94 182 
9.94175 
9.94 168 
9.94161 
9.94 154 



9.94 147 
9 . 94 140 
9.94 133 
9.94 126 
9.94 118 



0.25 208 9.94 083 



0.25 179 
0.25 149 
0.25 120 
0.25 090 
0.25 060 



0.25 031 
0.25 001 
0.24 972 
0.24 942 
0.24 913 



0.24 88'o 
0.24 854 
0.24 824 
0.24 795 
0.24 765 

0.24 736 
0.24 706 
0.24 677 
. 24 647 
0^24J1J 

0.24 588 
0.24 559 
0.24 529 
0.24 500 
0.24 471 



. 24 441 
0.24 412 
0.24 383 
0.24 353 
0.24 324 



0.24 
0.24 
0.24 
0.24 
0.24 



0.24 
0.24 
0.24 
0.24 
0.24 



295 
265 
236 
207 
177 
14g 
119 
09G 
060 
031 



0.24 002 
0.23 973 
0.23 943 
0-23 914 
0.23 885 



n ?3 8b6 



Log. Tan. 



9.94 111 
9.94 104 
9.94 097 
9.94 090 



9.94 076 
9.94 069 
94 062 
9.94 055 
9.94 048 



9.94 041 
9.94 034 
9.94 026 
9.94 019 
9.94^12 

9.94 005 
9.93 998 
9.93 991 
9.93 984 
9.93 977 
9.93 969 
9.93 962 
9.93 955 
9.93 948 
9.93 941 



9.93 934 
9.93 926 
9.93 919 
9.93 912 
9.93 905 



9.93 898 
9.93 891 
9.93 883 
9.93 876 
9.93 869 



93 862 
93 854 
93 847 
93 840 
93 833 



93 826 
93 818 
93 8ll 
93 804 
93 796 



93 789 
93 782 
9.93 775 
9.93 767 

9-93 753 
Log. Sin. 



60 

59 
58 
57 

li. 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46. 

45 
44 
43 
42 
41 

40 

39 
38 
37 
_36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

r^o 

19 
18 
17 

JLi 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 
2 

O 



P. P. 





30 


29. 


2i 


6 


3.0 


2.9 


2. 


7 


8.5 


3 


4 


3. 


8 


4.0 


3 


9 


3 


9 


45 


4 


4 


4. 


10 


5.0 


4 


9 


4- 


20 


10.0 


9 


g 


9. 


30 


15.0 


14 


7 


14. 


40 


20.0 


19 


6 


19. 


50 


25.0 


24 


6 


24. 



6 


2 


7 


2 


8 


3. 


9 


3 


10 


3. 


20 


7. 


30 


11. 


40 


15. 


50 


19. 



S3 

3 

7 

4 
8 
6 
5 
3 
I 





33 


33 


21 


6 


2.2 


2.2 


2.1 


7 


2.6 


2 


5 


2 


5 


8 


3.0 


2 


9 


2 


3 


9 


3.4 


3 


3 


3 


2 


10 


3.7 


3 


g 


3 


6 


20 


7.5 


7 


3 


7 




30 


11.2 


11 





10 




40 


15.0 


14 


6 


14 


3 


50 


18-7 


18 


3 


17 


9 



6 
7 
8 

9 
10 
20 
30 
40 
60 



7 1 

0.7|0.7 
0.9 0.8 
l.OjO.9 
l.lll.O 
1.2,1.1 
2.5^2 3 
3. 7,3 5 
5.04 6 
6-2 5-8 



P. P 



iia*" 



676 



60*' 



30* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



149* 



5 
6 
7 
8 
9F9 



L^g. Sin, 



9 
9 

9 
9 
9 
9 

30 9 

31 

32 

33 

34 

85 
36 
37 
38 
39 9 



897 
919 
940 
962 
98i 



006 
028 
050 
07l 
093 

115 
137 
158 
180 

202 

223 
245 
267 
288 
310 



331 
353 
375 
396 
418 



439 
461 
482 
5041 
525 



547 
568 
590 
611 

632 



654 
675 
696 
718 
739 



760 
782 
803 
824 
846 



867 
888 
909 
930 
952 



9 
9 
9 
9 

^•71 184 



973 
994 
015 
036 
057 
078 
099 
121 
142 
163 



Log. Cos. d 



22 
21 
22 
22 
2l 
22 
22 
21 
22 
2l 
22 
21 
21 
22 

2l 
2l 
22 
21 
21 

2l 
22 
2l 
21 
21 

21 
21 
21 
21 
21 

2l 
21 
21 
21 
2l 

21 
21 
21 
21 
21 

21 
2l 
21 
2l 
2l 

21 
21 
2l 
21 
21 

21 
21 
2l 
21 
21 

21 
21 
2l 
21 
21 

21 



Log. Tan. 



9.76 144 
9.76 173 
9.76 202 
9.76 231 
76 260 



9.76 289 
9.76 319 
9.76 348 
9.76 377 
9-76 406 



9.76 435 
9.76 464 
9.76 493 
9.76 522 
9.76 55l 



9.76 580 
9-76 609 
9.76 638 
9-76 667 
9-76 696 



9.76 725 
9.76 754 
9.76 783 
9.76 812 
9-76 84l 



9.76 870 
9.76 899 
9.76 928 
9.76 957 
9.76 986 



9.77 015 
9.77 043 
9.77 072 
9.77 101 
9-77 130 



9.77 159 
9.77 188 
9.77 217 
9.77 245 
.9. 77 274 



77 303 
9.77 332 
9.77 361 
9.77 389 
9-77 418 



9.77 447 
77 476 
9-77 504 
9.77 533 
9-77 562 



9-77 591 
9.77 619 
9-77 648 
9.77 677 
9.77 705 



9.77 734 
9.77 763 
9.77 791 
9.77 820 
9.77 849 



9-77 877 



c. d. 

29 
29 
29 
29 
29 
29 
29 
29 
29 

2? 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
28 
29 
29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
28 
29 

29 
28 
29 
28 
29 

28 
29 
28 
29 
28 
29 
28 
28 
29 
28 

28 
29 
28 
28 
29 

28 



Logs Cot. 



0-23 856 
0.23 827 
0.23 797 
0.23 768 
0.23 739 



Log. Cot 



0.23 710 
0.23 681 
0.23 652 
0.23 623 
0.23 594 



0.23 565 
0.23 535 
0.23 506 
0.23 477 
0-23 44E 



C.d 



23 41G 
23 390 
23 361 
23 332 
23 303 



Log. Cos. 



93 753 
93 746 
93 738 
93 731 
93 7^4 



93 716 
93 709 
93 702 
93 694 
93 687 



93 680 
93 672 
93 665 
93 658 
93 650 



93 643 
93 635 
93 628 
93 621 
93 613 



0-23 274 
0.23 245 
0.23 216 
0-23 187 
0- 23 158 

0-23 129 
0.23 101 
0.23 072 
0.23 043 
0-23 014 



9. 



0-22 985 
0.22 956 
0.22 927 
0.22 89£ 
0-22 869 



0-22 841 
0.22 812 
0.22 783 
0-22 754 
0.22 725 



0-22 696 
0-22 668 
0.22 639 
0.22 610 
0.22 581 



0.22 553 
0.22 524 
0.22 495 
0.22 466 
0.22 438 



0.22 
0.22 
0.22 
0-22 
0.22 



0.22 
0-22 
0.22 
0.22 
22 



409 
380 
352 
328 
294 

266 
237 
208 
180 
151 



0.22 122 



Log. Tan 



93 606 
93 599 
93 591 
93 584 
93 576 



93 569 
93 562 
93 554 
93 547 
93 539 



93 532 
93 524 
93 517 
93 509 
93 502 



93 495 
93 487 
93 480 
93 472 
93 465 



93 457 
93 450 

93 442 
93 435 
93 427 



9-93 420 
9-93 412 
9-93 405 
9-93 397 
9-93 390 



93 382 
93 374 
93 367 
93 359 
93 352 



93 344 
93 337 
93 329 
93 321 
93 314 



9 
9 

9-93 306 



Log. Sin 



60 

59 
58 
57 
56_ 

55 
54 
53 
52 
11 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 



40 

39 
38 

37 
36 

35 
34 
33 
32 
II 
30 
29 
28 
27 
26_ 

25 
24 
23 
22 
21 

20 

19 
18 

17 

15 

14 

13 

12 

Jl 

10 

9 

8 

7 

_i 

5 
4 
3 
2 

O 



P.P. 





39 


29 


6 


2-9 


2.9 


7 


3 


4 


3-4 


8 


3 


9 


3-8 


9 


4 


4 


4.3 


10 


4 


9 


4-8 


20 


9 


8 


9-6 


30 


14 


7 


14-5 


40 


19 


6 


19.3 


50 


24 


6 


24.1 



38 
2-8 

3.3 
38 

4.3 

4.7 

9.5 

14.2 

19.0 

23.7 



6 

7 

8 

9 

10 

20 

30 

40 

50 



22 

2-2 



21 

2-1 



21 

2.1 

2.i 
2.S 
3.1 
3.5 
7.0 
7.10.5 
3,14.0 
9I17.S 



8 
0-8 

0.9 
1-0 
1-2 
1-3 
2-6 
4.C 
5.1 
6.6 



7_ 7 
0-710.7 
0.90.8 
1.00.9 

1.1 1.0 

1.2 1. 1 



2.5 
3.7 
5.0 
6.2 



2.3 
3.5 
4.6 
5.8 



P.P. 



IW 



Q77 



6ff 



31" 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



148' 



Log. Sin. 



184 
205 
226 
247 
268 



289 
310 
331 
351 
372 



393 
414 
435 
456 
477 

498 
518 
539 
560 
581 



601 
622 
643 
684 
684 

705 
726 
746 
767 
788 



808 
829 
849 
870 
891 



911 
932 
952 
973 
993 



014 
034 
055 
075 
098 



116 
136 
157 
177 
198 



218 
238 
259 
279 
299 



319 
340 
360 
380 
400 



72 421 



Log, Cos 



21 
21 
21 
21 

21 
21 
21 
20 
21 

21 
21 
20 
21 
21 

21 
20 
21 
20 
21 

25 
21 

20 
21 
20 

21 
20 
20 
21 
20 

20 
20 
20 
21 
20 

20 
20 
20 
20 
20 

20 

20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 

"37 



Log. Tan. 



77 877 
77 906 
77 934 
77 963 
77 992 



78 020 
78 049 
78 077 
78 106 
78 134 



9.78 163 
9-78 191 
9.78 220 
9-78 248 
9-78 277 



9.78 305 
9.78 334 
9-78 362 
9.78 391 
9.78 419 



9.78 448 
9.78 476 
9-78 505 
9.78 533 
9-78 561 



9-78 590 
9.78 618 
9.78 647 
9.78 675 
9-78 703 



9.78 732 
9.78 780 
9.78 788 
9.78 817 
78 845 



78 873 
78 902 
9.78 930 
78 958 
78 987 



79 015 
79 043 
79 071 
79 100 
79 128 



79 156 
79 184 
79 213 
79 241 
79 269 



79 297 
79 325 
79 354 
79 382 
79 410 



9-79 438 
9.79 466 
9.79 494 
9.79 522 
9.79 551 



9.79 579 



Log. Cot. 



c.d. 



28 
28 
28 
29 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 

28 
28 
28 
28 

28 

28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 

C.d. 



Logr Cot. 



0-22 122 
0-22 094 
0-22 065 
0-22 037 
0. 22 00 8 

0.21979 
0-21 951 
0-21 922 
0-21 894 
0.21 865 



0-21 837 
0-21 808 
0-21 780 
0-21 751 
0.21 723 



0-21 694 
0.21 666 
0.21 637 
0-21 609 
0-21 580 



0-21 552 
0-21 523 
0-21 495 
0-21 467 
0-21 438 



0.21 410 
0.21 381 
0.21 353 
0.21 325 
0-21 296 



0-21 268 
0-21 239 
0-21 211 
0.21 183 
0-21 154 



0-21 126 
0.21 098 
0-21 070 
0.21 04l 
0.21 013 



0-20 985 
0-20 956 
0-20 928 
0.20 900 
0-20 872 



20 843 
20 815 
20 787 
20 759 



0.20 731 



0.20 702 
0.20 674 
0-20 646 
0.20 618 
0.20 590 



0.20 561 
0.20 533 
0.20 505 
0.20 477 
0.20 449 



0.20 421 
Log. Tan. 



Log. Cos, 



93 306 
93 299 
93 291 
93 284 
93 276 

93 268 
93 261 
93 253 
93 245 
93 238 



93 230 
93 223 
93 215 
93 207 
93 200 



93 192 
93 184 
93 177 
93 169 
93 161 



93 153 
93 146 
93 138 
93 130 
93 123 



93 115 
93 107 
93 100 
93 092 
93 084 



93 076 
93 069 
93 061 
93 053 
93 045 



93 038 
93 030 
93 022 
93 014 
93 006 



92 999 
92 99] 
92 983 
92 975 
92 967 



92 960 
92 952 
92 944 
92 936 
92 928 



92 920 
92 913 
92 905 
92 897 
92 889 



92 88i 
92 873 
92 865 
92 858 
92 850 



92 842 



1»1* 



Log. Sin. 
678 



d. 



60 

59 
58 
57 
51 
55 
54 
53 
52 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 
40 
39 
38 
37 

li 

35 
34 
33 
32 
_3i 
30 
29 
28 
27 
2§_ 

25 
24 
23 
22 
21 

20 

19 
18 
17 
11 
15 
14 
13 
12 
11 

10 

9 

8 

7 

_6. 

5 
4 
3 

2 

O 



P. P. 





29 


28 


28 


6 


2.9 


2-8 


2.8 


7 


3 


4 


3.3 


3.2 


8 


3 


8 


38 


3 7 


9 


4 


3 


43 


4-2 


10 


4 


3 


4.7 


4.6 


20 


9 


6 


9 5 


9.3 


30 


14 


5 


14.2 


14.0 


40 


19 


3 


19.0 


18. 6 
23.3 


50 


24 


1 


23.7 





21 


20 


6 


2.1 


2.0 


7 


2 


4 


2.4 


8 


2 


8 


2.7 


9 


3 


1 


3.1 


10 


3 


5 


3.4 


20 


7 





68 


30 


10 


5 


10.2 


40 


14 





13.6 


50 


17 


5 


17.1 



20 

n 

2.6 

3-0 

3-3 

6-6 

10-0 

13.3 

16.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 






8 





g 


1 





1 


2 


1 


3 


2 


6 


4 




5 


3 


6 


6 



0^ 

0.9 

1.0 

1.1 

1.2 
2-5 
3.7 
5.0 
6.2 



P. P. 



68' 



32' 



TABLE VIL— LOGARITHMIC SINES, COSINES. TANGENTS, 

AND COTANGENTS. I470 



O 

1 
2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29^ 

30 

31 
82 
33 
34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
4£ 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 

55 
56 
57 
58 
51 
60 



Log. Sin 



9-72 421 
9.72 441 
9.72 461 
9 . 72 481 
9-72 501 

9.72 522 
9.72 542 
9.72 562 
9.72 582 
9.72 602 



72 622 
72 642 
72 662 
72 682 
72 702 



72 723 
72 743 
72 763 
72 783 
72 802 



72 822 
72 842 
72 862 
72 882 
72 902 



72 922 
72 942 
72 962 

72 982 

73 002 



73 021 
73 041 
73 061 
73 081 
73 101 



73 120 
73 140 
73 160 
73 180 
73 199 



73 219 
73 239 
73 258 
73 278 
73 298 



73 317 
73 337 
9.73 357 
9.73 376 
9.73 396 



73 415 
73 435 
73 455 
73 474 
73 494 



9.73 513 
9.73 533 
9.73 552 
9.73 572 
73 591 



9.73 611 



Log, Cos. 



20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 

2g 

19 

20 
20 
20 
20 
20 

20 
19 
20 
20 
20 

19 
20 
20 
19 
20 

19 
20 
19 
20 
19 

20 

19 
19 
20 
19 

19 
20 
19 
19 
19 

19 
20 
19 
19 
19 

19 
19 
19 
19 
19 

19 

T 



Log, Tan. c. d. Log. Cot. Log. Cos. 



9 



79 579 
79 607 
79 635 
79 663 
79 691 



79 719 
79 747 
79 775 
79 803 
79 831 



79 859 
79 887 
79 915 
79 943 
79 971 



79 999 

80 027 
80 055 
80 083 
80 111 



80 139 
80 167 
80 195 
80 223 
80 251 



80 279 
80 307 
80 335 
80 363 
80 391 



80 418 
80 446 
80 474 
80 502 
80 530 



80 558 
80 586 
80 613 
80 64l 
80669 



80 697 
80 725 
80 752 
80 780 
80 808 



80 836 
80 864 
80 81)1 
80 919 
80 947 



80 975 

81 002 
81 030 
81 058 
81 085 



81 113 
81 141 
81 168 
81 196 
81 224 



81 251 



Log. Cot. 



28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

27 
28 
28 
28 
28 

27 
28 
28 
28 
27 

28 
28 
27 
28 
28 

97 
28 
27 
28 
28 

27 
28 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
27 



0.20 421 
0.20 393 
0.20 365 
0.20 337 
0.20 308 



0.20 280 
0.20 252 
0-20 224 
0-20 196 
0.20 168 



0.20 140 
0.20 112 
0.20 084 
0.20 056 
0-20 028 



0.20 000 
0.19 972 
0.19 944 
0.19 916 
0-19 888 



0.19 860 
0-19 832 
0.19 804 
0.19 776 
0.19 748 



0.19 721 
0.19 693 
0.19 665 
0.19 637 
0.19 609 



0.19 581 
0.19 553 
0-19 525 
0.19 497 
0-19 470 



92 842 
92 834 
92 826 
92 818 
92 810 



92 802 
92 794 
92 786 
92 778 
92 771 



92 763 
92 755 
92 747 
92 739 
92 731 



92 723 
92 715 
92 707 
92 699 
92 691 



92 683 
92 675 
92 667 
92 659 
92 651 



92 643 
92 635 
92 627 
92 619 
92 611 



92 603 
92 595 
92 587 
92 579 
92 570 



0.19 442 
0.19 414 
0.19 386 
0.19 35P 
0.19 330 



19 303 
19 27F 
19 247 
19 219 
19 191 



0.19 164 
0.19 136 
0.19 108 
0.19 080 
0.19 053 



19 025 
18 997 
18 970 
18 942 
18 914 



0.18 886 
0.18 859 
0.18 831 
0.18 803 
0-18 776 



0.18 748 



92 562 
92 554 
92 546 
92 538 
92 530 



92 522 
92 514 
92 506 
92 498 
92 489 



92 481 
92 473 
92 465 
92 457 
92 449 



92 441 
92 433 
92 424 
92 416 
92 408 



92 400 
92 392 
9.92 383 
9.92 375 
9. 92 367 



9.92 359 



i22r 



Log. Tan. Log. S 
679 



m, 



d. 



60 

59 
58 

57 
_56 

55 
54 
53 
52 
51 

50 

49 
48 

47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 

35 
34 
33 
32 
-31 
30 
29 
28 
27 
-26 
25 
24 
23 
22 
21 

20 

19 
18 
17 
11 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 

5 
4 



P.P. 





38 


28 


6 


2.8 


2.81 


7 


3.3 


3 


2 


8 


3.8 


3 


7 


9 


4.3 


4 


2 


10 


4.7 


4 


6 


20 


9.5 


9 


3 


30 


14.2 


14 





40 


19.0 


18 


b 


50 


23.7 


23 


3 



27„ 

2.7 

3.2 

3.6 

4.1 

4.6 

9.1 

.3.7 

i8.3 

^2.9 





20 


20 


6 


2.0 


2.0 


7 


2.4 


2 


3 


8 


2.7 


2 


6 


9 


3.1 


3 





10 


3.4 


3 


3 


20 


6.8 


6 


6 


30 


10.2 


10 





40 


13.6 


13 


3 


50 


17.1 


16 


6 



19^ 
1.9 
2.3 
2.6 
2.9 
3.2 
6.5 
9.7 
13-0 
16.2 





8 


8 , 


e 


o.p 


0.8 


7 


l.f 


0.9 


8 


1.1 


1.0 


9 


1.3 


1 • 2 


10 


1.4 


1 .3 


20 


2-8 


2.6 


SO 


4.2 


4.0 


40 


5.6 


5.3 


50 


7.1 


6.6 



0.9 
0.9 

1.0 

1.1 

1.2 
2.5 
3.7 
5.0 
6.2 



P.P. 



B'f 



33' 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS, 

AND COTANGENTS. 146® 



7 

8 

^ 

10 

11 
12 
13 
li 
15 
16 
17 
18 

ii 
20 
21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 

46 

47 

48 

ii 

50 

51 

52 

53 

51 

55 

56 

57 

58 

59. 

60 



Log. Sin. 



9.73 611 
9.73 630 
9.73 650 
9.73 66? 
9.73 688 



9.73 708 
9.73 727 
73 746 
9.73 766 
9 .73 785 
9-73 805 
73 824 
73 843 
73 862 
73 882 



73 901 
73 920 
73 940 
73 95? 
73 978 



73 997 

74 016 
74 036 
74 055 
74174 

74 093 
74 112 
74 131 
74 151 
74 170 



9.74 189 
9.74 208 
9.74 227 
9-74 246 
9-74 265 



9-74 284 
9-74 303 
9-74 322 
9-74 341 
9-74 360 



9-74 379 
9-74 398 
9-74 417 
9-74 436 
9-74 455 



9-74 474 
9-74 493 
9-74 511 
9 - 74 53?) 
9 - 74 549 



9-74 568 
9-74 587 
9-74 606 
9-74 625 
9 . 74 643 



9-74 662 
9-74 681 
9.74 700 
9-74 718 
9-74 737 



9-74 756 



Log. Cos 



1? 
19 

1? 
19 

19 
19 
19 
1? 
19 

19 

1? 
19 
19 
19 
19 
19 
19 
19 
19 

19 

1? 
19 
19 
•19 

19 
19 

1? 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
18 

19 
19 
19 
19 
19 

19 
19 
18 
19 
19 

19 
18 
19 
19 
18 
19 
18 
19 
18 
19 

18 



Log. Tan. 



81 251 
81 279 
81 307 
81 334 
81 362 



c.d, 



81 390 
81 417 
81 445 
81 473 
81 500 



81 528 
81 555 
81 583 
81 610 
81 638 



81 666 
81 693 
81 721 
81 748 
81776 



81 803 
81 831 
81 858 
81 886 
81 913 



81 941 
81 968 

81 996 

82 023 
82 051 



82 078 
82 105 
82 133 
82 160 
82 188 



82 215 
82 243 
82 270 
82 297 
82 325 



82 352 
82 380 
82 407 
82 434 
82 462 



82 48? 
82 516 
82 544 
82 571 
82 598 



82 626 
82 653 
82 680 
82 708 
82 735 



82 762 
82 789 
82 817 
82 844 
82 871 



82 898 



d. Log. Cot 



28 
27 
27 
28 

27 
27 
27 
28 
27 

27 
27 
27 
27 
27 
28 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 



Log. Cot. 



c.d 



18 748 
18 720 
18 693 
18 665 
18 637 



18 610 
18 582 
18 555 
18 527 
18 499 



18 472 
18 444 
18 417 
18 389 
18 362 



18 334 
18 306 
18 27? 
18 251 
18 224 



18 196 
18 169 
18 141 
18 114 
18 086 



Log. Cos. d- 



18 05? 
18 031 
18 004 
17 976 
17 949 



17 921 
17 894 
17 867 
17 839 
17 812 



17 784 
17 757 
17 729 
17 702 
17 675 



17 647 
17 620 
17 593 
17 565 
17 538 



17 510 
17 483 
17 45f 
17 42P 
17 4CI 



17 374 
17 347 
17 319 
17 292 
17 265 



17 237 
17 210 
17 183 
17 156 
17 128 



17 lOl 



Log. Tan, 



92 359 
92 351 
92 342 
92 334 

92 326 

92 318 
92 310 
92 301 
92 293 
92 285 



92 277 
92 268 
92 260 
92 252 
92 244 



92 235 
92 227 
92 21? 
92 210 
92 202 



92 194 
92 185 
92 177 
92 16? 
92 160 



92 152 
92 144 
92 135 
92 127 
92 119 



92 110 
92 102 
92 094 
92 085 
92 077 



92 06? 
92 060 
92 052 
92 043 
92 035 



92 027 
92 018 
92 010 
92 001 
91 993 



91 984 
91 976 
91 967 
91 959 
91 951 



91 942 
91 934 
91 925 
91 917 
91 908 



91 900 
91 89l 
91 883 
91 874 
91 866 



991 857 
Log. Sin 



60 

59 
58 
57 
-56. 
55 
54 
53 
52 
IL 
50 
49 
48 
47 

Ji 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
JA 
15 
14 
13 
12 
11 



10 

9 

8 

7 

_6 

5 
4 
3 
2 
_1 
O 



P.P. 





28 


27 


27 


6 


2-8 


2.7 2-7 


7 


3 


2 


3 


2 3 


1 


8 


3 


7 


3 


6 3 


6 


9 


4 


2 


4 


1 


4 





10 


4 


6 


4 


6 


4 


5 


20 


9 


3 


9 


1 


9 





30 


14 





13 


7 


13 


5 


40 


18 


6 


18 


3 


18 





50 


23 


3 


22 


9122 


5 





19 


19 


18. 


6 


1.9 


1.9 


18 


7 
8 


2.3 
2.6 


2 
2 


1 


2 
2 


I 


9 


2.9 


2 


8 


2 


8 


10 


3.2 


3 


1 


3 


1 


20: 6-5 


6 


3 


6 


1 


30 9-7 


9 


5 


9 


2 


40 130 


12 


6 


12 


3 


50 


16.2 


15 


8 


15 


4 



8_ 8 



0.8 


0. 


1.0 


0. 


1.1 


1- 


1.3 


1- 


1.4 


1- 


2-8 


2. 


4.2 


4. 


5.6 


5. 


7.1 


6. 



P.P. 



133' 



680 



5e« 



34' 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS, 
AND COTANGENTS. 



145' 



Log. Sin. 



74 756 
74 775 
74 793 
74 812 
74 831 



74 849 
74 868 
74 887 
74 905 
74 924 



74 943 
74 961 
74 980 

74 998 

75 017 



75 036 
75 054 
75 073 
75 091 
75 110 



75 128 
75 147 
75 165 
75 184 
75 202 



75 221 
75 239 
75 257 
75 276 
75 294 



75 313 
75 331 
7.5 349 
75 368 
75 386 



75 404 
75 423 
75 441 
75 459 
75 478 



75 496 
75 514 
75 532 
75 551 
75 569 



75 587 
75 605 
75 623 
75 642 
75 660 



75 678 
75 696 
75 714 
9. 75 732 
9.75 750 



9-75 769 
9.75 787 
9.75 805 
9.75 823 
9.75 841 



975 859 



Log. Cos. 



1? 
18 
19 
18 

18 
19 
18 
18 
19 

18 
18 
18 
18 
18 

1? 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 



d. 



Log. Tan, c. d. Log. Cot, Log. Cos 



9-82 898 
9.82 926 
9.82 953 

9.82 980 
9^83_007 

9.83 035 
9.83 062 
9.83 089 
9.83 116 
9.83 143 



83 171 
83 198 
83 225 
83 252 
83 279 



9.83 307 
9.83 334 
9.83 361 
9.83 388 

9..83415 



9.83 442 
83 469 
9.83 496 
9.83 524 
83 551 



9.83 578 
9.83 605 
9.83 632 
83 659 
9.83 686 



83 713 
83 740 
83 767 
9.83 794 
9.83 821 



83 848 
9.83 875 
9.83 902 

83 929 
9-83 9-^7 



83 984 

84 011 
84 038 
84 065 
84 091 



•84 118 

• 84 145 

• 84 172 
•84 199 

9.84 226 



9-84 253 
9-84 280 
9-84 307 
9-84 334 
9.84 361 



9-84 388 
9-84 415 
9 . 84 44? 
9.84 469 
9.84 496 



9 - 84 522 



27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
26, 

27 
27 
27 
27 
27 

27 
97 
27 
^7 
26 

27 
27 
27 
27 
27 
26 



0-17 101 
0.17 074 
0.17 047 
0.17 019 
0.16 992 



0.16 965 
0^.16 938 
0.16 910 
0.16 883 
0.16 856 



0.16 55v 
0.16 530 
0.16 503 
16 476 
0.16 449 

0.16 422 
0.16 395 
0.16 368 
16 340 
0.16 313 



0.16 829 
0.16 802 
16 774 
16 747 
0.16 720 



0.16 693 
0.16 666 
16 639 

16 612 
16 584 



9.91 857 
9-91 849 
9-91 840 
9-91 832 
9.91 823 
9.91 814 
9 91 806 
9.91 797 
9.91 78? 
91 780 



9.91772 
91 763 
9.91 755 
9.91 746 
9.91 737 



9.91 686 
91 677 
9.91 668 
9.91 660 
9.91 651 



9.91 642 
9.91 634 
91 625 
91 616 
9-91 608 



0.16 286 
0.16 259 
0.16 232 
0.16 205 
0.16 178 



0.16 151 
0.16 124 
0-16 097 
0.16 070 
0-16 043 



0.16 016 
0.15 989 
0.15 962 
0.15 935 
0-15 90P 



91 72? 
91 720 
91 712 
91 703 
91 694 



9-91 59? 
9-91 590 
9-91 582 
9.91 573 
9-91 564 



91 556 
91 547 
91 538 
91 529 
91 521 



9-91 512 
9 . 91 50§ 
9 . 91 495 
9-91 486 
9 - 91 47^ 



15 88T 
15 85? 
15 827 
15 800 
15 773 



0-15 746 
0-15 715 
0.15 695 
0.15 66F 
0.15 639 



0.15 61? 
0.15 58P 
0.15 558 
0.15 531 
0.15 504 
0.15 477 



Log. Cot.lc.d. Iig. Tan 



9-91468 
9-91 460 
9.91 45T 
9 . 91 449 
9 . 91 433 



9.91 424 
9.91 416 
9.91 407 
9.91 398 
91 389 



9. 91 38(5 
9.91 372 
9.91 363 
9-91 354 
9-91 345 



9. 91 336 



VZ4f 



Log, Sin 
681 



60 

59 
58 
57 
56 



55 
54 
53 
52 
_51 
50 
49 
48 
47 
16 

45 
44 
43 
42 

-ii. 

40 
39 
38 

37 
_36 
35 
34 
33 
32 
11 
30 
29 
28 
27 
21 
25 
24 
23 
22 
21 

20 

19 
18 
17 

li 

15 
14 
13 
12 
\1 

10 

9 

8 

7 

_6 

5 
4 
3 
2 
_1 




P. P. 





27 


27 


6 


2.7 


2.7 


7 


3.2 


3.1 


8 


3.6 


3.6 


9 


4.1 


4.0 


10 


4.6 


4.5 


20 


9.1 


9.0 


30 


13.7 


13.5 


40 


18.3 


18.0 


50 


22-9 


22.5 



26 

2.6 

3.1 

3.5 

4.0 

4.4 

8.8 

13.2 

17.6 

22.1 





19 


18 


6 


1.9 


1.8 


7 


2 


2 


2.1 


8 


2 


5 


2.4 


9 


2 


3 


2.8 


10 


3 


1 


3.1 


20 


6 


3 


6.1 


30 


9 


5 


9.2 


40 


12 


6 


12.3 


50 


15 


8 


15.4 



18 

1.8 
2.1 
2.4 
2.7 
30 
6.0 
9.0 
12.0 
15.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 






9 


0. 


1 





1 


1 


2 


1. 


1 


3 


1. 


1 


5 


1- 


3 





2. 


4 


5 


4- 


6 





5- 


7 


5 


7. 



p. p. 



6tf> 



35*= 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



144' 



Log. Sin. d. Log. Tan. c. d. Log. Cot. Log. Cos 



9.75 
75 



859 
877 
895 
913 
931 

949 
967 
985 
003 
021 



039 
057 
075 
092 
110 



128 
146 
164 
182 
200 



217 
235 
253 
271 
289 



306 
324 
342 
360 
377 



395 
413 
431 
448 
466 



484 
501 
519 
536 
554 



572 
589 
607 
624 
642 



660 
677 
695 
712 
730 



747 
765 
782 
800 
817 



835 
852 
869 
887 
904 



76 922 



Log. Cos 



84 522 
84 549 
84 576 
84 603 
84 630 



9-84 657 
9.84 684 
9.84 711 
9.84 737 
9.84 764 



9.84 791 
9.84 818 
9 . 84 845 
9.84 871 
9.84 898 



9.84 925 
9.84 952 

9.84 979 

9.85 005 
9.85 032 



9.85 059 
9.85 086 
9.85 113 
9.85 139 
9.85 166 



9.85 193 
9.85 220 
9.85 246 
9.85 273 
9.85 300 



9.85 327 
9.85 353 
9.85 380 
9.85 407 
9.85 433 



9.85 460 
9.85 487 
9.85 513 
9.85 540 
9.85 567 



85 594 
85 620 
8? 647 
85 673 
85 700 



85 727 
85 753 
85 780 
85 807 
85 833 



9.85 860 
9.85 887 
9.85 913 
9-85 940 
9.85 966 



9.85 993 

9.86 020 
9.86 046 
9.86 073 
9.86 099 



9.86 126 



Log. Cot. 



27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
26 
27 
26 
27 

27 
26 
27 
26 
27 

27 

26 
27 
26 
27 
26 
27 
26 
27 
26 

27 
26 
26 
27 
26 

27 
26 
26 
27 
26 

27 
26 
26 
26 
27 

26 
26 
27 
26 
26 

26 
27 
26 
26 
26 

26 
27 
26 
26 
26 

26 



c d 



15 477 
15 450 
15 423 
15 39b 
15 370 



15 343 
15 316 
15 289 
15 262 
15 235 



15 20b 
15 182 
15 155 
15 12b 
15 101 



074 
048 
021 
994 
967 



940 
914 
887 
860 
833 



14 807 
14 780 
14 753 
14 726 
14 700 



14 673 
14 646 
14 620 
14 593 
14 566 



14 539 
14 513 
14 486 
14 459 
14 433 



14 406 
14 379 
14 353 
14 326 
14 299 



14 273 
14 246 
14 219 
14 193 
14 166 



14 140 
14 113 
14 086 
14 060 
14 033 



14 007 
13 980 
13 953 
13 927 
13 900 



0.13 874 



Log. Tan, 



91 336 
91327 
91 318 
91 310 
91 301 



91 292 
91 283 
91 274 
91 265 
91 256 



91247 
91 239 
91230 
91 221 
91 212 



91 203 
91 194 
91 185 
91 176 
91 167 



91 158 
91 149 
91140 
91131 
91 122 



91 113 
91 104 
91 095 
91 086 
91 077 



91 068 
91 059 
91 050 
91041 
91 032 



91023 
91014 
91005 
90 996 
90 987 



90 978 
90 969 
90 960 
90 951 
90 942 



90 933 
90 923 
90 914 
90 905 
90 896 



90 887 
90 878 
90 869 
90 860 
90 850 



90 841 
90 832 
90 823 
90 814 
90 805 



9.90 796 



125' 



Log, Sin. 

682 



d. 



60 
59 
58 
57 
56 



55 
54 
53 
52 
11 
50 
49 
48 
47 

li 

45 
44 
43 
42 
41 



40 

39 
38 
37 
16 

35 
34 
33 
32 
31 



30 

29 
28 

27 
26 



25 
24 
23 
22 
21 

30 

19 
18 

17 

ii 

15 
14 
13 
12 
il 

10 

9 
8 

7 
6 

5 
4 
3 

2 

_JL 




P. P. 





27 


26 


6 


2 7 


2.6 


7 


3 


1 


3.1 


8 


3 


e 


3.5 


9 


4 


C 


4-0 


IC 


4 


5 


4.4 


20 


9 


C 


88 


30 


13 


5 


13.2 


40 


18 





17.6 


50 


22 


5 


I22.I 



18 


17 


17 


1.8 


1.7 


1. 


2 


1 


2.0 


2. 


2 


4 


2.3 


2. 


2 


7 


2-6 


2. 


3 





2.9 


2. 


6 





5.8 


5 


9 


C 


8.7 


8. 


12 


C 


11.6 


11. 


15 





14.6 


14. 





9 


9 


6 


0.9 


0.9 


7 


1.1 


1.0 


8 


1.2 


1.2 


9 


1.4 


1.3 


IC 


1.6 


1.5 


20 


3-1 


3.0 


30 


4.7 


4.5 


40 


6.3 


6.0 


50 


7.9 


7.5 



8 
0.8 
1.0 
1.1 
1-3 
1-4 
2-8 
4.2 
5.6 
7.1 



P.P. 



54= 



36' 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 

AND COTANGENTS. 143* 



Log. Sin. d 



o 

1 

2 
3 

5 
6 
7 
8 

10 

11 

12 

13 

11 

15 

16 

17 

18 

19 



9.76 922 
9.76 939 
76 956 
76 974 
76 991 



9.77 008 
9.77 026 
9-77 043 
77 060 
77 078 



30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



77 095 
77 112 
77 130 
77 147 
77 164 



77 181 
77 198 
77 216 
77 233 
77 250 



77 267 
77 284 
77 302 
77 319 
77 336 



77 353 
77 370 
77 387 
77 404 
77 421 



77 439 
9-77 456 
9-77 473 
9-77 490 
9-77 507 



40 

41 
42 
43 
4^ 

45 
46 
47 
48 
49_ 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



77 524 

77 541 

77 558 

9-77 575 

9-77 592 



77 609 
77 626 
77 643 
77 660 
77 677 



77 693 
77 710 
77 727 
77 744 
77 76l 



77 778 
9-77 795 
9-77 812 
9-77 828 
9.77 845 



77 862 
9.77 879 
9.77 896 
9-77 913 
9-77 929 



9-77 946 



17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
16 
17 
17 
17 
17 

16 
17 
17 
16 
17 

17 
16 
17 
17 
16 

17 



86 126 
86 152 
9-86 179 
9-86 206 
9-88 232 



9-86 259 
9-86 285 
9-86 312 
9-86 338 
86 365 



Log. Cos, 



Log. Tan. 



86 391 
9-86 418 
9-86 444 

86 471 
9-86 497 



9-86 524 
9-86 550 
9-86 577 
86 603 
9-86 630 



9-36 656 
9-86 683 
9-86 709 
9.86 736 
86 762 



9.86 788 
9.86 815 
9.86 841 
9-86 868 
9-86 894 



9.86 921 
9.86 947 

9.86 973 

9.87 000 
9j_87JD26 

9.87 053 
9.87 079 
9.87 105 
9.87 132 
9.87 158 



9.87 185 
87 211 
9.87 237 
9.87 264 
9-87 290 



87 316 
87 343 
9-87 369 
9-87 395 
9-87 422 



9-87 448 
9-87 474 
9-87 501 
9-87 527 
9-87 553 



9-87 580 
9-87 606 
9-87 632 
9-87 659 
9-87 685 



9-87 711 
Log. Cot. 



c.d. 

26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 

26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 

cTJ 



Log. Cot, 



13 874 

13 847 

0-13 821 

0-13 794 

0-13 767 



0-13 741 
0.13 714 
0-13 688 
0-13 66l 
0-13 635 



0-13 60£ 
0-13 582 
0.13 555 
0.13 528 
0.13 502 



Log. Cos, 



9-90 796 
9-90 786 
9-90 777 
90 768 
9-90 759 



9-90 750 
9-90 740 
9.90 731 
9.90 722 
9-90 713 



9-90 703 
9 . 90 694 
9.90 685 
9.90 676 
9.90 666 



0.13 476 
0.13 44£ 
0.13 423 
0.13 391 
0-13 37C 



13 343 
0-13 317 
0.13 290 
0.13 264 
0.13 237 

. 13 2ll 
0.13 185 
0.13 15£ 
0.13 132 
0.13 10" 



90 657 
90 648 
90 639 
90 629 
90 620 

90 611 
90 602 
90 592 
90 583 
90 574 



9-90 564 
9-90 555 
9.90 546 
9-90 536 
9-90 527 



0-13 078 
0-13 052 
0.13 026 
0-13 OOC 
0-12 973 



0-12 947 
0.12 92C 
0.12 894 

0.12 see 

0.12 841 



0.12 BIZ 
0.12 78e 
0.12 762 
0.12 73C 
0.12 7CG 



0.12 e8c 
0.12 657 

0.12 esr 
0.12 ec< 

0.12 57C 



0.12 551 
0.12 52e 
0.12 49r 
0.12 47? 

0-12 ue 



0-12 420 
0-12 393 
0-12 367 
0-12 341 
0-12 315 



0-12 288 
Log. Tan. 



9-90 518 
9-90 508 
9-90 499 
9 . 90 490 
9 - 90 480 



90 471 
90 461 
90 452 
90 443 
90 433 



9-90 424 
9-90 414 
9-90 405 
9.90 396 
9.90 386 



9.80 377 
9.90 367 
9.90 358 
9.80 348 
9_.90 339 
9 
9 
9 



90 330 
90 320 
90 311 
90 301 
90 292 



9-90 282 
9-90 273 
9-90 263 
9-90 254 
9 . 90 244 



136' 



9-90 235 
Log. Sin 

683 



60 

59 
58 
57 
56 



55 
54 
53 

52 

11 
50 

49' 
48 
47 
46 

45 
44 
43 
42 
IL 
40 
39 
38 
37 
36 

35 
34 
S3 
32 
31 
30 
29 
28 
27 
26 



25 
24 
23 
22 
21 

30 

19 
18 
17 
16 

15 
14 
13 
12 
11 



10 

9 

8 

7 

_6 

5 
4 
3 

2 
1 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



37 


36 


2.7 


2-6 


3.1 


3.1 


3.6 


3.5 


4.0 


4.0 


4.5 


4.4 


9.0 


8-t 


13.5 


13-2 


18.0 


i7.e 


22-5 


22.1 



36 

n 

3.5 

3.9 

4.3 

8.6 

13. 

17-1 

21.6 





17 


1*^ 


6 


1-7 


1-71 


7 


2-0 


2 





8 


2-3 


2 


2 


9 


2.6 


2 


5 


10 


2.9 


2 


8 


20 


5-8 


5 


6 


30 


8.7 


8 


5 


40 


11.6 


11 


3 


50 


14.6 


14 


• 1 



16_ 

1.6 
1.9 
2.2 
2.5 
2.7 
5.5 
8.2 
11.0 
13.^ 





S 


9 


6 


0.9 


0.9 


7 


1.1 


1.0 


8 


1.2 


1 "2 


9 


1.4 


1.3 


10 


1.6 


1.5 


20 


3-1 


3.0 


30 


4.7 


4.5 


40 


6.3 


6.0 


50 


7.9 


7.5 



P.P. 



63'= 



37* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 143' 



Log. Sin. 



9.77 946 
9-77 963 
9.77 980 

9.77 996 

9.78 013 



78 030 
78 046 
9.78 063 
9.78 080 
78 097 



9.78 113 
9.78 130 
9.78 147 
78 163 
9.78 180 



9-78 196 
78 213 
9.78 230 
9.78 246 
9-78 263 



78 279 
78 296 
78 312 
78 329 
78 346 



78 362 
78 379 
78 395 
78 412 
78 428 



78 444 
78 461 
78 477 
78 494 
78 510 



78 527 
78 543 
78 559 
78 576 
78 592 



9-78 609 
9.78 625 
78 641 
9.78 658 
9-78 674 



78 690 
78 707 
78 723 
78 739 
78 755 



78 772 
78 788 
78 804 
78 821 
78 837 



78 853 
78 869 
78 885 
78 902 
78 918 



9. 78 934 



Log, Cos, 



16 
17 
16 
17 

16 
16 

IZ 
16 
17 

16 
16 
17 
16 
16 

16 
16 
17 
16 
16 

16 
16 
16 
16 
17 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 



d. 



Log. Tan, 



89 



711 
737 
764 
790 
816 

843 
869 
895 
921 
948 

974 
000 
026 
053 
079 

105 
131 
157 
184 
210 

236 
262 
288 
315 
341 

367 
393 
419 
445 
472 

498 
524 
550 
576 
602 

629 
655 
681 
707 
733 

759 
785 
811 
838 
864 

890 
916 
942 
968 
994 

020 
046 
072 
098 
124 

150 
177 
203 
229 
255 
281 



Log. Cot. c.d 



c.d, 



26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 



Log. Cot 



0.12 288 
0.12 262 
0.12 236 
0.12 209 
0.12 183 



0.12 157 
0.12 131 
0.12 104 
0.12 078 
0.12 052 



0.12 026 
0.11 999 
0.11 97'b 
0.11 947 
0.11 921 



0.11 895 
0.11861 
0.11 842 
0.11816 
0.11 7GC 



0.11 7ec 
0.11 737 
0.11 7ll 
0.11 68£ 
0.11 658 



0-11 633 
0.11 6C6 
0.11 58C 
0.11 554 
0.11 52 



0.11 502 
0.11476 
0.11 44£ 
0.11 423 
0.11 397 



0.11 371 
0.11 345 
0.11 319 
0.11 293 
0.11 266 



0.11 24C 
0.11 214 
0.11 188 
0.11 162 
0.11 136 



0.11 110 
0.11084 
0.11 058 
0.11 032 
0.11 005 



0.10 97£ 
0.10 95? 
0.10 927 
0.10 9C1 
0.10 87f 



0.10 84P 
0.10 823 
0-10 797 
0.10 771 
10 745 



0-10 719 
Log. Tan. 



Log. Cos 



90 235 
90 225 
90 216 
90 206 
90 196 



90 187 
90 177 
90 168 
90 158 
90 149 



90 139 
90 180 
90 120 
90 110 
90 101 



90 091 
90 082 
90 072 
90 062 
90 053 



90C43 
90 033 
90 024 
90 014 
90 004 



89 995 
89 985 
89 975 
89 966 
89 956 



89 946 
89 937 
89 927 
89 917 
89 908 



89 898 
89 888 
89 878 
89 869 
89 859 



89 849 
89 839 
89 830 
89 820 
89 81C 



89 SCO 
89 791 
89 781 
89 771 
89 761 



89 751 
89 742 
89 732 
89 722 
89 712 



89 702 
89 692 
89 683 
89 673 
89 663 



9-89 653 



Log. Sin, 



d. 



60 

59 
58 
57 
56 

55 
54 
53 
52 
_51 

50 

49 
48 
47 
M. 
45 
44 
43 
42 
41 



40 

39 
38 
37 
_36 

35 
34 
33 
32 
11 
SO 
29 
28 
27 
11 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 

8 

7 

__6 

5 
4 
3 
2 
_1^ 





P.P. 





26 


26 


6 


2.6 


2.6 


7 


3 


1 


3 





8 


3 


5 


3 


4 


9 


4 





3 


9 


10 


4 


4 


4 


3 


20 


8 


8 


8 


6 


30 


13 


2 


13 





40 


17 


6 


17 


3 


50 


22 


1 


21 


6 





17 


16 


6 


1.7 


1.6 


7 


2.0 


1-9 


8 


2.2 


2.2 


9 


2-5 


2.5 


IC 


2-8 


2.7 


20 


5.6 


5-5 


30 


85 


8.2 


40 


11.3 


11. C 


50 


14.1 


13.7 



16 

1.6 
1.8 
2.1 
2.4 
2.6 
5-3 
8-0 
10.6 
13.3 



10 


9 


61.0!0.9 


71 


11 


1 


81 


3'l 


2 


91 


5 1 


4 


10 1 


6 1 


6 


20 3 


33 


1 


30 5 


4 


7 


40 6 


66 


3 


50 8 


37 


9 



P. p. 



1S7* 



^ 684 



52* 



38' 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS. 
AND COTANGENTS. 



141' 



Log. Sin, 



78 934 
78 950 
78 966 
78 982 
78 999 



79 015 
79 031 
79 047 
79 063 
79 079 



79 095 
79 111 
79 127 
79 143 
79 159 



79 175 
79 191 
79 207 
79 223 
79 239 



79 255 
79 271 
79 287 
79 303 
79 319 



79 335 
79 351 
79 367 
79 383 
79 399 



79 415 
79 431 
79 446 
79 462 
79 478 



79 494 
79 510 
79 526 
79 541 
79 557 



79 573 
79 589 
79 605 
79 620 
79 636 



79 652 
79 668 
79 683 
79 699 
79 715 



79 730 
79 746 
79 762 
79 777 
79 793 



79 809 
79 824 
79 840 
79 856 
79 871 



79 887 



Log. Cos, 



Log. Tan 



89 281 
89 307 
89 333 
89 359 
89 385 



89 411 
89 437 
89 463 
89 489 
89 515 



89 541 
89 567 
89 593 
89 619 
89 645 



89 671 
89 697 
89 723 
89 749 
89 775 



89 801 
89 827 
89 853 
89 879 
89 905 



89 931 
89 957 

89 982 

90 008 
90 034 



90 060 
90 086 
90 112 
90 138 
90 164 



90 190 
90 216 
90 242 
90 268 
90 294 



90 319 
90 345 
90 371 
90 397 
90 423 



90 449 
90 475 
90 501 
90 526 
90 552 



90 578 
90 604 
90 630 
90 656 
90 682 



90 707 
90 733 
90 759 
90 785 
90 811 



90 837 



d. Log. Cot, 



c.d 



26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
25 
26 
26 

26 
26 
26 
26 
25 

26 
26 
26 
26 
26 
25 
26 
26 
26 
25 

26 
26 
26 
25 
26 

26 
26 
25 
26 
26 

25 
26 
26 
26 
25 
26 



C.d, 



Log. Cot 



10 719 
10 693 
10 667 
10 641 
10 615 



10 589 
10 563 
10 537 
10 511 
10 485 



10 459 
10 433 
10 407 
10 381 
10 355 



10 329 
10 303 
10 277 
10 251 
10 225 



10 199 
10 173 
10 147 
10 121 
10 095 



10 069 
10 043 
10 017 
09 991 
09 965 



09 939 
09 913 
09 887 
09 861 
09 836 



09 810 
09 784 
09 758 
09 732 
09 706 



09 680 
09 654 
09 628 
09 602 
09 577 



09 551 
09 525 
09 499 
09 473 
09 447 



09 421 
09 395 
09 370 
09 344 
09 318 



09 292 
09 266 
09 240 
09 214 
09 189 



0-09 163 
Log. Tan. 



Log. Cos 



89 653 
89 643 
89 633 
89 623 
89 613 



89 604 
89 594 
89 584 
89 574 
89 564 



89 554 
89 544 
89 534 
89 524 
89 514 



89 504 
89 494 
89 484 
89 474 
89 464 



89 454 
89 444 
89 434 
89 424 
89 414 



89 404 
89 394 
89 384 
89 374 
89 364 



89 354 
89 344 
89 334 
89 324 
89 314 



89 304 
89 294 
89 284 
89 274 
89 264 



89 253 
89 243 
89 233 
89 223 
89 213 



89 203 
89 193 
89 182 
89 172 
89 162 



89 152 
89 142 
89 132 
89 121 
89 111 



89 101 
89 091 
89 081 
89 070 
89 060 



9-89 050 



138' 



Log. Sin, 
685 



d. 



60 

59 
58 

57 
56 



55 
54 
53 
52 
11 
50 
49 
48 
47 
j46 

45 
44 
43 
42 
_41 

40 

39 
38 
37 
16 

35 
34 
33 
32 
IL 
30 
29 
28 
27 
26_ 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16- 
15 
14 
13 
12 
11 



10 

9 
8 
7 
_6_ 

5 

4 
3 
2 
l_ 

O 



P. P. 





26 


6 


2.61 


7 


3 





8 


3 


4 


9 


3 


9 


10 


4 


3 


20 


8 


6 


30 


13 





40 


17 


3 


50)21 


6 



25 

2.5 

3-0 

3-4 

38 

4.2 

8.5 

12-7 

17-0 

21.2 





16 


16 


1^ 


6 


1.6 


1-6 


1. 


7 


1 


9 


1 


3 


1 


8 


2 


2 


2 


1 


2. 


9 


2 


5 


2 


4 


2. 


10 


2 


7 


2 


6 


2. 


20 


5 




5 


3 


5. 


30 


8 


2 


8 





7- 


40 


11 


10 


6 


10. 


50 


13 


7 


13 


3 


12. 



10 10 



00 

11 

31 

5 1 

6 1 
33 
0!4 
6,6 
317.9 



P. P. 



5r 



39* 



TABLE VIL— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



140^ 



O 

1 
2 
3 

5 
6 
7 
8 
_9^ 

10 

11 

12 

13 

li- 

15 

16 

17 

18 

19 



9.79 887 
79 903 

9.79 918 
79 934 

9.79 949 



Log. Sin. 



79 965 
79 980 

79 996 

80 011 
80 027 



9 . 80 042 
9. 80 058 
9. 80 073 
9-80 089 
9-80 104 



9-80 120 
9. 80 135 
9-80 151 
80 166 
9. 80 182 



30 

21 
22 
23 
24 

25 

26 

27 

28 

29L 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 

46 

47 

48 

41 

50 

51 

62 

53 

54 

55 
56 
57 
58 
59_ 
60 



80 197 
80 213 
80 228 
80 243 
80 259 



80 274 
80 289 
80 305 
80 320 
80 335 



80 351 
80 366 
80 381 
80 397 
80 412 



80 427 
80 443 
80 458 
80 473 
80 488 



9-80 504 
9.80 519 
9.80 534 
9 . 80 549 
9-80 564 



80 580 
80 595 
80 610 
80 625 
80 640 



80 655 
80 671 
9. 80 686 
9.80 701 
9.80 716 



80 731 
80 746 

9. 80 761 
80 776 

9. 80 791 



9. 80 806 



Log. Cos. 



— 

16 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
15 
15 
15 
15 
15 

15 



Log. Tan, 



c. di 



90 837 
90 863 
90 888 
90 914 
90 940 



90 966 

90 992 

91 017 
91043 
91 069 



91 095 
91 121 
91 146 
91 172 
91 198 



91 224 
91 250 
91 275 
91301 
91 327 



91 353 
91 378 
91 404 
91430 
91456 



91481 
91 507 
91 533 
91 559 
91 584 



91 610 
91 636 
91 662 
91 687 
91 713 



91 739 
91 765 
91 790 
91 816 
91 842 



91 867 
91893 
91919 
91945 
91 970 



91 996 

92 022 
92 047 
92 073 
92 099 

92 124 
92 150 
92 176 
92 201 
92 227 



92 253 
92 278 
92 304 
92 330 
92 355 



92 381 



d. Log. Cot, 



26 
25 
26 
25 

26 
26 
25 
26 
26 

25 
26 
25 
26 
25 
26 
26 
25 
26 
25 

26 
25 
26 
25 
26 

25 
26 
25 
26 
25 
26 
25 
26 
25 
26 

25 
26 
25 
26 
25 

25 
26 
25 
26 
25 

25 
26 
25 
26 
25 

25 
26 
25 
25 
26 

25 
25 
26 
25 
25 
26 



c.d, 



Log. Cot. Log. Cos 



09 163 
09 137 
09 111 
09 085 
09 060 



09 034 
09 008 
08 982 
08 956 
08 930 



08 905 
08 879 
08 853 
08 827 
08 802 



08 776 
08 750 
08 724 
08 698 
08 673 



08 647 
08 621 
08 595 
08 57C 
08 544 



08 518 
08 492 
08 467 
08 441 
08 415 



08 389 
08 364 
08 338 
08 312 
08 286 



08 261 
08 235 
08 209 
08 183 
08 158 



08 132 
08 106 
08 081 
08 055 
08 029 



08 004 
07 978 
07 952 
07 926 
07 901 



07 875 
07 849 
07 824 
07 798 
07 772 



07 747 
07 721 
07 695 
07 67C 
07 644 



0.07 618 



Log. Tan. 



89 050 
89 040 
89 030 
89 019 
89 009 



88 999 
88 98? 
88 978 
88 968 
88 958 



88 947 
88 937 
88 927 
88 917 
88 906 



88 896 
88 886 
88 875 
88 865 
88 855 



88 844 
88 834 
88 823 
88 813 
88 803 



88 792 
88 782 
88 772 
88 761 
88 751 



88 740 
88 730 
88 720 
88 709 
88 699 



88. 688 
88 678 
88 667 
88 657 
88 646 



88 636 
88 625 
88 615 
88 604 
88 594 



88 583 
88 573 
88 562 
88 552 
88 541 



88 531 
88 520 
88 510 
88 499 
88 489 



88 478 
88 467 
88 457 
88 446 
88 436 



9 
9 
9 
9 

9. 88 425 



Log. Sin. 



d. 



60 

59 
58 
57 
56 

55 
54 
53 
52 
11 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 



40 

39 
38 
37 



35 
34 
33 
32 
II 
30 
29 
28 
27 

25 
24 
23 
22 
21 

20 

19 
18 
17 

li 

15 
14 
13 
12 
U 

10 

9 
8 

7 
6 

5 
4 
3 

2 
_1 

O 



P. p. 





26 


25_ 


6 


2.6 


2.5 


7 


3 





3 





8 


3 


4 


3 


4 


8 


3 


9 


3 


8 


10 


4. 


3 


4 


2 


20 


8 


6 


8 


5 


30 


13 





12 


7 


40 


17 


3 


17 





50 


21 


6 


21 


2 





16 


15 


6 


1.6 


1.51 


7 


1 


8 


1 


8 


8 


2 


1 


2 





9 


2 


4 


2 


3 


10 


2 




2 


6 


20 


5 


3 


5 


1 


30 


8 





7 


7 


40 


10 


6 


10 


3 


50 


13 


3 


12 


9 



15 

1.5 
1.7 
20 
2.2 
2.5 
5.0 
7.5 
10.0 
12.5 





11 


10 


10 


6 


1.1 


1.0 


1.0 


7 


1 


3 




2 


1 


1 


8 


1 


4 




4 


1 


3 


9 


1 


6 




6 


1 


5 


10 


1 


g 




7 


1 


6 


20 


3 


6 


3 




3 


3 


30 


5 


5 


5 


25 





40 


7 


3 


7 


06 6 


50 


9 


1 


8 


7 


8 


3 



P. p. 



139' 



686 



50*^ 



40^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



139' 



Log. Sin 



80 806 
80 822 
80 837 
80 852 
80 867 



80 882 
80 897 
80 912 
80 927 
80 942 



80 957 
80 972 

80 987 

81 001 
81 016 



81031 
81 046 
81061 
81 076 
81 091 



81 106 
81 121 
81 136 
81 150 
81 165 



81 180 
81 195 
81 210 
81 225 
81 239 



81 254 
81 269 
81 284 
81 299 
81 313 



81 328 
81 343 
81 358 
81 372 
81 387 



81 402 
81416 
81431 
81 448 
81 460 

81475 
81490 
81 504 
81 519 
81 534 



81 548 
81 563 
81 578 
81 592 
81 607 



81 621 
81 636 
81 650 
81 665 
81 680 



9-81 694 



Log, Cos. 



15 
15 
15 
15 

15 
15 
15 
15 
15 
15 
15 
15 
14 
15 

15 
15 
15 
15 
14 

15 
15 
15 
14 
15 

15 
14 
15 
15 
14 

15 
14 
15 
15 
14 

15 
14 
15 
14 
14 

15 
14 
15 
14 
14 

15 
14 
14 
15 
14 

14 
14 
15 
14 
14 

14 
14 
14 
14 
15 
14 



d. 



Log. Tan. c. d 



92 381 
92 407 
92 432 
92 458 
92 484 



92 509 
92 535 
92 581 
92 586 
92 612 



92 638 
92 663 
92 689 
92 714 
92 740 



92 766 
92 791 
92 817 

92 842 
92 838 



92 894 
92 919 
92 945 
92 971 
92 996 



93 022 
93 047 
93 073 
93 098 
93 124 



93 150 
93 175 
93 201 
93 226 
93 252 



93 278 
93 303 
93 329 
93 354 
93 380 



93 405 
93 431 
93 456 
93 482 
93 508 



93 533 
93 559 
93 584 
93 610 
93 635 



93 661 
93 686 
93 712 
93 737 
93 763 



93 788 
93 814 
93 840 
93 865 
93 891 



93 916 



Log. Cot. c. d 



25 
25 
26 
25 

25 
25 
26 
25 
25 

26 
25 
25 
25 
26 

25 
25 
25 
25 
26 

25 
25 
25 
26 
25 

25 
25 
25 
25 
26 

25 
25 
25 
25 
25 

26 
25 
25 
25 
25 

25 
25 
25 
25 
26 

25 

25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
26 
25 
25 
25 



Log. Cot, 



07 618 



07 593 
07 567 
07 541 
07 516 9 



Log. Cos. 



07 490 
07 465 
07 439 
07 413 
07 388 



07 362 
07 336 
07 311 
07 285 
07 259 



07 234 
07 208 
07 183 
07 157 
07 131 

07 106 
07 080 
07 055 
07 029 
07 003 



06 978 
06 952 
06 927 
06 901 
06 875 



06 850 
06 824 
06 799 
06 773 
06 748 



06 722 
06 696 
06 671 
06 645 
06 620 



06 594 
06 569 
06 543 
06 518 
06 492 



06 466 
06 441 
06 415 
06 390 
06 364 



06 339 
06 313 
06 288 
06 262 
06 237 



06 211 
06 186 
06 160 
06 134 
06 109 



0.06 08P 



Log. Tan. 



88 425 
88 415 
88 404 
88 393 
88 383 



88 372 
88 36l 
88 351 
88 340 
88 329 



88 319 
88 308 
88 297 
88 287 
88 276 



88 265 
88 255 
88 244 
88 233 
88 223 



88 212 
88 201 
88 190 
88 180 
88 169 



88 158 
88 147 
88 137 
88 126 
88 115 



88 104 
88 094 
88 083 
88 072 
88 061 



88 050 
88 039 
88 029 
88 018 
88 007 



87 996 
87 985 
87 974 
87 963 
87 953 



87 942 
87 931 
87 920 
87 909 
87 898 



87 887 
87 876 
87 865 
87 854 
87 844 



87 833 
87 822 
87 811 
87 800 
87 789 



P. 87 778 



Log. Sin. 



10 
11 

IQ 
10 

10 
11 
10 
10 
11 

IQ 
10 
11 

IQ 
10 

11 
10 
10 
11 
10 

11 
10 
11 
10 
11 
10 

11 

10 

11 

10 

11 

10 

11 
11 

10 

11 
11 

10 

11 
11 

10 

11 
11 
11 

10 

11 
11 
11 

10 

11 

11 
11 
11 
11 

10 

11 
11 
11 
11 
11 

11 



-I, 

60 

59 
58 
57 
_56 

55 
54 
53 
52 
51 



50 

49 
48 
47 
46 



40 

39 
38 
37 
36 



30 

29 
28 
27 
26 

25 
24 
23 
22 

£1 
20 

19 
18 
17 
16 



15 
14 
13 
12 
11 

10 

9 
8 
7 
6 



P. P. 



26 



7 
8 
9 
10 
20 
30 
40 
50 



2 


6 


2. 


3 





3. 


3 


4 


3. 


3 


9 


3. 


4 


3 


4. 


8 


6 


8. 


13 





12. 


17 


3 


17. 


21 


6 


21. 




4 

I 

5 
7 

2 





15 


15 


6 


1.5 


1.5 


7 


1.8 


1.7 


8 


2.0 


2.0 


9 


2.3 


2.2 


10 


2.6 


2.5 


20 


5.1 


.5.0 


30 


7.7 


7.5 


40 


10-3 


10.0 


50 


12.9 


12.5 



1 

1. 

1-7 
1.9 
2.2 
2-4 
4.8 
7.2 
9.6 
12.1 



11 



la 



61-1 

1.3 

1-4 

1-6 

1-8 
20 3.6 
305. 5 
407-3 7.0 
50'9.I8.7 



1.5 

1-2 
1.4 
1.6 
1.7 
35 
5.2 



P.P. 



687 



49^ 



TABLE Vn.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



Log. Sin. 



9 



81 694 
81 709 
81 723 
81 738 
81 752 



81 767 
81 781 
81 796 
81 810 
81 824 



81 839 
81 853 
81 868 
81 882 
81 897 



81911 
81 925 
81 940 
81 954 
81 969 



81 983 

81 997 

82 012 
82 026 
82 040 



82 055 
82 069 
82 083 
82 098 
82 112 



82 126 
82 140 
82 155 
82 169 
82 183 



82 197 
82 212 
82 226 
82 240 
82 254 



82 269 
82 283 
82 297 
82 311 
82 325 



82 339 
82 354 
82 368 
82 382 
82 396 



82 410 
82 424 
82 438 
82 452 
82 467 



82 481 
82 495 
82 509 
82 523 
82 537 



9. 82 551 



Log. Cos, 



14 
14 
14 
14 

14 
14 

li 
14 
14 
14 

1| 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
■14 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 
14 



d. I Log. Tan, 



93 916 
93 942 
93 967 

93 993 

94 018 



c.d. 



94 044- 
94 069 
94 095 
94 120 
94 146 



94 171 
94 197 
94 222 
94 248 
94 273 



94 299 
94 324 
94 350 
94 375 
94 400 



94 426 
94 451 
94 477 
94 502 
94 528 



94 553 
94 579 
94 604 
94 630 
94 655 



94 681 
94 706 
94 732 
94 757 
94 782 



94 808 
94 833 
94 859 
94 884 
94 910 



94 935 
94 961 

94 986 

95 Oil 
95 037 



95 062 
95 088 
95 113 
95 139 
95 164 



95 189 
95 215 
95 240 
95 266 
95 291 



95 316 
95 342 
95 367 
95 393 
95 Aie 



95 443 



Log. Cot, 



25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

C.d. 



Log. Cot, 



06 083 
06 058 
06 032 
06 007 
05 981 



05 956 
05 930 
05 90_ 
05 87G 
05 85^; 



9 
9 
9 
9 

05 828 9 
05 80S 9 
05 777 
05 752 
05 726 



Log. Cos, 



05 701 
05 675 
05 65C 

05 625 
05 599 



574 
548 
52S 
497 
472 

44e 
421 
39^ 
37C 

344 



31£ 
298 
268 
243 
217 
192 
166 
141 
115 
09r 



064 
039 
014 
988 
963 

937 
91? 
886 
861 
836 



81C 
78f 
759 
73£ 
708 

683 
65P 
632 
607 
5?^1 



0.04 556 



Log. Tan 



87 778 
87 767 
87 756 
87 745 
87 734 



87 723 
87 712 
87 701 
87 690 
87 679 



87 668 
87 657 
87 645 
87 634 
87 623 



87 612 
87 601 
87 590 
87 579 
87 568 

87 557 
87 546 
87 535 
87 523 
87 512 



87 501 
87 490 
87 479 
87 468 
87 457 



87 445 
87 434 
87 423 
87 412 
87 401 



87 389 
87 378 
87 367 
87 356 
87 345 



87 333 
87 322 
87 311 
87 300 
87 28R 



87 277 
87 266 
87 254 
87 243 
R7 232 



87 221 
87 209 
87 198 
87 187 
87 175 



87 164 
87 153 
87 14l 
87 130 
87 118 



9-87 107 
Log. Sin. 



d, 



11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
ll 
11 
11 
11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
ll 
11 
11 
ll 
11 
ll 
11 
11 

ll 
ll 
ll 
11 
11 

ll 
11 
ll 
11 
11 

ll 
11 
11 
11 
11 

11 
11 
11 
11 
ll 

ll 
11 
11 
11 
11 
11 



60 

59 
58 
57 
56^ 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 

26 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 

14 

13 

12 

-11 

10 

9 

8 

7 



138' 



P.P. 





25 


25 


6 


2.5 


2.5 


7 


3 





2 


g 


8 


3 


4 


8 


3 


9 


3 


3 


3 


7 


10 


4 


2 


4 


1 


20 


8 


5 


8 


3 


30 


12 


7 


12 




40 


17 





16 


5 


50 


21 


2 


20 


8 



6 
7 
8 

9 
10 
20 
30 
40 
50 



ll 



1 


4 


1 


7 


1 


9 


2 


2 


2 


4 


4 


8 


7 


2 


9 


6 


12 


11 



14 

1.4 
1.6 
1.8 
2.1 
2.S 
4.6 
70 
9.3 
1111.6 





11 


11 


6 


1.1 


1.1 


7 


1 


3 


1.3 


8 


1 


5 




9 


] 


7 


1 • 6 


10 


1 


9 


l.S 


20 


3 


8 


3.6 


30 


5 


7 


5.5 


40 


7 


(3 


7.3 


50 


9 


6 


9.1 



P.P. 



131' 



688 



48 



43' 



TABLE VII.— LOGARITHMIC SINES. COSINES. TANGENTS 
AND COTANGENTS. 



O 

1 
2 
3 

5 
6 
7 
8 
J^ 

10 

11 
12 
13 
14 

15 
16 
17 
18 

il 
20 

21 
22 
23 
24 

25 

26 

27 

28 

21 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 

46 

47 

48 

it 

50 

51 

52 

53 

54 

55 
56 
57 
58 
59 



Log. Sin. d. 



9-82 551 
9-82 565 
9-82 579 
9. 82 593 
9-82 607 
9 
9 



82 621 
82 635 
9-82 649 
9-82 663 
9.82 677 



• 82 691 
82 705 

• 82 719 

• 82 733 
■82 746 



82 760 
82 774 
82 788 
82 802 
82 816 



• 82 830 
82 844 
82 858 
82 871 
82 885 



82 899 
82 913 

• 82 927 

• 82 940 
82 954 



• 82 968 

• 82 982 

• 82 996 

• 83 009 
•83 023 



83 037 
83 051 
83 064 
83 078 
83 092 



• 83 106 

■ 83 119 

83 133 

83 147 

83 160 



14 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 
14 
14 
13 
14 
14 
14 
14 
13 

14 
14 
14 
13 
14 

14 
13 
14 
13 
14 

14 
13 
14 
13 
14 

13 
14 
13 
14 

13 



Log. Tan. c. d. Log. Cot. 



9.95 443 
9-95 469 
9.95 494 
9-95 520 
9-95 545 



9.95 571 
9.95 596 
9.95 621 
95 647 
9.95 672 



9.95 697 
95 723 
9.95 748 
9.95 774 
9 -95 799 

9 95 824 
9.95 850 
9.95 875 
95 901 
9.95 926 



9.95 951 
95 977 

9.96 002 
9.96 027 
9.96 053 



9-83 174 
9-83 188 
9.83 201 
9.83 215 
9.83 229 



9.83 242 
9.83 256 
9-83 269 
83 283 
9.83 297 



9-83 310 
9.83 324 
9.83 337 
9.83 351 
9. 83 365 



60 9-83 378 
(Log. Cos. 



14 
13 
13 
14 
13 

13 
14 
13 
13 
14 

13 
13 
13 
14 
13 

13 
13 
13 
13 
14 

13 



9.96 078 
9.96 104 
9.96 129 
9-96 154 
96 180 
9.96 205 
9.96 230 
9.96 256 
9. 96 281 
9.96 306 



9.96 332 
9.96 357 
96 383 
9.96 408 
9.96 433 



9.96 459 

9.96 484 

9.96 509 

9.98 535 

9.96 560 

9 

9 

9 

9. 

9. 



96 585 
96 611 
96 636 
96 66l 
96 687 



96 712 
9.96 737 
9.96 763 
96 788 
96 813 



96 839 
96 864 
96 889 
96 915 
96 940 



9.96 965 



Log, Cot. 



25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 

25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 



04 556 
04 531 
04 505 
04 480 
04 454 







0_ 

0-04 429 
0-04 404 
0-04 378 
0-04 353 
0-04 327 



Log. Cos. 



87 107 
87 096 
87 084 
9-87 073 
9.87 062 



0.04 302 
0.04 277 
0.04 251 
0.04 226 
0.04 200 



0.04 175 
0.04 150 
0.04 124 
0.04 099 
0.04 074 



0.04 048 
0.04 023 
0.03 997 
0.03 972 
0.03 947 



■87 050 
87 039 
87 027 
87 016 
87 004 



86 993 
86 982 
86 970 
86 959 
86 947 



d. 



86 936 
86 924 
86 913 
86 901 
86 890 



0.03 921 
0.03 896 
0.03 871 
0.03 845 
0-03 820 

0.03 795 
0.03 769 
0.03 744 
0.03 718 
0.03 693 



86 878 
86 867 
86 855 
86 844 
86 832 



0-03 668 
0-03 642 
0.03 617 
0.03 592 
0.03 566 



c.d. 



OS 341 
03 516 
03 490 
03 465 
03 440 



•86 821 

• 86 809 

• 86 798 

• 86 786 
^86 774 

■ 86 763 
86 751 
86 740 
86 728 
86 716 



86 705 
86 693 
9. 86 682 
9-86 670 
9-86 658 



03 414 
03 389 
03 364 
03 338 
03 313 



03 287 
03 262 
03 237 
03 211 
03 186 



0-03 161 
0.03 135 
0-03 110 
0.03 085 
0.03 059 



0.03 034 



Log, Tan. 



9.86 647 
9-86 635 
9.86 623 
9.86 612 
9-86 600 



86 588 
86 577 
86 565 
86 553 
86 542 



9-86 530 
86 518 
86 507 
9.86 495 
9. 86 483 



9-86471 
9-86 460 
9-86 448 
9 •86 436 
9^86 424 



9.86 412 



Log. Sin. 



11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 

11 
11 
11 
11 
11 
11 

11 
11 
12 
11 

11 
11 
11 
11 
12 

11 
11 
11 
11 
12 

11 
11 
12 
11 
11 

12 
11 
11 
12 
11 

12 
11 
11 
12 
11 

12 
11 
12 
11 
12 

12 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 
50 
49 
48 
47 
46. 
45 
44 
43 
42 
_41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 
30 
29 
28 
27 
-26 
25 
24 
23 
22 
2i 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 

7 

_6 

5 
4 



P. P, 





25 


25 


6 


2.5 


2.5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 


7 


10 


4 


2 


4 


I 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 





16 




50 


21- 


2 


20. 


8 



6 


1.4 


7 


1-6 


8 


1-8 


9 


2-1 


10 


2-3 


20 


4-6 


30 


7-0 


40 


9-3 


50 


11-6 



13 

1-3 
16 
18 
2-0 

2-2 
4.5 
6-7 
9-0 
11.2 





12 


IT 


6 


1-2 


1.1 


7 


1 


4 




3 


8 


1 


6 




5 


9 


1 


8 




7 


10 


2 







g 


20 


4 





3 


8 


30 


6 





5 


7 


40 


8 





7 


6 


50 


10 





9 


6 



11 

1.1 

1-3 
1-4 
1.6 
1-8 
36 
5.§ 
7.3 
9.1 



P, P. 



137* 



689 



47^ 



43^ 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



136' 



Log. Sin. 



83 378 
83 392 
83 405 
83 419 
83 432 



83 446 
83 459 
83 473 
83 486 
83 500 



83 513 
83 527 
83 540 
83 554 
83 567 



83 580 
83 594 
83 607 
83 621 
83 634 



83 647 
83 661 
83 674 
83 688 
83 701 



83 714 
83 728 
83 741 
83 754 
83 768 



83 781 
83 794 
83 808 
83 821 
83 834 



83 847 
83 861 
83 874 
83 887 
83 900 



83 914 
83 927 
83 940 
83 953 
83 967 



83 980 

83 993 

84 006 
84 019 
84 033 



84 046 
84 059 
84 072 
84 085 
84 098 



84 111 
84 124 
84 138 
84 151 
84 164 



84 177 



d. 



Log. Cos, 



13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 



Log. Tan, 



96 965 

96 991 

97 016 
97 041 
97 067 



97 092 
97 117 
97 143 
97 168 
97 193 



97 219 
97 244 
97 269 
97 295 
97 320 



97 345 
97 370 
97 396 
97 421 
97 446 



97 472 
97 497 
97 522 
97 548 
97 573 



97 598 
97 624 
97 649 
97 674 
97 699 



97 725 
97 750 
97 775 
97 801 
97 826 



97 851 
97 877 
97 902 
97 927 
97 952 



97 978 

98 003 
98 028 
98 054 
98 079 



98 104 
98 129 
98 155 
98 180 
98 205 



98 231 
98 256 
98 281 
98 306 
98 332 



98 357 
98 382 
98 408 
98 433 
98 458 



9-98 483 
Log. Cot. 



c. d. 



25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 



Log. Cot. 



03 034 
03 009 
02 984 
02 958 
02 933 



0.02 908 
0.02 882 
0.02 857 
0.02 832 
0.02 806 



0-02 781 
0.02 756 
0-02 730 
0.02 705 
0.02 680 



0.02 654 
0.02 629 
0.02 604 
0.02 578 
0.02 553 



02 528 
02 502 
02 477 
02 452 
02 427 



02 401 
02 376 
02 351 
02 325 
02 300 



0.02 
0.02 
0.02 
0.02 
0.02 



275 
24G 
224 
199 
174 



0.02 
0.02 
0.02 
0.02 
0.02 



14e 
123 
098 
072 

047 



0.02 
0.01 
0.01 
0.01 
0.01 



02_ 
996 
971 
94e 

921 



0.01 
0.01 
0.01 
0.01 
0.01 



0.01 
0.01 
0.01 
0.01 
0.01 



89 

870 

845 

819 
794 

769 
744 
718 
693 
668 



0.01 642 
0.01 617 
0.01 592 
0.01 567 
0.01 54l 



Q.Ol 516 
Log. Tan 



Log. Cos, 



86 412 
86 401 
86 389 
86 377 
86 365 



86 354 
86 342 
86 330 
86 318 
86 306 



86 294 
86 282 
86 271 
86 259 
86 247 



86 235 
86 223 
86 211 
86 199 
86 187 



86 176 
86 164 
86 152 
86 140 
86 128 



86 116 
86 104 
86 092 
86 080 
86 068 



86 056 
86 044 
86 032 
86 020 
86 008 



85 996 
85 984 
85 972 
85 960 
85 948 



85 936 
85 924 
85 912 
85 900 
85 887 



85 875 
85 863 
85 851 
85 839 
85 827 



85 815 
85 803 
85 791 
85 778 
85 766 



85 754 
85 742 
85 730 
85 718 
85 705 



9-85 693 
Log. Sin, 



60 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 

27 

25 
24 
23 
22 
21 

20 

19 
18 
17 
16 

15 
14 
13 
12 
11 

10 

9 
8 

7 
6 

5 
4 
3 
2 
1 





P.P. 





25 


25 


6 


2.5 


2.5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 




10 


4 


2 


4 


J 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 





16 


6 


50 


21 


2 


20 


8 





13 


13 


6 


1.3 


1.3 


7 


1 


6 


1.5 


8 


1 


8 


1-7 


9 


2 





1 . 9 


10 


2 


2 


2.1 


20 


4 


5 


4.3 


30 


6 


7 


6.5 


40 


9 





8.6 


50 


11 


2 


10.8 





12 


12 




1 


6 


1.2 


1.2 


1.1 


7 


1 


4 


1 


4 




3 


8 


1 


6 


1 


6 




5 


9 


1 


9 


1 


8 




7 


10 


2 


1 


2 







9 


20 


4 


1 


4 


03 


8 


30 


6 


2 


6 


05 


7 


40 


8 


3 


8 


07 


6 


50 


10 


4 


10 





9 


6 



P.P. 



133' 



690 



44^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



135' 



Log. Sin 



84 177 
84 190 

84 203 
84 216 
84 229 



84 242 
84 255 
84 268 
84 281 
84 294 



84 307 
84 320 
84 333 

84 346 
84 359 



84 372 
84 385 
84 398 
84 411 

84 424 



84 437 
84 450 
84 463 
84 476 
84 4^9 



84 502 
84 514 
84 527 
84 540 
84 553 



84 566 
84 579 
84 592 
84 604 
84 617 



84 630 
84 643 
84 656 
84 669 
84 681 



84 694 
84 707 
84 720 
84 732 
84 745 



84 758 
84 771 
84 783 
84 796 
84 809 



84 822 
84 834 
84 847 
84 860 
84 872 



84 885 
84 898 
84 910 
84 923 
84 936 



9-84 948 



Log. Cos. 



Log. Tan 



98 483 
98 509 
98 534 
98 559 
98 585 



98 610 
98 635 
98 660 
98 686 
98 711 



98 736 
98 762 
98 787 
98 812 
98 837 



98 863 
98 888 
98 913 
98 938 
98 964 



98 989 

99 014 
99 040 
99 065 
99 090 



99 115 
99 141 
99 166 
99 191 
99 216 



99 242 
99 267 
99 292 
99 318 
99 343 



99 368 
99 393 
99 419 
99 444 
99 469 



99 494 
99 520 
99 545 
99 570 
99 595 



99 621 
99 646 
99 671 
99 697 
99 722 



99 747 
99 772 
99 798 
99 823 
99 848 



99 873 
99 899 
99 924 
99 949 
99 974 



0-00 OOP 
Log. Cot. 



c.d. Log. Cot 



25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 



01 516 
01 491 
01 465 
01 440 
01 415 



01 390 
01 364 
01 339 
01 314 
01 289 



01 263 
01 238 
01 213 
01 187 
01 162 



01 137 
01 112 
01 086 
01 06l 
01 036 



01 010 
00 985 
00 960 
00 935 
00 909 



00 884 
00 859 
00 834 
00 808 
00 783 



00 758 
00 733 
00 707 
00 682 
00 657 



00 631 
00 606 
00 581 
00 556 
00 530 



00 505 
00 480 
00 455 
00 429 
00 404 



00 379 
00 353 
00 328 
00 303 
00 278 



00 252 
00 227 
00 202 
00 177 
00 151 



00 126 
00 101 
00 076 
00 050 
00 025 



00 000 



C.d. Log, Tan, 



Log. Cos, 



85 693 
85 681 
85 669 
85 657 
85 644 



85 632 
85 620 
85 608 
85 595 
85 583 



85 571 
85 559 
85 546 
85 534 
85 522 



85 509 
85 497 
85 485 
85 472 
85 460 



85 448 
85 435 
85 423 
85 411 
85 398 



85 386 
85 374 
85 361 
85 349 
85 336 



85 324 
85 312 
85 299 
85 287 
85 274 



85 262 
85 249 
85 237 
85 224 
85 212 



85 199 
85 187 
85 174 
85 162 
85 149 



85 137 
85 124 
85 112 
85 099 
85 087 



85 074 
85 062 
85 049 
85 037 
85 024 



85 Oil 
84 999 
84 986 
84 974 
84 961 



9.84 94& 



134' 



Log. Sin. 
691 



d. 



60 

59 
58 
57 

1§. 
55 
54 
53 
52 

-51 

50 
49 
48 
47 

j46^ 

45 
44 
43 
42 
41 

40 

39 
38 
37 
11 
35 
34 
33 
32 
11 
30 
29 
28 
27 
26, 
25 
24 
23 
22 

2L 
20 

19 
18 
17 
li 
15 
14 
13 
12 
11 



10 

9 

8 

7 

_6 

5 
4 
3 
2 
_1 





P. P. 





25 


25 


6 


2.5 


2.5 


7 


3 





2.9 


8 


3 


4 


33 


9 


3 


8 


3.7 


10 


4 


2 


4.1 


20 


8 


5 


8.3 


30 


12 


7 


12.5 


40 


17 





16.6 


50 


21 


2 


20.6 



13 



1 


3 


1 


6 


1 


8 


2 





2 


2 


4 


5 


6 


7 


9 





11 


2 



13 

1.3 
1.5 
1-7 
l.p 
2.1 
4.3 
6.5 
8.6 
10.8 





12 


12 


6 


1.2 


1.2 


7 


1.4 


1.4 


8 


1-6 


1.6 


9 


1.9 


1.8 


10 


2.] 


2.0 


20 


4.1 


4.0 


30 


6.2 


6.0 


40 


8.3 


8.0 


50 


10.4 


10. 



P. p. 



TABLE VIIi:— LOGARITHMIC VERSED SINES AND EXTERNAL 
0° SECANTS. 1° 



Log. Vers, 



2-62642 
3-22848 
3-58066 
3- 83054 



4-02436 
.18272 
•31662 
.43260 
•53490 



4-62642 
.70920 
.78478 
.85431 
.91868 



4-97880 

5-03466 

.08732 

•13696 

-18393 



5-22848 
•27086 
•31126 
•34987 
•38684 



5-42230 
•45636 
•48915 
•52073 
•55121 



5-58066 
-60914 
•63372 
•63344 
-68337 



5-71455 
•73902 
.76282 
.78598 
•80354 



.83053 
.85198 
.87291 
.89335 
.91332 



5.93284 

.95193 

•97061 

5-98890 

6-00680 



6-02435 
•04155 
.05842 
.07496 
•09120 



6-10714 
-12279 
.13816 
-15327 
.16811 



6. 18271 



Log. Vers. 



2> 



60206 
35218 
24987 
19382 
15836 
13389 
11598 
10230 

9151 
8278 
7558 
6953 
6437 

5992 
5605 
5266 
4964 
4696 

4455 
4238 
4040 
3861 
3697 

3545 
3406 
3278 
3158 
3048 

2944 
2848 
2757 
2672 
2593 

2518 

2447 
2379 
2316 
2256 

2199 
2145 
2093 
2044 
1996 

1952 
1909 
1868 
1829 
1790 

1755 
1720 
1686 
1654 
1623 

1594 
1565 
1537 
1511 
1484 

1460 



n 



Log. Exsec. 



2-62642 
3-22848 
3-58066 
3-83054 



4-02436 
.18272 
.31662 
.43260 
.53491 



•62642 
•70921 
•78478 
.85431 
.91868 



4-97861 

5.03466 

.08732 

.13697 

.18393 



5.22849 
•27087 
•31127 
•34988 
•38685 



5^42231 
•45638 
.48916 
.52075 
.55123 



.58068 
.60916 
.63674 
.66346 
.68940 



5.71457 
.73904 
.76284 
.78801 
.80857 



5-83056 
.85201 
.87295 
.89338 
.91335 



5. 93288 

.95197 

.97065 

5.98894 

6-00685 



6-02440 
.04160 
•05847 
.07501 
•09125 



6-10719 
•12284 
•13822 
•15333 
-16818 

6.18278 



Log. Exsec. 



I> Log. Vers, 



60206 
35218 
24987 
19382 
15836 
13389 
11598 
10230 

9151 
8279 
7557 
6952 
6437 
5993 
5605 
5266 
. 4964 
4696 

4456 
4238 
4040 
3861 
3697 

3545 
3407 
3278 
3159 
3048 

2945 
2848 
2758 
2672 
2593 

2517 
2447 
2380 
2316 
2256 

2199 
2145 
2093 
2043 
1997 

1952 
1909 
1868 
1829 
1791 

1755 
1720 
1687 
1654 
1623 
1594 
1565 
1537 
1511 
1485 
1460 



2> 



18271 
19707 
21119 
22509 
23877 



25223 
26549 
27856 
29142 
30410 



31660 
32892 
34107 
35305 
36487 



37653 
38803 
39938 
41059 
42165 



43258 
44337 
45403 
46455 
47496 



48524 
49539 
50544 
51536 
52518 



53488 
54448 
55397 
56336 
57265 



58184 
59093 
59993 
60884 
61766 



62639 
63503 
64359 
65206 
66045 



66876 
67700 
68515 
69323 
70124 



70917 
71703 
72482 
73254 
74019 



74777 
75529 
76275 
77014 
77747 



78474 



Log. Vers. 



1435 
1412 
1389 
1368 
1346 
1326 
1306 
1286 
1268 

1250 
1232 
1214 
1198 
1182 
1166 
1150 
1135 
1121 
1106 

1093 
1078 
1066 
1052 
1040 
1028 
1016 
1004 
992 
981 

970 
960 
949 
939 
929 
919 
909 
900 
891 
882 

872 
864 
855 
847 
839 

831 
823 
815 
808 
800 

793 
786 
779 
772 
765 

758 
752 
745 
739 
733 

726 



Log. Exsec, 



18278 

19714 

21126 

22516 

23884^ 

25231 

26557 

27864 

29151 

30419 



31669 
32901 
34116 
35315 
36497 



37663 
38814 
39949 
41070 
42177 



43270 
44349 
45415 
46468 
47509 



48537 
49553 
50557 
51550 
52532 



53503 
54463 
55413 
56352 
57281 



58201 
59110 
60011 
60902 
61784 



62657 
63522 
64378 
65226 
66065 



66897 
67720 
68536 
69345 
70145 



70939 
71725 
72505 
73277 
74043 



74802 
75554 
76300 
77040 
77773 



78500 



Log. Exsec, 



692 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL 
2° SECANTS. 3° 



' Log. Vers. 



6-78474 
.79195 
.79909 
.80618 
_. 81322^ 
6.82019 
.82711 
.83398 
•84079 
.84755 



6 



85425 
86091 
86751 
87407 
88057 



88703 
89344 
89980 
90612 
91239 



91862 
92480 
93093 
93703 
94308 



6-94909 
.95506 
.96099 
.96688 
-97272 



6-97853 

.98430 

-99004 

6.99573 

7.00139 



7.00701 
-01259 
-01814 
-02366 
.0^914 



7 .,03458 
-03999 
•04537 
•05071 
.nPROS 



7.06130 
•06655 
•07177 
•07695 
•08211 



7.087?.^ 
-09232 
-09739 
-10242 
•10743 



7-11240 
-11735 
.12227 
-12716 
.13203 



7.13R87 



Log. Vers. 



2> 



721 
714 
709 
703 

697 
692 
686 
681 
676 

670 
665 
660 
655 
650 

646 
641 
636 
631 
627 
622 
618 
613 
609 
605 

601 
597 
592 
589 
584 

581 
577 
573 
569 
565 

562 
558 
555 
55l 
548 

544 
541 
537 
534 
53l 

527 
525 
521 
518 
515 

519 
509 
506 
503 
500 

497 
495 
492 
489 
486 
484 



n 



Log. Exsec, 



6-78500 
.79221 
.79937 
.80646 
.81350 

6 •82048 
-82740 
.83427 
.84109 
-84785 



6-85457 
.86123 
.86783 
.87439 
-88090 



6-88737 
.89378 
.90015 
.90647 
-91275 



6-91898 
.92516 
.93131 
.93741 
-94346 



6-94948 
-95545 
.96139 
.96728 
-97313 



6-97895 

-98472 

-99046 

6.99616 

7.00182 



7.00745 
-01304 
-01860 
•02412 
.09960 



703505 
-04047 
-04585 
-05120 
•05652 



7^06180 
-06706 

'. 07228 
-07747 
•08263 



708776 
•09286 
.09793 
-10297 
-10798 



7-11297 
.11792 
-12285 
-12775 
•13262 

7.13746 



Log. Exsec 



721 
715 
709 
703 
698 
692 
687 
682 
676 

67l 
666 
660 
656 
651 

646 
641 
636 
632 
628 

623 
618 
614 
610 
605 

601 
597 
593 
589 
585 
581 
577 
574 
570 
566 

563 
559 
555 
552 
548 

545 
54l 
538 
535 
53l 

528 
525 
522 
519 
516 

513 
509 
507 
503 
501 

498 
495 
493 
490 
487 
484 



2> 



Log. Vers, 



13687 
14168 
14646 
15122 
15595 



16066 
16534 
17000 
17463 
17923 



18382 
18837 
19291 
19742 
20191 

20637 
21081 
21523 
21963 
22400 



22836 
23269 
23700 
24129 
24555 



24980 
25402 
25823 
26241 
26658 



27072 
27485 
27895 
28304 
28711 



29116 
29518 
29919 
30319 
30716 



31112 
31505 
31897 
32288 
32676 



33063 
33448 
33831 

34213 
34593 



34971 
35348 
35723 
36097 
36468 



36839 
37207 
37574 
37940 
38304 



3RR67 



Log. Vers. 



481 
478 
475 
473 
470 
468 
466 
463 
460 

458 
455 
453 
451 
448 

446 
444 
442 
440 
437 
435 
433 
431 
429 
426 

424 
422 
420 
418 
416 

414 
412 
410 
409 
406 

405 
402 
401 
399 
397 

395 
393 
392 
390 
388 

386 
385 
383 
382 
380 

378 
377 
375 
373 
37l 
370 
368 
367 
366 
364 

362 



n 



Log, Exsec, 



13746 
14228 
14707 
15183 
15657 



16129 
16598 
17064 
17528 
17989 



18448 
18905 
19359 
19811 
20260 



20707 
21152 
21595 
22035 
22473 



22909 
23343 
23775 
24204 
24632 



25057 
25480 
25902 
26321 
26738 



27153 
27567 
27978 
28387 
28795 



29200 
29604 
30006 
3G406 
30804 



31201 
31595 
31988 
32379 
32768 



33156 
33542 
33926 
34309 
34689 



3506? 
35446 
35822 
36196 
36569 



36940 
37310 
37678 
38044 
38409 



38773 



Log. ExseCi 



481 
47? 
476 
474 

471 
469 
466 
464 
46l 
45? 
456 
454 
452 
449 

447 
445 
442 
440 
438 

436 
434 
431 
42? 
427 
425 
423 
42l 
41? 
417 

415 
413 
411 
40? 
407 

405 
404 
402 
400 
398 
396 
394 
393 
391 
389 

388 
385 
384 
382 
380 

37? 
377 
376 
374 
373 

371 
369 
368 
366 
365 

363 



2> 



693 



TABLE VIII.— LOGARITHMIC VERSED SINES AKu EXTERNAL SECANTS. 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
1^ 

15 
16 
17 
18 
19 



30 

21 

22 

23 

21 

25 

26 

27 

28 

29_ 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



Lg. Vers, 



7-38667 
-39028 
-39387 
-39745 
-40102 



7-40457 
-40810 
-41163 
-41513 
-41863 



7-42211 
-42557 
-42903 
-43246 
-43589 



7-43930 
44270 
44608 
44946 
45281 



7-45616 
-45949 
-46281 
-46612 
-46941 

7-47270 
-47597 
-47922 
-48247 
-48570 



7-48892 
-49213 
-49533 
-49852 
-50169 



7. 50485 
-50800 
-51114 
-51427 
•51739 



7-52050 
-52359 
-52667 
-52975 
•53281 



7-53586 
•53890 
-54193 
-54495 
-54796 



7-55096 
•55395 
-55692 
-55989 
.56285 



7-56580 
•56873 
-57166 
• 57458 
•57749 



7 58039 
Lg. Vers 



361 
359 
358 
356 

355 
353 
352 
350 
849 
348 
346 
345 
343 
342 

341 
339 
338 
337 
335 

334 
333 
332 
330 
329 
328 
327 
325 
324 
323 

322 
321 
320 
318 
317 

316 
315 
314 
313 
311 

311 
309 
308 
307 
306 

305 
304 
303 
302 
300 

300 
299 
297 
297 

295 

295 
293 
293 
292 
290 

290 

Id 



Log.'Exs. 



7-38773 
39134 
39495 
39854 
40211 



40567 
40922 
41275 
41627 
41977 



42326 
42673 
43019 
43364 
43708 



44050 
44390 
44730 
45068 
45405 



45740 
46075 
46407 
46739 
47070 



47399 
47727 
48054 
48379 
48703 



49026 
49348 
49669 
49989 
50307 



50624 
50941 
51256 
51569 
51882 



D 



52194 
52504 
52814 
53122 
53429 



53735 
54041 
54345 
54648 
54950 



55251 
55550 
55849 
56147 
56444 



56740 
57035 
57329 
57621 
57913 



58204 
Log. Exs, 



361 
360 
359 
357 
356 
354 
353 
352 
350 

349 
347 
346 
345 
343 

342 
340 
339 
338 
337 

335 
334 
332 
332 
330 

329 
328 

327 
325 
324 

323 
322 
321 
319 
318 

317 
316 
315 
313 
313 

311 
310 
309 
308 
307 
306 
305 
304 
303 
302 

301 
299 
299 
298 
296 

296 
295 
294 
292 
292 

291 
1o 



Lg. Vers, 



7-58039 
•58328 
-58615 
-58902 
-59188 



7-59473 
•59758 
-60041 
-60323 
-60604 



7-60885 
-61164 
-61443 
-61721 
•61998 



7^62274 
62549 
62823 
63096 
63369 



7-63641 
-63911 
-64181 
-64451 
•64719 



7-64986 
-65253 
-65519 
-65784 
-66048 



7-66311 
-66574 
-66836 
-67097 
-67357 



7-67617 
-67875 
-68133 
-68390 
•68647 



7-68902 
-69157 
-69411 
-69665 
•69917 



7-70169 
•70421 
.70671 
.70921 
-71170 



7-71418 
-71666 
-71913 
-72159 
- 72404 



7-72649 
•72893 
•73137 
•73379 
•73621 



7-73883 
Lg. Vers 



28? 
287 
287 
286 

285 
284 
283 
282 
281 

280 

279 
27? 
277 
277 

276 
275 
274 
273 
272 

272 
270 
27C 
269 
268 

267 
266 
266 
265 
264 

263 
263 
261 
261 
260 

259 
258 
258 
257 
256 

255 
255 
254 
253 
252 

252 
25l 
25C 
25C 
249 

248 
247 
247 
246 
245 

245 
244 
243 
242 
242 

24l 



Log. Exs, 



7^58204 
-58494 
-58783 
-59071 
•59358 



7-59645 
-59930 
-60214 
-60498 
-60780 



7-61062 
-61342 
-61622 
-61901 
•62179 



62456 
62733 
63008 
63282 
63556 



7-6382? 
•64101 
-64372 
-64643 
•64912 



7^65181 
-65449 
-65716 
-65982 
-66247 



7-66512 
•66776 
•67039 
•67301 
•67562 



67823 
68083 
68342 
68601 
68858 



7^69115 
-69371 
•69627 
-69881 
-70135 



70388 
70641 
70893 
71144 
71394 



7-71644 
-71892 
-72141 
-72388 
•72635 



7 •72881 
-73126 
•73371 
•73615 
■-73859 



7-7410] 



Log. Exs 



290 
289 
288 
287 

286 
285 
284 
283 
282 

28l 
280 
280 
279 
278 

277 
276 
275 
274 
274 

273 
272 
271 
270 
269 

269 
268 
267 
266 
265 
264 
264 
263 
262 
261 

261 
260 
259 
258 
257 

257 
256 
255 
254 
254 

253 
252 
252 
251 
250 

250 

248 
248 
247 
246 

246 
245 
245 
244 
243 
242 



O 

1 
2 
3 
4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
_59 
60 



P.P. 





360 


350 


6 


36^0 


35^0| 


7 


42 





40 


8 


8 


48 





46 


6 


9 


54 





51 


5 


10 


60 





58 


3 


20 


120 





116 


g 


30 


180 





175 





40 


240 





233 


3 


50 


300 





291 


6 



340 

34^0 

39^6 

45^3 

51-0 

56^6 

113^3 

170^0 

226^6 

2833 





330 


330 


310 


6 


33^0 


32-0 


31-0 


7 


38 


5 


37 


3 


36 


1 


8 


44 





42 


6 


41 


3 


9 


49 


5 


48 





46 


5 


10 


55 





53 


3 


51 


g 


20 


110 





106 


6 


103 


3 


30 


165 





160 





155 





40 


220 





213 


3 


206 


6 


50 


275 





266 


6 


258 


3 





300 


390 


6 


30-0 


29^01 


7 


35 





33 


8 


8 


40 





38 


6 


9 


45 





43 


5 


10 


50 





48 


3 


20 


100 





96 


6 


30 


150 





145 





40 


200 





193 


3 


50 


250 





241 


6 





370 


360 


35< 


6 


27.0 


26^0 


25. 


7 


31 


5 


30 


3 


29. 


8 


36 





34 


6 


33- 


9 


40 


5 


39 





37- 


10 


45 





43 


3 


41- 


20 


90 





86 


6 


83- 


30 


135 





130 





125^ 


40 


180 





173 


3 


166^ 


50 


225 





216 


6 


208 • 



380 

28.0 

32.6 

37^3 

42-0 

46-6 

933 

140 •O 

1866 

233-3 





340 


330 


33( 


6 


24-0 


23^0 


22- 


7 


28 





26 • 8 


25- 


8 


32 





30^6 


29- 


9 


36 





34^5 


33- 


10 


40 





38^3 


36- 


20 


80 





78^6 


73- 


30 


120 





115^0 


110- 


40 


160 





153^3 


146- 


50 


200 





191-6 


183 • 



I 



6 

7 

8 

9 

10 

20 

30 

40 

50 



310 


300 


190 


21^0 


20^0 


19-0 


24 


5 


23 


3 


22 


1 


28 





26 


g 


25 


3 


31 


5 


30 





28 


5 


35 





33 


3 


31 


6 


70 





66 


6 


63 


3 


105 





100 





95 





140 





133 


3 


126 


6 


175 





166 


6 


158 


3 



P.P. 



694 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
6° 7° 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 

12 

13 

il 

15 

16 

17 

18 

19 



Lg. Vers. I> Log.Exs. 2>:. Lg. Vers. D Log.Exs. » 



7.73863 

• 74104 
. 74344 
.74583 

• 74822 



7.75060 
•75297 
.75534 
•75770 
.76006 



7.76240 
•76475 
•76703 
•76941 
•77173 



30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

39 



^0 

41 
42 
43 
4£ 

45 
48 
47 
48 
49^ 

50 

51 

52 

53 

51 

55 

56 

57 

58 

59_ 

60 



7.77405 
•77636 
•77867 
•78097 
.78326 



7-78554 
•78783 
•79010 
.79237 
•79463 



7-79689 
•79914 
•80138 
•80362 
-80586 



7-80808 
•81031 
•81252 
•81473 
-81694 



7-81914 
•82133 
•82352 
•82570 
-82788 



7.83005 
.83222 
.83438 
.83653 
.83868 



7-84083 
-84297 
-84510 
-84723 
•84935 



7.85147 
-85359 
-85570 
.85780 
•85990 



7-86199 
.86408 
.86616 
.86824 
•87031 



7-87238 



241 
240 
239 
239 

238 

237 
236 
236 
235 

234 
234 
233 
233 
232 

232 
231 
230 
230 
229 

228 
228 
227 
227 
226 

225 
225 
224 
224 
223 
222 
222 
221 
221 
220 
220 
219 
219 
218 
217 
217 
217 
216 
215 
215 
214 
214 
213 
213 
212 

212 
211 
211 
210 
210 
209 
209 
208 
208 
207 
206 



7 •74101 
.74343 
.74585 
•74826 
.75066 



7.75305 
•75544 
•75782 
•76019 
•76256 



7.76492 
.76728 
.76963 
•77197 
•77431 



Lg. Vers. I> 



7^77664 
•77897 
•78128 
.78360 
•78590 



7^78820 
.79050 
.79279 
.79507 
.79735 



7^79962 
•80188 
•80414 
•80639 
•80864 



7^81088 
•81312 
•81535 
•81758 
.81980 



7.82201 
.82422 
.82642 
.82862 
•8308T 



7.83300 
.83518 
.83735 
•83952 
•84169 



7.84385 
.84600 
.84815 
•85030 
•85243 



7.8M:57 
.85670 
•85882 
.86094 
.86305 



7.86516 
.86726 
.86936 
.87146 
•87354 



7-87563 



Log.Exs 



242 
241 
241 
240 
239 
239 
238 
237 
237 

236 
235 
235 
234 
233 

233 
232 
231 
231 
230 

230 
229 
229 
228 
228 

227 
226 
226 
225 
225 

224 
224 
223 
222 
222 
221 
221 
220 
219 
219 

219 
218 
217 
217 
216 

216 

215 
215 
214 
213 

213 
213 
212 
211 
211 

211 
210 
210 
209 
208 
208 



7.87238 
.87444 
.87650 
.87855 
-88060 



7-88264 
.88468 
.88672 
.88875 
.8907 7 

7.89279 
.89481 
.89682 
•89882 
•90082 



7^90282 
•90481 
•90680 
•90878 
.91076 



JD 



7.91273 
.91470 
.91667 
.91863 
.92058 



7.92253 
.92448 
.92642 
.92836 
.93029 



7.93222 
.93415 
.93607 
.93799 
.93990 



7.94181 
.94371 
.94561 
•94751 
•94940 



7^95129 
.95317 
.95505 
.95693 
•95880 



7.96066 
.96253 
.96439 
.96624 
.96809 



7.96994 
.97178 
.97362 
.97546 
.97729 



7.97912 
•98094 
•98276 
•98458 
-98639 



7-98820 



Lg. Vers, 



206 
205 
2u5 
204 

204 
204 
203 
203 
202 

202 
201 
201 
200 
200 

199 
199 
198 
198 
197 

197 
197 
196 
196 
195 
195 
195 
194 
194 
193 

193 
192 
192 
191 
191 

190 
190 
190 
189 
189 

189 
188 
187 
188 
187 

186 
186 
186 
185 
185 
184 
184 
184 
183 
183 

183 
182 
182 
182 
181 

181 



7-87563 
-87771 
•87978 
•88185 
-88391 



7-88597 
•88803 
•89008 
•89212 
•89416 



7-89620 
•89823 
•90025 
•90228 
-90429 



7-90630 
•90831 
•91032 
.91231 
•91431 

7-91630 
•91828 
•92027 
•92224 
•92421 



7^92618 
•92815 
•93010 
•93206 
•93401 



7^93596 
•93790 
•93984 
•94177 
-94370 



7-94562 
•94754 
•94946 
•95137 
.95328 



7.95519 
•95709 
•95898 
•96088 

.96276 



7.96465 
•96653 
•96841 
•97028 
.97215 



7.97401 
.97587 
.97773 
•97958 
.98143 



7.98327 
•98512 
.98695 
.98879 
-99062 



7-99244 



Log.Exs. 



208 
207 
207 
206 

206 
205 
205 
204 
204 

203 
203 
202 
202 
201 

201 
201 
200 
199 
199 
199 
198 
198 
197 
197 

197 
196 
195 
195 
195 
195 
194 
194 
193 
193 
192 
192 
192 
191 
191 

190 
190 
189 
189 
188 

188 
188 
188 
187 
187 

186 
186 
185 
185 
184 

184 
184 
183 
183 
183 
182 



O 

1 
2 
3 
4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
Ik 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
_29 

30 

31 
32 
33 

M. 

35 
36 
37 
38 
39 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 

56 

57 

58 
31. 
60 



P.P. 





180 


9 


6 


18.0 


0.9 


7 


21 





1-1 


8 


24 





1-2 


9 


27 





1-4 


10 


30 





1-6 


20 


60 





3.1 


30 


90 


C 


4.7 


40 


120 


G 


6.3 


5C 


150 


C 


7-9 





8 


8 


6 


0-8 


0-8 


7 


1-0 


0-9 


8 


1-1 


1-0 


9 


1.3 


1.2 


10 


1.4 


1.3 


20 


2.8 


2.6 


30 


4-2 


4.0 


40 


5-6 


5.3 


50 


7.1 


6.6 





7 


6 


6 


0-7 


0.6 


7 


0-8 


0.7 


8 


0-9 


0.8 


9 


1.0 


1.0 


10 


1-1 


1.1 


20 


2-3 


2.:. 


30 


3-5 


3-2 


40 


4.6 


4-3 


50 


5.8 


5-4 





5 


5 


5 


6 


0-5 


0-5 


0.4 


7 


0-6 


0.6 


0.5 


8 


0-7 


0.6 


0.6 


9 


0-8 


0.7 


0.7 


10 


0-9 


0.8 


0.7 


20 


1-8 


1.6 


1.5 


30 


2-7 


3-5 


2.2 


40 


3-6 


3-3 


3.0 


50 


4-6 


4.1 


3.7 





3 


3 


6 


0.3 


0.3 


7 


0.4 


0.3 


8 


0.4 


0-4 


9 


0.5 


0-4 


10 


0.6 


0-5 


20 


1.1 


1-0 


30 


1.7 


1-5 


40 


2.3 


2-0 


50 


2.9 


2-5 



9 

0-9 
1-0 
1-2 
1-3 
1-5 
3.0 
4.5 
6-0 
7.5 

7. 
0-7 
0-9 
1.0 
1.1 
1.2 
2.5 
3.7 
5.0 
6.2 

6 

0.6 
0-7 
0-8 
0-9 
1-0 
2-0 
3-0 
4-0 
5-0 

4 

0-4 
0-4 
0-5 
0-6 
0.6 
1.3 
2.0 
2.6 
3 



3 2 

0-210-2 
0-3i0-2 
0-3,0-2 
0.4'0.3 
0.40.3 
0-80.6 
1.2 1.0 
1.611.3 
2.1Jl.g 



1 


1 


( 


f) 


e'o.i 


0-1 


0-6 


70^2 


0-1 





d 


80.2 


0-1 





d 


90.2 


0-1 





1 


100.2 


0-1 







200-5 


0-3 





\ 


300-^ 


0-5 





2 


401.0 


0^6 





3 


50'l-5 


0-8 





4 



P.P. 



695 



TABLE VHI.—LOGaRITHMIC VERSED SINES AND EXTERNAL SECANTS. 
8° 9** 



Lg. Vers. 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



7-98820 
.99000 
.99180 
.99360 
.99539 



D Log.Exs. 



7.99718 

7.99897 

8.00075 

•00253 

.00431 



8.00608 
.00784 
.009 31 
.01^37 
.01313 

8-01488 
.01663 
.01838 
.02012 
-02186 



8-02359 
.02533 
.02706 
.02878 
.03050 



8-03222 
.03394 
.03565 
.03736 
.03906 



8-04076 
.04246 
.04416 
.04585 
-04754 



8-04922 
.05090 
.05258 
.05426 
-05593 



8.C5760 
-05926 
-06093 
.06259 
-06424 



8-06589 
-06754 
.06919 
.07083 

-07247 



8 07411 
.07575 
-07738 
.07900 
-08063 



8-08225 
.08387 
.08549 
.08710 
-08871 



809031 
Lg. Vers, 



K180 
180 
179 
179 
179 
178 
178 
177 
178 
177 
176 
17'6 
176 
176 

175 
175 
175 
174 
174 

173 
173 
173 
172 
172 
172 
171 
171 
171 
17b 
17C 
17C 
169 
169 
169 

168 
168 
168 
167 
167 
167 
166 
166 
166 
165 
165 
165 
165 
164 
164 

164 
163 
163 
162 
162 

162 
161 
162 
161 
161 

160 



7-99244 
.99427 
.99609 
.99790 

7-99971 



8-00152 
.00332 
.00512 
.00692 
.00871 



8-01050 
.01229 
.01407 
.01585 
.01763 



8-01940 
.02117 
.02293 
.02469 
.02645 



8-02820 
.02995 
.03170 
.03345 
J)3519 

8.03692 
.03866 
.04039 
.04212 
-04384 



n 



8-04556 
.04728 
.04899 
.05070 
-05241 



8-05411 
.05581 
•05751 
.05921 
.OR090 



8-06259 
-06427 
.06595 
.06763 
.06931 



8-07098 
-07265 
.07431 
.07598 
.07764 



8.07929 
.08095 
.08260 
.08424 
.08589 



8-08753 

.08917 

.09081 

.09244 

_^09407 

809569 

Log.Exs. 



D 

182 

182 

18i 

181 

180 

180 

180 

180 

179 

179 

178 

178 

178 

177 

177 

177 

176 

176 

175 

175 

175 

17 

174 

174 

173 
17& 
173 
173 
172 
172 
171 
17l 
171 
170 

170 
170 
170 
169 
189 

169 
168 
168 
168 
167 

167 
167 
166 
166 
166 

165 
165 
165 
164 
164 

16? 
163 
164 
16^ 
163 
162 



Lg. Vers, 



8-09031 
-09192 
.09352 
.09512 
.09671 



n 



8-09830 
.09989 
.10148 
.10306 
.10464 



8.10622 
.10779 
.10936 
.11093 
.11250 



8.11406 
.11562 
.11718 
.11873 
-12029 



Log.Exs, 



8-12184 
.12338 
.12492 
.12647 
.12800 



8-12954 
.13107 
.13260 
.13413 
.13565 



8.13717 
.13869 
.14021 
14172 
14323 



8 . 14474 
.14625 
.14775 
.14925 
-15075 



8 15225 

15374 

-15523 

.15672 

.15820 



8.15968 
.16116 
.16264 
.16412 
.16559 



8.16706 
.16852 
.16999 
.17145 
.17291 



8-17437 
.17582 
.17728 
.17873 
.18017 



8181P? 
Lg. Vers. 



160 
160 
160 
159 
159 
159 
158 
158 
158 

157 
157 
157 
157 
156 
156 
156 
155 
155 
155 

155 

154 
154 
154 
153 

153 
153 
153 
152 
152 
152 
152 
151 
151 
151 

151 
150 
15C 
15C 
149 
150 
14S 
149 
149 
148 

148 
148 
148 
147 
147 
147 
146 
146 
146 
146 

145 
145 
145 
145 
144 
144 

"d 



8-09569 
-09732 
.09894 
.10056 
.10217 

8.10378 
.10539 
.10700 
.10860 
.11020 



8.11180 
•11340 
.11499 
.11658 
.11816 



8.11975 
.12133 
.12291 
.12448 
-12605 



8-12762 
-12919 
.13075 
.13232 
-13387 



8-13543 
•13698 
•13854 
.14008 
.14163 



8-14317 
•14471 
•14625 
.14778 
-14932 



8-15085 
.15237 
.15390 
-15542 
.15694 



•15846 
-15997 
-16148 
.16299 
.16450 



8-16600 
-16750 
.16900 
.17050 
-17199 



8-17349 
.17497 
.17646 
.17795 
.17943 

8.18091 
.18238 
.18386 
.18533 
-18680 



8.188C>7 



Log.Exs, 



P. P. 



162 
162 
162 
16i 

161 
161 
160 
160 
160 

160 
159 
159 
159 
158 

158 
158 
158 
157 
157 
157 
157 
156 
15e 
155 

156 
155 
155 
154 
154 

154 
154 
153 
153 
153 
158 
152 
152 
152 
152 
152 
151 
151 
15] 
15C 

150 
15C 
15C 
149 
149 

149 
148 
149 
148 
148 
148 
147 
147 
147 
14'7 

146 



O 

1 

2 

3 
j4 

5 
6 
7 
8 
_i 

10 

11 
1-2 
13 
li 
15 
16 
17 
18 

li 
3© 

21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 

35 
36 
37 
88 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
51 
60 





IbU 


1/ 


l> 


16 


6 


.18-0 


17-0 


16- 


7 


21-0 


19 


8 


18. 


8 


24-0 


22 


6 


21- 


9 


27-0 


25 


5 


24. 


10 


30-0 


28 


3 


26- 


20 


60-0 


56 


6 


53- 


30 


90-0 


85 





80. 


40 


120-0 


113 


3 


106- 


50 


1150.0 


141 


• 6 


133- 



150 



9 

60-9 
7|1-1 
8 1-2 
911-4 
lOil-6 



15 





14- 


17 


5 


16- 


20 





18- 


22 


5 


21- 


25 





23- 


50 





46- 


75 





70- 


100 





93- 


125 





116- 



140 


3 
6 

3 
6 

f; 
% 
6 



3-1 
4-7 
6-3 
7-9 



9 

0-9 
1-0 
1-2 
1-3 
1-5 
3-0 
4-5 
6-0 
7.5 





8 


7 


: 


6 


0-8 


0-7 


0- 


7 


0-9 


0-9 


0- 


8 


1-0 


1-0 


0- 


9 


1-2 


1.1 


1- 


10 


1-3 


1-2 


1- 


20 


2-6 


2-5 


2- 


30 


4-0 


3-7 


3- 


40 


5-3 


5-0 


4- 


50 


6-6 


6-2 


5- 





6 


6 


0-61 


7 





7 


8 





8 


9 


1 





10 


1 


1 


20 


2 


1 


30 


3 


2 


40 


4 


3 


50 


5 


4 



6 

7 

8 

9 

10 

20 

30 

40 



5 

0-5 
0-6 



6 

0.6 
0-7 
0-8 
0-9 
1-0 
2-0 
3-0 
4-0 
50 

5 

0-5 



P. P. 



696 



TABLE VIIT.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



10= 



11= 



o 

1 

2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 

li 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29_ 
30 
31 
32 
33 
3£ 

35 
36 
37 
38 
39 



Lg. Vers. -D 



8.18162 
.18306 
.18450 
.18594 
•18738 



40 

41 
42 
43 
4£ 

45 
46 
47 
48 
4S^ 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 



8.18881 

.19024 
•19167 
.19309 
.19452 



8.19594 
.19736 
.19878 
.20019 
.20160 



8.20301 
. 20442 
•20582 
.20723 
.20863 



8.21003 
•21142 
• 21282 
•21421 
•21560 



8.21698 
.21837 
.21975 
.22113 
•22251 



22389 
22526 
.22663 
22800 
22937 



23073 
23209 
23346 
23481 
23617 



23752 
23888 
24023 
24158 
24292 



24426 
24561 
24695 
24828 
24962 



8.25095 
•25228 
• 25361 
•25494 
•25627 



8-25759 

• 25891 
•26023 

• 26155 

• 26286 



8-26417 
Lg. Vers. 



144 
144 
144 
143 
143 
143 
142 
142 
142 

142 
142 
142 
141 
14l 
141 
140 
140 
140 
140 
140 
139 
139 
139 
139 

138 
138 
138 
138 

137 
138 
137 

137 
136 
137 
136 
136 
136 
135 
136 
135 
135 
135 
135 
134 

134 
134 
134 
133 
133 

133 
133 
133 
132 
133 

132 
132 
132 
132 
131 
131 

"d" 



Log.Exs. T> Lg. Vers 



8.18827 
•18973 
•19120 
•19266 
.19411 



8-19557 
.19702 
.19847 
.19992 
.20137 



8-20281 
•20425 
•20569 
•20713 
•20857 



8.21000 
•21143 
•21286 
•21428 
•21571 



8.21713 
•21855 
•21996 
•22138 
•22279 



8 .2242 J 
•22561 
•22701 
•22842 
.22932 



8.23122 
.23262 
.23401 
.23540 
.23679 



8.23818 
•23957 
.24095 
•24234 
.2437!:^ 



8-24509 
.24647 
.24784 
.24922 
.25059 



8-25195 
25332 
25468 
25604 
25740 

8-25876 
-26012 
-26147 
.26282 
■26417 

8-26552 
-26686 
-26821 
-26955 
-27089 



8-27:?2.S 
Log.Exs. 



146 
146 
146 
145 

145 
145 
145 
145 
144 

144 
144 
144 
144 
143 
143 
143 
143 
142 
142 

142 
142 
141 
141 
141 

141 
140 
140 
140 
140 

140 
140 
139 
139 
139 
139 
138 
138 
138 
138 

137 
138 
137 
137 
137 

136 
136 
136 
136 
136 

136 
135 
135 
135 
135 

134 
134 
134 
134 
134 

134 



8-26417 
•26548 
•26679 
•26810 
.26941 



8.27071 
•27201 
•27331 
•27461 
-27590 



8-27719 
•27849 
•27977 
•28106 
.28235 



8.28363 
•28491 
•28619 
•28747 
.28875 



8.29002 
•29129 
•29256 
•29383 
•29510 



8^29636 
•29763 
•29889 
•30015 
.30140 



8.30266 
.30391 
.30516 
.30642 
.30766 



8.30891 

.31015 

.31140 

31264 

-3138<^ 



8-31511 

31635 

.31758 

.31882 

.32005 



8.32128 
32250 
32373 
32495 
32617 



8.32739 
.32861 
.32983 
.33104 
-33225 



8-33347 
33468 
33588 
33709 
33829 



8.33950 
Lg. Vers 



131 
131 
131 
130 

130 
130 
130 
130 
129 
129 
129 
128 
129 
128 
128 
128 
128 
128 
127 

127 
127 
127 
127 
126 
126 
126 
126 
126 
125 

125 
125 
125 
125 
124 
124 
124 
124 
124 
124 

123 
124 
123 
123 
123 

123 
122 
122 
122 
122 

122 
122 
121 
121 
121 

121 
121 
120 
120 
120 

120 



Log.Exs. I> 



8-27223 
•27356 
•27490 
•27623 
•27756 



8^27889 
•28021 
.28153 
.28286 
.28418 



8.28550 
•28681 
•28813 
•28944 
.29075 



8.29206 
•29336 
•29467 
•29597 
•29727 



8.29857 
•29987 
•30117 
•30246 
.30375 



8.30504 
•30633 
•30762 
•30890 
•31019 

8^31147 
•31275 
•31402 
.31530 
•31657 



8^31785 
•31912 
•32039 
•32165 
•32292 



8-32411 
•32544 
•32670 
.32796 
.32922 



8.33047 
.33173 
•33298 
.33423 

^335_47 

8.33672 
.33797 
•33921 
.34045 

^34169 

8 • 34293 
•34417 
•34540 
.34663 
•34786 



8.34909 
Log.Exs. 



133 
133 
133 
133 

133 
132 
132 
132 
132 
132 
131 
131 
131 
131 

131 
130 
130 
130 
130 

130 

130 
129 
129 
129 

129 
129 
128 
128 
128 

128 
128 

127 
127 
127 

127 
127 
127 
126 
126 

126 
126 
126 
126 
125 

125 
125 
125 
125 
124 

125 
124 
124 
124 
123 
124 
124 
123 
123 
123 

123 



10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 

ii 
40 

41 
42 
43 
44 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 

55 
56 
57 
58 
_59_ 
«0 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 
7 
8 

9 
10 
20 
30 
40 
50 



130 



130 

12.0 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
800 
100.0 



4_ 4 3 

0.40^4i0^3 
0^50^40^4 



13 





15 


1 


17 


3 


19 


5 


21 


6 


43 


3 


65 





86 


6 


108 


3 



0.60.5 
0.70.6 
0.7,0.6 
1.5,1.3 
2.2:2.0 
3.02.6 
3.7i3.3 



O.i 
0.5 
0.6 
l.I 

1-7 

2.3 
2.9 





3 


6 


0.3 


7 


0-3 


8 


0^4 


9 


0^4 


10 


0^5 


20 


1^0 


30 


1^5 


40 


2^0 


50 


2.5 



2 

0.2 
0-3 
0.3 
0.4 
0-4 
08 
1.2 
1.6 
2.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 






2 





2 





2 





3 





3 





6 


1 





1 


3 


1 


6 



1 




o.ii 





1 





1 





1 





1 





3 





5 





6 





8 



0-7 
1.0 
1.2 



0.9 
o^o 

0^1 
0.1 
O.I 
0.2 
0^3 
0.4 



P. P. 



697 



^BLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
13° 13° 



Lg. Vers. I> Log. Exs. D Lg. Vers. I> Log. Exs. 2> 



33950 
34070 
34190 
34309 
34429 



34549 
34668 
34787 
34906 
35025 



35143 
35262 
35380 
35498 
35616 



35734 
35852 
35969 
36086 
36204 



36321 
36437 
36554 
36671 
36787 



36903 
37019 
37135 
37251 
37366 



37482 
37597 
37712 
37827 
37942 



38057 
38171 
38286 
38400 
38514 



38628 
38741 
38855 
38969 
39082 



39195 
39308 
39421 
3953^ 
39646 



39758 
39871 
39983 
40095 
40207 



40318 
40430 
40541 
40652 
40764 



9.A0«75 
Lg. Vers, 



120 
120 
119 
120 

119 
119 
119 
119 
119 

118 
118 
118 
118 
118 

117 
118 
117 
117 
117 
117 
116 
117 
116 
116 
116 
116 
116 
115 
115 

115 
115 
115 
115 
115 
114 
114 
114 
114 
114 

114 
113 
114 
113 
113 

113 
113 
113 
113 
112 

112 
112 
112 
112 
112 

111 
111 
111 
111 
111 

111 



34909 
35032 
35155 
35277 
35399 



35522 
35644 
35765 
35887 
36009 



36130 
36251 
36372 
36493 
36614 



36734 
36855 
36975 
37095 
37215 



37335 
37454 
37574 
37693 
37812 



37931 
38050 
38169 
38287 
38406 



38524 
38642 
38760 
38878 
38995 

39113 
39230 
39347 
39464 
39581 



39698 
39814 
39931 
40047 
40163 



40279 
40395 
40511 
40626 
40742 



40857 
40972 
41087 
41202 
41317 



41431 
41546 
4166f) 
41774 
41888 



A^on9 



Log. Exs 



123 
122 
122 
122 

122 
122 
121 
122 
121 

121 
121 
121 
120 
121 

120 
120 
120 
120 
120 
120 
119 
119 
119 
119 
119 
118 
119 
118 
118 

118 
118 
118 
118 
117 
117 
117 
117 
117 
117 

116 
116 
116 
116 
116 

116 
116 
115 
115 
115 

115 
115 
115 
115 
114 

114 
114 
114 
114 
114 
114 



40875 
40985 
41096 
41206 
41317 



41427 
41537 
41647 
41757 
41867 



41976 
42086 
42195 
42304 
42413 



42522 
42630 
42739 
42847 
42956 



8. 



43064 
43172 
43280 
43388 
43495 
43603 
43710 
43817 
43924 
44031 



44138 
44245 
44351 
44458 
44564 



44670 
.44776 
44882 
44988 
45093 



45199 
45304 
45409 
45514 
45619 



45724 
45829 
45934 
4603P 
46142 



46247 
46351 
4645F 
46558 
46662 



4676P 
4686P 
46972 
47076 
4717P 



«.A79P9 
Lg. Vers. 



110 
110 
110 
110 
110 
110 
110 
109 
110 

109 
109 
109 
109 
109 

109 
108 
109 
108 
108 
108 
108 
108 
108 
107 

107 
107 
107 
107 
107 
106 
107 
106 
106 
106 

106 

loe 

105 
106 
105 

105 
105 
105 
105 
105 

105 
104 
105 
104 
104 

104 
104 
104 
103 
104 

10? 
10? 
lOR 

TO? 

TO? 
103 



42002 
42116 
42229 
42343 
42456 

42569 
42682 
42795 
42908 
43021 

43133 
43246 
43358 
43470 
43582 



43694 
43805 
43917 
44028 
44139 



44251 
44362 
44473 
44583 
44694 



44804 
44915 
.45025 
45135 
45245 



45355 
45465 
45574 
45684 
45793 



45902 
46011 
46120 
46229 
46338 



46446 
46555 
46663 
46771 
46879 



46987 
47095 
47203 
47310 
47417 



47525 
47632 
4773P 
47846 
47953 



48060 
48166 
48273 
4837P 
48485 



fi.ARRPl 



Log. Exs. 



113 
113 
113 
113 

113 
113 
113 
113 
112 
112 
112 
112 
112 
112 

112 
111 
111 
111 
111 

111 
111 
111 

lie 
lie 
lie 
lie 
lie 
lie 

109 

lie 
lie 

109 
109 
109 

lOG 
109 
109 

loe 

109 

108 
108 
108 
108 
108 

108 
107 
108 
107 
107 

107 
107 
107 
107 
106 

107 
lOF 

loe 

106 

loe 
loe 





1 

2 
3 

4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
2^ 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 

55 
56 
57 
58 
59 

BO 



P.P. 





120 


119 


6 


12.0 


11.9 


7 


14 





13.9 


8 


16 





15.8 


9 


18 





17.8 


10 


20 





19.8 


20 


40 





39-6 


30 


60 





59.5 


40 


80 





79-3 


50 


100 





99.1 





117 116 


6 


11.7 


11.61 


7 


13.6 


13 


5 


8 


15.6 


15 


4 


9 


17.5 


17 


4 


10 


19.5 


19 


3 


20 


39.0 


38 


6 


30 


58-5 


58 





40 


78.0 


77 


3 


50 


97.5 


96 


6 





114 113 


6 


11.4 


11.31 


7 


13.3 


13 


2 


8 


15.2 


15 





9 


17.1 


16 





10 


19-0 


18 


8 


20 


38.0 


37 


6 


30 


57.0 


56 


5 


40 


76.0 


75 


3 


50 


95.0 


94 


1 



111 110 



6 


11 


1 


11 





7 


12 


9 


12 


8 


8 


14 


8 


14 


6 


9 


16 


6 


16 


5 


10 


18 


5 


18 


3 


20 


37 





36 


g 


30 


55 


5 


55 





40 


74 





73 


3 


50 


92 


5 


91 


6 



118 

11.8 
13-2 
15.7 
17. Z 
19.6 
39.3 
59.0 
78-6 
98.3 

115 

11.5 
13.4 
15.3 
17.2 
19.1 
38.3 
57.5 
76.6 
95.8 

113 

11.2 
13. Q 
14.9 
16.8 
18-6 
37.3 
56.0 
74.6 
93.3 

109 

10.9 

12 

14 

16 

18 

36 

54 

72 

90 



108 107 


10-8 


10.71 


12 


6 


12 


5 


14 


4 


14 


2 


16 


2 


16 





18 





17 


8 


36 





35 


6 


54 





53 


5 


72 





71 


3' 


90 





89 


1. 



106 

10.6 
12.3 
14.1 
15.9 
17.6 
35.3 
53.0 
70.6 
88.3 



105 


104 


0_ 


10.5 


10.4 


0.0 


12.2 


12 


] 


0.0 


14.0 


13 


8 


0.0 


15.7 


15 


6 


0.1 


17.5 


17 


3 


0.1 


35.0 


34 


6 


O.T 


52-5 


52 





0.2 


70.0 


69 


3 


0.3 


87-5 


86 


6 


0.4 



?.?. 



698 



•TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL' SECANTS. 
14° 15° 



Lg.Vers, 1> Log.Exs. I> Lg.Vers 



47282 
47384 
47487 
47590 
47692 



47795 
47897 
47999 
48101 
48203 



48304 
48406 
48507 
48609 
48710 



48811 
48912 
49013 
49114 
49215 



49315 
49415 
49516 
49616 
49716 



49816 
49916 
50015 
50115 
50215 



50314 
50413 
50512 
50611 
50710 



50809 
50908 
51006 
51105 
51208 



51301 
51399 
51497 
51595 
51693 



51791 
51888 
51986 
52083 
52180 



52277 
52374 
52471 
52568 
52665 



52761 
52858 
52954 
53050 
53146 



8.53242 



Lg. Vers, 



102 
103 
102 
102 

102 
102 
102 
102 
102 

101 
101 
101 
101 
101 

101 
101 
101 
100 
101 

100 

log 

100 

100 

100 

100 

100 

99 

100 

99 

99 

99 

99 

99 

99 

98 
99 
98 
98 
98 
98 
98 
98 
98 
97 
98 
97 
97 
97 
97 
97 
97 
97 
96 
97 
96 
96 
96 
96 
96 

96 



8.48591 
.48697 
.48803 
.48909 
.49014 



8.49120 
.49225 
.49331 
.49436 
.49541 



8.49646 
.49750 
.49855 
.49960 
.50064 



8.50168 
.50273 
.50377 
.50481 
.50585 



8.50688 
.50792 
.50896 
.50999 
.51102 



8.51205 
.51309 
.51412 
.51514 
.51617 



8.51720 
.51822 
.51925 
.52027 
.52129 



8.52231 
.52333 
.52435 
.52537 

.5263« 



8.5274C 
.52841 
.52943 
.53044 
.53145 

8.53246 
.53347 
.53448 
.53548 
.53649 



8.53749 
.53850 
.53950 
.54050 
.54150 



8.54250 
.54350 
. 54449 
.54549 
.54649 

8-54748 
Log. Exs 



106 
106 
105 
105 
105 
105 
105 
105 
105 

105 
104 
105 
104 
104 

104 
104 
104 
104 
104 

103 
104 
103 
103 
103 

103 
103 
103 
102 
103 
102 
102 
102 
102 
102 
102 
102 
102 
101 
101 

101 
101 
101 
101 
101 

101 
101 
101 
100 
100 

100 
100 
100 
100 
100 

100 

100 

99 

100 

99 

99 



8.53242 
.53338 
.53434 
.53530 
.53625 



8.53721 
.53816 
.53911 
.54007 
.54102 



8.54197 
.54291 
.54386 
.54481 
. 54575 

8.54670 
.54764 
.54868 
.54952 
.55046 



8.55140 
.55234 
.55328 
.55421 
.55515 



.55608 
.55701 
.55795 
.55888 
.55981 



8.56074 
.56166 
.56259 
.56352 
. 56444 

8.56536 
.56629 
.56721 
.56813 
.56905 



8.56997 
.57089 
.57180 
.57272 
.57363 



8.57455 
.57546 
.57637 
.57728 
.57819 



8.57910 
.58001 
.58092 
.58182 
.58273 



8-58363 
.58453 
.58544 
.58634 
.58724 



8.58814 



Lg. Vers, 



96 
95 
96 
95 

95 
95 
95 
95 
95 

95 
94 
95 
94 
94 

94 
94 
94 
94 
94 

94 
93 
94 
93 
93 

93 
93 
93 
93 
93 

93 
92 
92 
93 
92 

92 
92 
92 
92 
92 
92 
92 
91 
91 
91 

91 
91 
91 
91 
91 

91 
90 
91 
90 
90 

90 
90 
90 
90 
90 

90 



Log.Exs. 2> 



8.54748 
.54847 
.54946 
.55045 
.55144 



8.55^'!'3 
.55342 
.55141 
.55539 
.55638 



8.55736 
.55834 
.55933 
.56031 
.56129 



8.56226 
.56324 
.56422 
.56519 
-56617 



8-56714 
.56812 
.56909 
.57006 
.57103 



8.57200 
.57296 
.57393 
.57490 
.57586 



8.57682 
•57779 
.57875 
.57971 
.58067 



8.58163 
.58259 
.58354 
.58450 
.58546 



8.58641 
.58736 
.58832 
•58927 
.590 22 

8.59117 
.59211 
.59306 
.59401 
.59495 



8.59590 
.59684 
.59779 
•59873 
-59967 



8-60061 
.60155 
.60249 
.60342 
-60436 



8.60530 



Log. Exs 



u 



10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
2^ 

25 
26 
27 
28 
_29 

30 

31 
32 
33 
_34 

35 
36 
37 
38 
31 
40 
41 
42 
43 
j44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 

55 
56 
57 
58 
-59 
60 



P.P. 





103 


102 


6 


10.3 


10.2 


7 


12.0 


11.9 


8 


13.7 


13.6 


9 


15. -S: 


15-3 


10 


17.:. 


17.0 


20 


34.3 


34-0 


30 


51^5 


51.0 


40 


68.6 


68.0 


50 


85.8 


85.0 



101 

10. 1 
11.8 
13.4 
15.1 
16.8 
33.6 
50.5 
67.3 
84. i 



100 

610.0 
711.6 
8 13.3 
9115.0 



10 16-6 
2033-3 
30 50.C 
40 66-e 
50183. S 



99 98 

9.9| 9.8 

11.5 11-4 



13.2 
14.8 
16.5 
33.0 
49.5 
66.0 
82-5 





97 


96 


6 


9.7 


9.6 


7 


11.3 


11.2 


8 


12.9 


12.8 


9 


14-5 


14.4 


10 


16.1 


16.0 


20 


32.3 


32.0 


30 


48.5 


48.0 


40 


64.6 


64.0 


50 


80.8 


80.0 





94 


93 


6 


9.4 


9.3 


7 


10.9 


10.8 


8 


12.5 


12.4 


9 


14-1 


13.9 


10 


15.6 


15.5 


20 


31.3 


31.0 


30 


47.0 


46.5 


40 


62.6 


62.0 


50 


78.3 


77.5 





91 


90 


6 


9.1 


9-0 


7 


10.6 


10-5 


8 


12.1 


12.0 


9 


13.6 


13.5 


10 


15.1 


15.0 


20 


30.3 


30.0 


30 


45.5 


45.0 


40 


60.6 


60.0 


50 


75.8 


75.0 



13.0 
14.7 
16.3 
32.6 
49.0 
65.3 
82.6 



95 

9.5 
11.1 
12.6 
14.2 
15.8 
31.6 
47.5 
63.3 
79.1 



93 

9.2 
10.7 
12.2 
13.8 
15.3 
30.6 
46.0 
61.3 
76.6 



O 

0.0 
0.0 

0.9 

0.1 
0.1 
O.I 
0.2 
0.3 
0.4 



P.P. 



699 



^'ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
16° 17° 



Lg. Vers, 



8-58814 
58904 
58993 
59083 
59173 
59262 
59351 
59441 
59530 
59619 



59708 
59797 
59886 
59974 
60063 



60152 
60240 
60328 
60417 
60505 



60593 
60681 
60769 
60857 
60944 



61032 
61119 
61207 
61294 
61381 



61469 
61556 
61643 
61730 
61816 



61903 
61990 
62076 
62163 

62249 



62336 
62422 
62508 
62594 
62680 



62766 
62852 
62937 
63023 
63108 



63194 
63279 
63364 
63449 
63534 



63619 
63704 
63789 
63874 
63959 



I> Log.Exs. X> Lg.Vers.|2> Log.Exs. D 



64043 



' jLg.Vers, 



89 
90 
89 
89 
89 
89 
89 
89 

89 
89 
89 
88 
89 

88 
88 
88 
88 
88 

88 
88 
88 
88 
87 

87 
87 
87 
87 
87 
87 
87 
87 
87 
86 

87 
86 
86 
86 
86 
86 
86 
86 
86 
86 

86 
86 
85 
85 
85 

85 
85 
85 
85 
85 

85 
85 
85 
84 
85 
84 



8.60530 
.60623 
.60716 
.60810 
.60903 



8.60996 
.61089 
.61182 
.61275 
.61368 



8.61460 
.61553 
.61645 
.61738 
.61830 



8.61922 
.62014 
.62106 
.62198 
.62290 



8.62382 
.62474 
.62565 
.62657 
.62748 



8.62840 
•62931 
.63022 
.63113 
.63204 



8.63295 
.63386 
.63477 
.63567 
.63658 

8.63748 
.63839 
.63929 
.64019 
.64109 



8.64199 
.64289 
.64379 
.64469 
.64559 



8.64649 
.64738 
.64828 
.64917 
.65006 

8.65096 
.65185 
.65274 
.65363 
.65452 



8.65541 
.65629 
.65718 
.65807 
.65895 



8-65984 
Log.Exs. 



93 
93 
93 
93 

93 
93 
93 
92 
93 
92 
92 
92 
92 
92 

92 
92 
92 
92 
92 

91 
92 
91 

91 
91 

91 
91 
91 
91 
91 

90 
91 
91 
90 
90 
90 
90 
90 
90 
90 

90 
90 
90 
90 
89 
90 
89 
89 
89 
89 

89 
89 
89 
89 
89 

89 
88 
88 
89 
88 
88 



8 . 64043 
.64128 
.64212 
.64296 
.64381 



8.64465 
.64549 
.64633 
.64717 
.64801 



8.64884 
.64968 
.65052 
.65135 
.65218 



8.65302 
.65385 
.65468 
.65551 
•65634 

8.65717 
.65800 
.65883 
.65965 
.66048 



8.66131 
.66213 
.66295 
.66378 
.66460 



8.66542 
.66624 
.66706 
.66788 
.66870 



8.66951 
.67033 
.67115 
.67196 
.67277 



8.67359 
.67440 
.67521 
.67602 
.67683 

8.67764 
.67845 
.67926 
.68007 
.68087 



8.68168 
.68248 
.68329 
.68409 
.68489 

8.68569 
.68650 
.68730 
.68810 
.68889 



8-68969 
Lg. Vers. 



8.65984 
.66072 
.66160 
.66248 
.66336 



8.66425 
.66512 
.66600 
.66688 
.66776 



8.66863 
.66951 
.67039 
.67126 
.67213 



8.67301 
.67388 
.67475 
.67562 
-67649 



8-67736 
.67822 
.67909 
.67996 
.68082 



8.68169 
.6825_ 
.68341 
.68428 
.68514 

8 . 6*8600 
.68686 
.68772 
.68858 
.68944 



8.69029 
.69115 
.69201 
.69286 
- 69372 



8.69457 
.69542 
.69627 
.69712 
.69798 



8.69883 
.69967 
.70052 

.70137 

.7U222 



8 . 70306 
.70391 
.70475 
.70560 
-70644 



8-70728 
.70813 
.70897 
.70981 
.71065 



8.71149 



Log.Exs 





1 
2 
3 

4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 

-£i 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
Z^ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
_49 

50 

51 
52 
53 
54 

55 
56 
57 
58 

60 



P. P, 





93 


93 


6 


9-3 


9-2 


7 


10-8 


10-7 


8 


12-4 


12-2 


9 


13-9 


13-8 


10 


15-5 


15.3 


20 


31.0 


30.6 


30 


46-5 


46.0 


40 


62.0 


61.3 


50 


77.5 


76.6 



6 
7 
8 

9 
10 
20 
3C 
40 
50 



90 

9.0 

10.5 
12.0 
13-5 
15-0 
30-0 
45-0 
60-0 
75-0 



89 

8.9 

10.4 
11-8 
13.3 
14.8 
29.6 
44.5 
59.3 
74.1 



91 

9-1 
10-6 
12.1 
13-6 
15-1 
30.5 
45.5 
60-6 
75-8 

88 
8-8 





87 


86 


6 


8-7 


8.6 


7 


10-1 


10.0 


8 


11-6 


11.4 


9 


13.0 


12-9 


10 


14.5 


14-3 


20 


29.0 


28-6 


30 


43.5 


43.0 


40 


58.0 


57-8 


50 


72.5 


71.6 



84 

8.4 
9-8 

11.2 
12.6 
14.0 
28-0 
42-0 
56-0 
70.0 



83 

83 



9-7 
11.0 
12.4 
13.8 
27.6 
41.5 
55.3 
69-1 





81 


80 


6 


8-1 


8-0 


7 


9-4 


9-3 


8 


10.8 


10.6 


9 


12.1 


12-0 


10 


13.5 


13.3 


20 


27.0 


26.6 


30 


40.5 


40-0 


40 


54.0 


53.3 


50 


67.5 


66.6 



85 
85 
9-9 

11-3 
12-7 
14-1 
28-3 
42-5 
56-6 
70.8 

83 
8-2 
9-5 
10-9 
12-3 
13-6 
27-3 
41-0 
54-6 
68-3 

79 

79 
9.2 
10.5 
11-8 
13.1 
26.3 
39.5 
52.6 
65.8 



6 

7 
8 

9 
IC 
20 
30 

4C 

pr 



h 

0.0 
0-0 
0.1 
0.1 
O-I 
0-2 
0.3 
n. A 



r. P. 



700 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
18° 19° 



Lg. Vers. 



68969 
69049 
69129 
69208 
69288 



69367 
69446 
69526 
69605 
69684 



Z> 



69763 
69842 
69921 
70000 
70079 



70157 
70236 
70314 
70393 
7047T 



7055o 
70628 
70706 
70784 
70862 



70940 
71018 
71096 
71174 
71251 



71329 
71406 
71484 
71561 
L1.639 

71716 
71793 
71870 
71947 
72024 



72101 
72178 
72255 
72331 
72408 



72485 
72561 
72637 
72714 
72790 



72886 
72942 
73018 
73094 
73170 



73246 
73322 
73398 
73473 
73549 



.60 R.7?^^^^ 
' Lg, Vers. 



79 
80 
79 
79 

79 
79 
79 
79 
79 
79 
79 
79 
78 
79 

78 
78 
78 
78 
78 

78 
78 
78 
78 
78 
78 
78 
77 
78 
77 

77 
77 
77 
77 
77 
77 
11 
77 
77 
77 

77 

76 
77 
76 
77 

76 
76 
76 
76 
76 

76 
76 
76 
76 
76 

76 
76 
75 
75 
76 
75 



Log.Exs. 



8.71149 
•71232 
•71316 
•71400 
.71484 



8.71567 
•71651 
•71734 
•71817 
•71901 



8.71984 
•72067 
•72150 
•72233 
•72316 



8.72399 
•72481 
•72564 
•72647 
•72729 



8.72812 
•72894 
•72977 
.73059 
•73141 

8.73223 
•73306 
•73388 
•73470 
•73551 



8 •73633 
•73715 
•73797 
•73878 
•73960 



8 . 74041 
•74122 
. 74204 
•74286 

.748 "7 

8 . 7444C 
•74529 
•74610 
•74691 
.74772 



8.74853 
•74934 
•75014 
•75095 
.75175 

8.75256 
•75336 
•75417 
.75497 
.75577 



8.75658 
•75738 
•75818 
•75898 
.75978 



8.7Pn58 
Log.Exs. 



83 
84 
83 
84 

83 
83 
83 
83 
83 

83 
83 
83 
83 
83 

83 
82 
83 
82 
82 
82 
82 
82 
82 
82 

82 
82 
82 
82 
81 

82 
82 
81 
81 
81 

81 
81 
81 
81 
81 

81 
81 
81 
81 
80 

81 
81 
80 
80 
80 

80 
80 
80 
80 
80 

80 
80 
80 
80 
80 
80 



Lg. Vers. ^^ Log.Exs 



8.73625 
.73700 
.73775 
.73851 
•73926 



8.74001 
.74076 
.74151 
.74226 
•74301 



8.74376 
.74451 
.74526 
.74600 
•74675 



8.74749 
.74824 
•74898 
•74973 
•75047 



8 •75121 
.75195 
.75269 
.75343 
•75417 



8.75491 
.75565 
•75839 
.75712 
•75786 

8^75860 
•75933 
.76006 
.76080 
.76153 



8.76226 
.76300 
.76373 
.76446 

•76519 

8.76592 
.76664 
.76737 
.76810 
•76883 



8.76955 
.77028 
.77100 
.77173 
•77245 

8 •77317 
•77390 
•77462 
.77534 
•77606 



8.77678 
.77750 
.77822 
.77893 
•77965 



R- 78037 
Lg. Vers. 



75 
75 
75 
75 
75 
75 
75 
75 
75 

75 
74 
75 
74 
74 

74 
74 
74 
74 
74 
74 
74 
74 
74 
74 

74 
73 
74 
73 
73 
74 
73 
73 
73 
73 

73 
73 
73 
73 
73 

73 

72 
73 
72 
73 
72 
72 
72 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
71 
72 

71 



76G5& 
7613V 
76217 
76297 
76376 

76456 
76536 
76615 
76694 
76774 

76853 
76932 
77011 
.77090 
77169 



77248 
77327 
77406 
77485 
77563 



8. 



77642 
77720 
77799 
77877 
77956 



78034 
78112 
78191 
78269 
78347 



78425 
78503 
78581 
78659 
78736 



8. 



78814 
78892 
78969 
79047 
79124 
79202 
79279 
79357 
79434 
133M 
79588 
79665 
79742 
79819 
79896 



79973 
80050 
80126 
80203 
80280 



80356 
80433 
80509 
.80586 
80662 



8.8073R 



Log.Exs. 



P.P. 



10 

11 
12 
13 

Jki 
15 
16 
17 
18 

29 

30 

21 
22 
23 

_24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
4i 

50 

51 
52 
53 
54 
55 
56 
57 
58 

«0 





84 


83 


6 


8.4 


8.3 


7 


9.8 


9.7 


8 


11.2 


11.0 


9 


12.6 


12.4 


10 


14.0 


13.8 


20 


28.0 


27-6 


30 


42.0 


41-5 


40 


56^0 


55^3 


50 


70.0 


69^1 





81 


8C 


6 


8.1 


8.0 


7 


9.4 


9.3 


8 


10.8 


10-6 


9 


12^1 


12.0 


10 


13-5 


13.3 


20 


27.0 


26.6 


30 


40.5 


40^0 


40 


54.0 


53^3 


50 


67-5 


66.6 



83 
8.2 
9.5 
10. § 
12.3 
13.6 
27.3 
41.0 
54.6 
68. S 



79 

7-9 

9.2 

10.5 

ii.a 

13^I 
26.? 
39^5 
52^6 
35^3 





78 


77 


6 


7^8 


7.7 


7 


9^1 


9.C 


8 


10.4 


10.2 


9 


11^7 


11.5 


10 


13.0 


12.8 


20 


26.0 


25.6 


30 


39.0 


38.5 


40 


52.0 


51.3 


50 


65.0 


64.1 





75 


74 


6 


7.5 


7.4 


7 


8.7 


8.6 


8 


10.0 


9.8 


9 


11.2 


11.1 


10 


12.5 


12.3 


20 


25^0 


24.6 


30 


37^5 


37.0 


40 


50^0 


49.3 


50 


62^5 


61.6 





72 


71 


e 


7.2 


7^1 


7 


8.4 


8.3 


8 


9.6 


9.4 


9 


10.8 


10.6 


10 


12.0 


11.6 


20 


24.0 


23.6 


80 


36.0 


35.5 


40 


48.0 


47.3 


50 


60.0 


59 .ll 



0% 

o^5 
0^(5 
0.1 

0.1 

o-I 

0.2 
0.3 
0^4 



P.P. 



701 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECAKXS. 
20** 21° 



Lg. Vers 



78037 
78108 
78180 
78251 
78323 



78394 
78466 
78537 
78608 
78679 



78750 
78821 
78892 
78963 
79034 



79105 
79175 
79246 
79317 
79387 



79458 
79528 
79598 
79669 
79739 



79809 
79879 
79949 
80019 
80089 



80159 
80229 
80299 
80369 
80438 
80508 
80577 
80647 
80716 
80786 



80855 
80924 
80993 
81063 
81132 



81201 
81270 
81339 
81407 
81476 



81545 
81614 
81682 
81751 
81819 



81888 
81956 
82025 
82093 
82161 



8-82229 
Lg. Vers 



Log.Exs. 



80738 
80814 
80891 
80967 
81043 



81119 
81195 
81271 
81346 
81422 



81498 
81573 
81649 
81725 
81800 



81876 
81951 
82026 
82102 
82177 



82252 
82327 
82402 
S2477 
32552 



82627 
82702 
82776 
82851 
82926 



83000 
83075 
83149 
83224 
83298 



83373 
83447 
83521 
83595 
83670 



83744 

83818 

83892 

8396 

8403 



84113 

84187 

8426 

8433 

84408 



84481 
84555 
84625 
84*702 
84775 

8484§ 
84922 
84995 
85068 
85141 



8.85214 
Log. ExSi 



I> Lg.Vers. J> Log.Exs 



76 
76 
76 
76 

76 
76 
76 
75 
76 

75 
75 
76 
75 
75 

75 
75 
75 
75 
75 

75 
75 

75 
75 
74 

75 
75 
74 
75 
74 

74 
74 
74 
74 
74 

74 
74 
74 
74 
74 

74 
74 
74 
74 
73 

74 

73 
74 
73 
7§ 

7| 
75 

73 
73 
73 
73 
73 
73 
73 
73 

73 



8.82229 
.82297 
.82366 
.82434 
.82502 



8.82569 
.82637 
.82705 
^82773 
.82841 



8.82908 
.82976 
.83043 
.83111 
.83178 



.83246 
.83313 
.83380 
.8344'? 
.83515 



8.83582 
.83649 
.83716 
.83788 
.83850 



8.83916 
.83988 
.84050 
.84117 
.84183 



8.84250 
.84316 
.84383 

.84449 
84515 



8.84582 
.84648 
.84714 
.84780 
• 84846 



8.84912 
.84978 
.85044 
.85110 
.8517g 



8.85242 
.85308 
.85373 
.85439 
.85506 



8.855''0 
.85626 
.85''0i 

.85832 



8.85897 
.85962 
.86027 
.86092 
.86158 



8.86223 
lg. Vers. 



u 



68 
68 
68 
68 
67 
68 
68 
67 
68 
67 
67 
67 
67 
67 

67 
67 
67 
67 
67 

67 

67 
67 
67 
67 

66 
67 
66 
67 
66 

66 
66 
66 
66 
66 

66 
66 
66 
66 
66 

66 
66 
66 
66 
66 

65 

66 
65 
65 
66 

65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 



8.85214 
.85287 
.85360 
.85433 
.85506 

8.85579 
.85651 
.85724 
.85787 
.85869 



8.85942 
.86014 
.86087 
.86159 
.86231 



8.86304 
.86376 
.86448 
.86520 
.86592 



8.86664 
.86736 



86952 



8.87024 
.87095 
.87167 
.87239 

-^7310 

8.87382 
.87453 
.87525 
.87596 
.87868 

8.87739 
.87810 
.87881 
.87953 
•88024 



8.88095 
.88166 
.88237 
.88308 
•88378 



8.88449 
.88520 
.88591 
.88661 
.88732 



8.88O03 
.888'73 
o 88944 
.89014 
.89085 



8.89155 
.89225 
.89295 
.89366 
.89436 



8.89506 



Log.Exs. 



1> 

73 
73 

72 
73 

73 
72 
73 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
72 
72 

72 
72 
72 
72 
7i 
72 
71 
72 
71 
71 

7l 
71 
71 
71 
71 

71 
71 
71 
71 
71 

71 
71 
71 
71 
70 

71 
71 
70 
70 

71 

70 
70 
70 
70 
70 

70 
70 
70 
70 
70 
70 



O 

1 

2 
3 
j4 

5 
6 
7 
8 
J. 

10 

11 

12 
13 

ii 
15 
16 
17 
18 

_19 

20 

21 
22 
23 
_24 

25 
26 
27 
28 

30 

31 
32 
33 

M. 
35 
36 
37 
38 

_39 

40 

41 
42 
43 
M. 
45 
46 
47 
48 
49 



50 

51 
52 
53 
54 

55 
56 
57 
58 

_59. 

60 



P.P. 





76 


75 


6 


7.6 


7.5 


7 


8.8 


8.7 


8 


10.1 


10.0 


9 


11.4 


11.2 


10 


12.6 


12.5 


20 


25-3 


25.0 


80 


38.0 


37.5 


40 


50.6 


50-0 


50 


63.3 


62.5 



73 

7.S 
8.5 
9.7 
10.8 
12.1 
24^3 
36-5 
48.6 
60.8 





70 


69 


6 


7.0 


6-9 


7 


8.1 


8.0 


8 


9.3 


9.2 


9 


10.5 


10.3 


10 


11.6 


11.5 


20 


23.3 


23.0 


30 


35-0 


34.5 


40 


46.6 


46.0 


50 


58.3 


57.5 



67 


66 


6.7 


6.6 


7 


8 


7-7 


8 


9 


88 


10 





9.9 


11 


\ 


11.0 


22 


3 


22.0 


33 


5 


33.0 


44 


6 


44.0 


55 


8 


55.0 



72 71 

7.2 7.1 

84 8. 

9.6 9._ 

10-810.6 

12.011.8 

24-023.6 

36-035.5 

48.0|47.3 

60.0i59.I 



68 

6.8 

9.5 

10.2 
11.3 
22-6 
34.0 
45.3 
56.6 



65 

6.5 

7-6 

8-6 

9.7 

10.8 

21.6 

32.5 

43.3 

54.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



0.0 

0.0 

0.1 

u 

0.2 
0.3 
0.4 



P.P. 



702 



TABLE VIII.—LOGARITHMIC VERSED SINES AND EXTERNAL SECAWTS. 
33° 



33' 



Lg, Vers. -D Log.Exs. 1> Lg. Vers. J> Log.Exs 



86223 
86287 
86352 
86417 
86482 



86547 
86612 
86676 
86741 
86805 



86870 
86934 
86999 
87063 
87127 



87192 
87256 
87320 
87384 
87448 



87512 
87576 
87640 
87704 
87768 



87832 
87895 
87959 
88023 
88086 



88150 
88213 
88277 
88340 
88404 



88467 
88530 
88593 
88656 
88720 



88783 
88846 
88909 
88971 
89034 



C9097 
89160 
89223 
89285 
89348 



89411 
89473 
89536 
89598 
89660 



89723 
89785 
89847 
89910 
89972 



8-00034 



Lg. Vers. 



J> 



8.89508 
.89576 
.89646 
.89716 
.89786 



8.89856 
.89926 
.89995 
.90065 
.90135 



8.90205 
.90274 
.90344 
.90413 
.90483 



8-90552 
.90622 
.90691 
.90760 
.90830 



8.90899 
.90968 
.91037 
.91106 
.91175 



8-91244 
.91313 
.91382 
.91451 
♦91520 

8-91588 
.91657 
.91726 
.91794 
.91863 



8.91932 
.92000 
.92068 
.92137 

-92205 



8-92274 
.92342 
.92410 
.92478 
-92546 



8-92615 
.92683 
.92751 
.92819 
-92887 



8-92955 
-93022 
-93090 
-93158 
-93226 



8.93293 
-93361 
-93429 
-93496 
-93564 



8.93631 



|Log.Exs 



70 
70 
70 
69 

70 
70 
69 
70 
69 
70 
69 
69 
69 
69 

69 
69 
69 
69 
69 

69 
69 
69 
69 
69 

69 
69 
68 
69 
69 

68 
69 
68 
68 
68 
69 
68 
68 
68 
68 

68 
68 
68 
68 
68 

68 
68 
68 
68 
68 

68 
67 
68 
67 
68 

67 
68 
67 
67 
67 

67 



2> 



90034 
90096 
90158 
90220 
90282 



90344 
90406 
90467 
90529 
90591 
90652 
90714 
90776 
90837 
90899 



90960 
91021 
91083 
91144 
91205 



91267 
91328 
91389 
91450 
91511 



91572 
91633 
91694 
91755 
91815 



91876 
91937 
91997 
92058 
92119 



92179 
92240 
92300 
92361 
92421 



92487 
92542 
92602 
92662 
92722 



92782 
92842 
92902 
92962 
93022 



93082 
93142 
93202 
93261 
93321 



93381 
93440 
93500 
93560 
93619 



8-93879 
Lg. Vers 



8.93631 
93699 
93768 
93833 
93901 



93968 
94035 
94102 
94170 
94237 



94304 
94371 
94438 
94505 
94572 



94638 
94705 
94772 
94839 
94905 



94972 
95039 
95105 
95172 
95238 



95305 
95371 
95437 
95504 
95570 



95636 
95703 
95769 
95835 
95901 



95967 
96033 
96099 
96165 
96231 



96297 
96362 
96428 
96494 
96560 



96625 
96691 
96757 
96822 
96888 



96953 
97018 
97084 
97149 
97214 



97280 
97345 
97410 
97475 
97540 



8-97606 
Log.Exs. 



O 

1 
2 
3 
_4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
14 



15 
16 
17 
18 
11 
30 
21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 

34 

35 
36 
37 
38 
.39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
_54 

55 
56 
57 
58 

60 



P.P. 



70 


69 


7-0 


6-9 


8-1 


8-0 


9-3 


9-2 


10-5 


10-3 


11-8 


11-5 


23-3 


23-0 


35-0 


34-5 


46-6 


46-0 


58-3 


57-5 





67 


66 


6 


6-7 


6-6 


7 


7-8 


7-7 


8 


8-9 


8-8 


9 


10.0 


9-9 


10 


11-1 


11.0 


20 


22-3 


22.0 


30 


33-5 


33.0 


40 


44.6 


44.0 


50 


55-8 


55.0 





64 


63 


6^ 


6 


6.4 


6. J! 


6- 


7 


7 


4 


7.3 


7- 


8 


8 


5 


8.4 


8- 


9 


9 


6 


9.4 


9. 


10 


10 


6 


10-5 


10. 


20 


21 


3 


21-0 


20. 


30 


32 





31-5 


31. 


40 


42 




42.0 


41. 


50 


53 


3 


52-5 


51- 



68 

6.8 

7-9 
9-0 
10.2 
11-3 
22-6 
34-0 
45-3 
56-6 



65 

6-5 

7-6 

8-6 

9-7 

10.8 

21-6 

32.5 

43.3 

54-1 





61 


60 


59 


6 


6.1 


6.0 


5.9 


7 


7-1 


7 





6 


9 


8 


8.1 


8 





7 


3 


9 


9.1 


9 





8 


3 


10 


10.1 


10 





9 


3 


20 


20-3 


20 





19 


5 


30 


30-5 


30 





29 


5 


40 


40.6 


40 





39 


f 


50 


50-8 


50 





49 



6 

7 

8 

9 

10 

20 

30 

40 

50 



0% 
0-9 
0-0 
0-1 

0.2 
0-3 
0.4 



P.P. 



703 



rABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAl. SECANTS. 

34° 25° 



Lg. Vers. 2> Log.Exs. I> Lg. Vers. I> Log.Exs. I> 



93679 
93738 
93797 
93857 
93916 



93975 
94034 
94094 
94153 
94212 



94271 
94330 
94389 
94448 
94506 



9450G 
94624 
94683 
94742 
94800 



94859 
94917 
94976 
95034 
95093 



95151 
95210 
95268 
95326 
95384 



95443 
95501 
95559 
95617 
95675 

95733 
95791 
95849 
95907 
95965 



96023 
9608G 
96138 
96196 
98253 



96311 
9636L 
96426 
96483 
96541 



9659£ 
96656 
96713 
9677C 
96827 



96885 
96942 
96999 
97056 
97113 



8-97170 



Lg. Vers 



97606 
97671 
97736 
97801 
97865 



97930 
97995 
98060 
98125 
98190 



98254 
98319 
98383 
98448 
98513 



98577 
98642 
98706 
98770 
98835 



98899 
98963 
99028 
99092 
99156 



99220 
99284 
9934L 
99412 
99476 



9954C 
99604 
99661 
99732 
99796 



99860 
99923 
99987 
00051 
00114 



0017& 
00242 
00305 
00369 
00432 

00495 
00559 
00622 
0068C 

00812 
00875 
00931: 
01002 
01065 



01128 
01191 
01254 
01317 
01380 



Q . 01 AA?? 



I> Log.Exs 



97170 
97227 
97284 
97341 
97398 



97455 
97511 
97568 
97625 
97681 



97738 
97795 
97851 
97908 
97964 



98020 
98077 
98133 
98190 
98246 



98302 
98358 
98414 
98470 
98527 



98583 
98638 
98695 
9875t 
98806 



65 
65 
65 
64 

65 
65 
64 
65 
65 

64 
64 
64 
65 
64 
44 
64 
64 
64 
64 

64 
64 
64 
64 
64 
64 
64 
64 
64 
64 

64 
64 
64 
64 
63 

64 
63 
64 
63 
63 
64 
63 
63 
63 
63 

63 
63 

63 
63 
63 

63 
63 
63 
63 
63 
63 
63 
63 
63 
63 

^^ Q-oo*^^- 
I> Lg. Vers, 



98802 
9891b 
98974 
9903C 

.Q9G85 



99141 
99197 
99252 
993C8 
99863 



99419 
99474 
99528 
995C5 
99640 



99695 
.99751 
.99806 
.99861 
•9991 6 

.99971 

.0G02C 
.00081 

.ooise 

•00191 

00246 
.00303 
00356 
0041] 
00466 



57 
56 
57 
57 

57 
56 
57 
56 
56 

56 
57 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 
56 
56 
56 
5o 
56 
56 
56 
55 
56 
55 

55 
56 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
65 
55 
55 

55 
55 
55 
54 
55 
54 



J> 



9.01443 
.01505 
.01568 
.01631 
.01694 



9.01756 
.01819 
.01882 
•01944 
.02007 



9.G207C 
.02132 
.02195 
•02257 
.02319 



9-02382 
•02444 
• 0250&. 
•02568 
.02631 



9.C269C 
•02755 
•02817 
•0288C 
•02942 



9.C3C04 
•03066 
•03128 
.03196 

.63252 



9.C331S 
•03375 
•03437 
•03498 
•03561 



9.0362 
•03684 
•03746 
•C38C7 

• csrf 8 
gTcsGsC 

•03892 
•04053 
•C411E 
•04176 



9.64238 
•04299 
•C436C 
•C442I 
. C^^83 

9.0454^ 
•04605 
.04666 
.04727 
.047Bf 



9.04850 
.04911 
•04972 
•C5033 
.0509 



9-0515^ 



Log.Exs. 



62 
63 
62 
63 
62 
63 
62 
62 
63 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

62 
62 
62 
62 
62 

61 
62 
62 
61 
62 

61 
61 
62 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

6l 
61 
61 
61 
60 

61 





1 

2 

3 

_4 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 
21 
22 
23 
2^ 

25 
26 
27 
28 



30 

31 
32 
33 
M. 
35 
36 
37 
38 

40 
41 
42 
43 

li 
45 
46 
47 
48 

j49 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 



P.P. 





65 


64 


6 


6.5 


6.4 


7 


7.6 


7.4 


8 


8^6 


8.5 


9 


9^7 


9.6 


IC 


10^8 


10.6 


2C 


21^6 


21^3 


3C 


32^5 


32^G 


4C 


43-3 


42^6 


5C 


54^1 


53.3 





62 


61 


6 


6-2 


6-1 


7 


7 


2 


7-1 


8 


8 


2 


8.1 


9 


9 


3 


9.1 


IC 


10 


3 


10.1 


26 


20 


6 


20.3 


3C 


31 


c 


30.5 


4C 


4] 


3 


40.6 


5C 


51 


6 


50.8 



6 

7 

8 

9 

10 

20 

36 

46 

50 



59 58 



5^8 


5.8 


6.8 


6.7 


7.8 


7.7 


8.E 


8.7 


9-8 


9.6 


19-6 


19-S 


29-5 


29.0 


39. £ 


38.6 


49.1 


48.3 



56 

5^6 

6-5 

7^4 

8^4 

9^3 

18^6 

28^C 

37.3 

46.6 



55 

5.5 
6.4 



63 

6.3 

7.3 

8^4 

9^4 

10.5 

21.0 

31.5 

42.0 

52-5 



60 

6^0 
7^0 
8^0 
9.0 

10-0 
20.0 
30.0 
40.0 
50-0 



57 

5.7 

6^8 

7.6 

8.5 

9.5 

19.0 

28.5 

38.0 

47.5 



54 

5^4 

6^3 

7.2 

8.1 

9.0 

18.0 

27.0 

36.0 

45.0 



O 

0.0 
0.0 
0.0 
CI 
0.1 
0.1 
0.2 
0.3 
0.4 



P. P. 



704 



TABLEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANXa 
26° 27° 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



20 

21 
22 
23 
ii 
2o 
26 
"87 
2C 
29 

30 

81 

C2 
83 
84 

85 
86 
87 
88 
39 

40 

41 
42 
43 

44 

45 
46 
47 
48 
49^ 

60 

61 
52 
53 

54 
«5 
66 
57 

8 

9 

O 



Lg.Vers, 



00520 
00575 
00630 
00684 
00739 



00794 
00848 
00903 
00957 
01011 



01066 
01120 
01174 
01229 
01283 



01337 
01391 
01445 
01499 
01554 



01608 
01662 
01715 
01769 
01823 



01877 
01931 
01985 
02038 
02092 



02146 
02199 
02253 
02307 
02360 



02414 
02467 
02521 
02574 
02627 



02681 
02734 
02787 
02840 
02894 



02947 
03000 
03053 
03106 
03159 



03212 
03265 
03318 
03371 
03423 



03476 
03529 
03582 
03634 
03687 



03740 
Lg. Vers. 



2> 

55 
54 
54 
54 

55 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
53 
54 
54 

54 
53 
54 
53 
54 

53 
53 
54 
53 
53 
53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
52 

53 
53 
52 
52 
53 
62 



Log.Exs, 



9.05154 
•05215 
.05276 
.05337 
.05398 



9.05458 
•05519 
.05580 
.05640 
.05701 



9.05762 
.05822 
.05883 
.05943 
.06004 



9.06064 
.06124 
.06185 
.06245 
.06305 



9.06366 
.06426 
.06486 
.06546 
.06606 

9.06667 
.06727 
.06787 
.06847 
•06907 



n 



9.06967 
.07027 
.07087 
.07146 
•072 06 

9.07266 
.07326 
•07386 
.07445 
.07505 

9.07565 
.07024 
.07684 
.07743 
.07803 



07863 
07922 
07981 
08041 
08100 



9. 08160 
.08219 
.08278 
.08338 
.08397 



9.08456 
•08515 
.08574 
.08634 
.08693 



9-08752 
Log. Exs 



61 
61 
60 
61 
60 
61 
60 
60 
60 

61 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
59 
60 

60 
59 
60 
59 
60 

59 
59 
59 
59 
60 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 

59 



Lg. Vers, 



03740 
03792 
03845 
03898 
03950 



04002 
04055 
04107 
04160 
04212 



04264 
04317 
04369 
04421 
04473 



04525 
04577 
04630 
04682 
04734 

04786 
04837 
04889 
04941 
04903 



05045 
05097 
05148 
05200 
05252 



05303 
05355 
05407 
05458 
05510 



05561 
05613 
05664 
05715 
05767 



05818 
05869 
05921 
05972 
06023 



06074 
06125 
06176 
06227 
06279 



06330 
06380 
06431 
06482 
06533 



06584 
06635 
06686 
06736 
06787 



06838 
Lg. Vers. 



52 
52 
53 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
52 
52 
52 
52 

52 
51 
52 
52 
52 

5l 

52 
51 
52 
52 

5l 

52 
51 
51 
51 

51 
51 
5l 
51 
51 

5l 
51 
51 
51 
51 

51 
51 
51 
51 
51 

51 
50 
51 
51 
51 

51 
50 
51 
50 
51 
50 



Log. Exs 



08752 
08811 
08870 
08929 
08988 

09047 
09106 
09164 
09223 
09282 



09341 
09400 
09458 
09517 
09576 

09634 
09693 
09752 
09810 
09869 



09927 
09986 
10044 
101^2 
10161 



10219 
10278 
10o36 
10394 
10452 



10511 
10569 
10627 
10685 
10743 



10801 
10859 
10917 
10975 
11033 



11091 
11149 
11207 
11265 
11323 



11380 
11438 
11496 
11554 
11611 



11669 
11727 
11784 
11842 
11899 



11957 
12015 
12072 
12129 
12187 



12244 



Log. Exs, 



10 

11 
12 
13 
L4 

15 
16 
17 
18 
11 
20 
21 
22 
23 
2^ 

25 
26 
27 
28 

30 

31 
32 
33 
11 
35 
36 
37 
38 

40 

41 
42 
43 
44 

45 
46 
47 
48 

50 

51 
52 
53 
_54 
55 
56 
57 
58 
31 
60 



P.P. 





61 


60 


6 


6.1 


6.0 


7 


7.1 


7.0 


8 


8.1 


8.0 


9 


9.1 


9.0 


10 


10.1 


10.0 


20 


20.3 


20.0 


30 


30.5 


30.0 


40 


40.6 


40.0 


50 


50.8 


50.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



58 

5.8 

6.7 

7.7 

8.7 

9.6 
19.3 
29.0 
38.6138.0 
48.3147.5 





55 


6 


5.5 


7 


6.4 


8 


7.3 


9 


8.2 


10 


9.1 


20 


18.3 


30 


27.5 


40 


36.6 


50 


45.8 





53 


6 


5.3 


7 


6.2 


8 


7.0 


9 


7.9 


10 


8.8 


20 


17.6 


30 


26.5 


40 


35-3 


50 


44.1 





51 


6 


5.1 


7 


5.9 


8 


6.8 


9 


7-6 


10 


8.5 


20 


17.0 


30 


25.8 


40 


34.0 


50 


42.5 



P.P. 



705 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



28' 



29' 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 

15 
16 
17 
18 
19 



Lg.Vers. 



9.06838 
.06888 
.06939 
.06990 
.07040 



9.07091 
.07141 
.07192 
.07242 
.07293 



9.07343 
.97393 
.07444 
.07494 
.07544 



9.07594 
.07644 
.07695 
.07745 
.07795 



20 

21 
22 
23 
24 

25 

26 

27 

28 

29, 

30 

31 

32 

33 

3i 

35 

36 

37 

38 

39 



9.07845 
.07895 
.07945 
.07995 
.08045 



D 



Log.Exs. 



9.08095 
.08145 
.08195 
.00244 
.08294 



9.08344 
.08394 
.08443 
.08493 
.08543 



9.08592 
.08642 
.08691 
.08741 
•08790 



40 

41 
42 
43 
44 

45 !9. 09087 



9.08840 
.08889 
.08939 
.08988 
.09087 



46 
47 
48 
49 

60 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



09136 
09185 
.09234 
.09284 



9-09333 
.09382 
.09431 
.09480 
.09529 



9.09578 
.09627 
.09676 
.09725 
.09774 



9 098^P- 
Lg.Vers. 



50 
51 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
49 
50 

49 
50 
49 
49 
50 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 

49 
49 
49 
48 
49 

49 



9.12244 
.12302 
.12359 
.12416 
.12474 



9.12531 
.12588 
.12645 
.12703 
.12760 



9.12817 
.12874 
.12931 
.12988 
.13045 

9.13102 
.131 9 
.13216 
.13273 
.13330 



D 

57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 



9.13387 
.13444 
.13500 
.13557 
■13614 

9.13671 
.13727 
.13784 
.13841 
.13897 



9.13954 
.14011 
.14067 
.14124 
•14180 



Lg.Vers. 2> 



9.09823 
.09872 
.09920 
.09969 
.10018 



9.10067 
.10115 
.10164 
.10213 
.10261 



9.10310 
.10358 
.10407 
.10455 
.10504 



9.14237 
.14293 
.14350 
.14406 
14462 



•14519 
•14575 
.14631 
.14688 
.14744 



9.14800 
.14856 
.14913 
.14969 
.15025 



9.15081 
.15137 
.15193 
.15249 
.15305 



J> 



9.15361 
.15417 
.15473 
.15529 
.15585 
9-15641 
Log.Exs. 



57 
57 
57 
56 
57 
56 

57 
56 
57 
56 
56 

57 
56 
56 
5§ 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 

56 
56 
56 
56 
56 
56 
56 
56 
56 
55 
56 



9.10552 
.10601 
.10649 
.10697 
.10746 



Log.Exs, 



•10794 
.10842 
.10890 
.10939 
.10987 



9.11035 
.11083 
.11131 
.11179 
•11227 



9-11270 
.11323 
.11371 
.11419 
.11467 

9-11515 
.11562 
.11610 
.11658 
-117 06 

9.11754 
.11801 
.11849 
.11897 
.11944 



9.11992 
.12039 
.12087 
.12134 
-12182 



9-12229 
.12277 
.12324 
.12371 
.12419 



9-12466 
-12513 
.12560 
-12608 
•12655 



9-12702 



49 
48 
49 
48 
49 
48 
48 
49 
48 
48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
47 
48 

48 
47 
48 
48 
47 
48 
47 
47 
48 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

47 



•15641 
-15697 
-15752 
.15808 
-15864 



9-15920 
-15975 
.16031 
.16087 
-16142 



-16198 
-16254 
-16309 
.16365 
-16420 



9-16476 
-16531 
-16587 
-1664. 
•16698 



9-16753 
-16808 
-16864 
-16919 
.16974 



9.17029 
-17085 
-17140 

-1719i; 

-17250 

9-17305 
-17361 
-17416 
-17471 
-17526 

9-17581 

-17636 

17691 

177'6 

17801 



Lg. Vers, 



9-17856 
-17910 
.17965 
.18020 
.18075 



9-18130 
-18185 
.18239 
.18294 
-18349 



9-18403 
-18458 
-18513 
-18567 
•18622 



9^18676 
-18731 
-18786 
-18840 
•18894 



9 18949 



2> 

56 
55 
56 
55 

56 
55 
56 
55 
55 

56 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
55 
55 
55 
55 

55 
54 
55 
33 
5i 
55 
55 
54 
55 
54 

54 
55 
54 
54 
54 

54 
54 
55 

54 
54 

54 



10 

11 
12 
13 
14 



P.P. 



20 

21 

22 

23 

24 

25 

26i 

27 

28 

29 



30 

31 
32 
33 
34 



35 
36 
37 

n. *> 

39 
40 

41 
42 
4? 
44 

46 
47 
48 
49 



Log.Exs, 



2> 



50 

51 
52 
53 
54 





57 


57 


6 


5-7 


5.7 


7 


6-7 


6.6 


8 


7.6 


7.6 


9 


8-6 


8.5 


10 


9-6 


9.5 


20 


19-1 


19.0 


3 


28-7 


28.5 


40 


38-3 


38-0 


50 


47-9 


47-5 



6 

7 

8 

9 

10 

20 

30 

40 

50 



56 


55 


5^6 


5.5 


6-5 


6-5 


7-4 


7-4 


8-4 


8-3 


9-3 


9-2 


18^6 


18^5 


28-0 


27-7 


37-3 


37-0 


46-6 


46-21 





54 


6 


5-4 


7 


6-3 


8 


7-2 


9 


8-2 


10 


9-1 


20 


18-1 


30 


27-2 


40 


36-3 


50 


45^4 



56 

5-6 

6-6 

7-5 

8.5 

9-4 

18-8 

28-2 

37-6 

47.1 

55 

5-5 

6-4 

7-3 

8-2 

9-1 

18-3 

27-6 

36-6 

45-8 



54 

5-4 

6-3 

7-2 

8-1 

3-0 

18-0 

27-0 

36-0 

[45-0 



49 


49 , 


i.9 


4-91 


0-8 




7 


6-6 


6 


5 


--4 


7 


3 


8-2 


8 


1 


16-5 


16 


o 


24^7 


24 


5 


3r>.c 


32 


• 6 


41-2 


40 


• 8 



48 
4-8 

5-6 
6-4 
7-3 
8-1 

16-1 
24-2 
32-3 
40-4 





51 


50 


50 


6 


5-1 


5-0 


5-0 


7 


5-9 


5.9 


5-8 


8 


6-8 


6.7 


6-6 


9 


7-6 


7.3 


7.5 


10 


8-5 


8.4 


8-3 


20 


17-0 


16^8 


16-6 


30 


25-5 


25^2 


25-0 


40 


34-G 


33.6 


33-3 


50 


42-5 


42-1 


41.6 





48 


47 


47 


6 


4-8 


4.7 


4-7 


7 


5.6 


5-5 


5-5 


8 


6-4 


6.3 


6-2 


9 


7-2 


7-1 


7-0 


10 


8.0 


7-9 


7-8 


^0 


16-0 


ir-d 


15-6 


30 


24-0 


23-7 


23- 5 


40 


32.0 


31.6 


31.3 


5014-0.0139-6 


39-1 



P. p 



706 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
30° 31° 



Lg. Vers 



12702 
12749 
12796 
12843 
12890 



12937 
12984 
13031 
13078 
13125 



13172 
13219 
13266 
13313 
13359 



13406 
13453 
13500 
13546 
13593 



13639 
13686 
13733 
13779 
13826 



13872 
13919 
13965 
14011 
14058 



14104 
14151 
14197 
14243 
14289 



14336 
14382 
14428 
14474 
14520 



14566 
14612 
14658 
14704 
14750 



14796 
14842 
14888 
14934 
14980 



15026 
15071 
15117 
15163 
15209 



15254 
15300 
15346 
15391 
15437 
15483 



Lg. Vers, 



Log.Exs. X) 



18949 
19003 
19058 
19112 
19167 
19221 
19275 
19329 
19384 
19438 
19492 
19546 
19601 
19655 
19709 

19763 
19817 
19871 
19925 
19979 



20033 
20087 
20141 
20195 
20249 



20303 
20357 
20411 
20465 
20518 



20572 
20626 
20680 
20733 
20787 



20841 
20894 
20948 
21002 
21055 



21109 
21162 
21216 
21269 
21323 



21376 
21430 
21483 
21537 
21590 



21643 
21697 
21750 
21803 
21857 



21910 
21963 
22016 
22070 
22123 



9-22176 
Log.Exs. 



Lg.Vers. D 



9.15483 
15528 
15574 
15619 
15665 



15710 
15755 
15801 
15846 
15891 



15937 
15982 
16027 
16073 
16118 

16163 
16208 
16253 
16298 
16343 

16388 
16434 
16479 
16523 
16568 



16613 
16658 
16703 
16748 
16793 



16838 
16882 
169?7 
16972 
17017 



17061 
17106 
17151 
17195 
172^0 



17284 
17329 
17373 
17418 
17462 



17507 
17551 
17596 
17640 
17684 



17729 
17773 
17817 
17861 
17906 



17950 
17994 
1803R 
18082 
18126 



18170 
Lg. Vers 



Log.Exs, T> 



9.22176 
22229 
22282 
22335 
22388 



22441 
22494 
22547 
22600 
22653 



22706 
22759 
22812 
22865 
22918 



22971 
23024 
23076 
23129 
23182 



23235 
23287 
23340 
23393 
23446 



23498 
23551 
23603 
23656 
23709 



23761 
23814 
23866 
23919 
23971 



24024 
24076 
24128 
24181 
24233 



24285 
24138 
24390 
24442 
24495 



24547 
24599 
24651 
24704 
24756 



24808 
24860 
24912 
24964 
25016 



25068 
25120 
25172 
25224 
25276 



25328 



Log.Exs, 



O 

1 
2 
3 

_4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
1^ 
15 
16 
17 
18 

Jl 
30 

21 
22 
23 
_24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
41 

50 

51 
52 
53 
54 

55 
56 
57 
58 

60 



P.P. 





51 


54 


5J 


6 


5.4 


5.4 


5. 


7 


6.3 


6 


3 


6. 


8 


7.2 


7 


2 


7. 


9 


8.2 


8 


1 


8. 


10 


9.1 


9 





8. 


20 


18.1 


18 





17. 


30 


27.2 


27 





26. 


40 


36.3 


36 





35. 


50 


45.4 


45 





44. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



53 


53 


5.3 


5.2 


6.2 


6.1 


7.0 


7.0 


7-9 


7.9 


8.8 


8.7 


17.6 


17.5 


26.5 


26.2 


35.3 


35.0 


44.1 


43.7 





47 


47 


6 


4.7 


4.7 


7 


5 


5 


5.5 


8 


6 


3 


6.2 


9 


7 


1 


7.0 


10 


7 


9 


7.8 


20 


15 


8 


15.6 


30 


23 


7 


23.5 


40 


31 


6 


31.3 


50 


39 


6 


39.1 





46 


45 


4^ 


6 


4.6 


4.5 


4. 


7 


5 


3 


5 


3 


5. 


8 


6 


1 


6 


C 


6. 


9 


6 


9 


6 


8 


6. 


10 


7 


6 


7 


6 


7. 


20 


15 


3 


15 


1 


15. 


30 


23 





22 


7 


22. 


40 


30 


6 


30 


3 


30. 


50 


38 


3 


37 


9 


37. 



53 

5.2 

6.0 

6.9 

78 

8.6 

17.3 

26.0 

34.6 

43.3 



46 

4.6 

5.4 

6.2 

7.0 

7.7 

15.5 

23.2 

31.0 

38.7 



44 

4.4 

5.2 

5.9 

6.7 

7.4 

14.8 

22.2 

29.6 

37.1 



44 

4.4 

5.1 

5.8 

6.6 

7.3 

14.6 

22.0 

29.3 

36.6 



tt: 



707 



-r^BLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
33° 33° 



1 

1 

2 

3 

_4 

5 

V, 

7 

8 

Ji 

10 

11 
12 
13 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
'is 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 

37 
88 

C9 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
60 
51 
52 
53 
54 

55 
56 
57 
58 
59 

go 



Lg. Vers, 



18170 
18214 
18258 
18302 
18346 



18390 
18434 
18478 
18522 
18566 

18610 
18654 
18697 
18741 
18785 



18829 
18872 
18916 
18959 
19003 



19047 
19090 
19134 
19177 
19221 



19264 
19308 
19351 
19395 
19438 
19481 
19525 
19568 
19611 
19654 



19698 
19741 
19784 
19827 
19870 



19914 
19957 
20000 
20043 
20086 



20129 
20172 
20215 
20258 
20301 



20343 
20386 
20429 
20472 
20515 

20558 
20600 
20643 
20686 
20728 



9. 20771 



Lg. Vers, 



J> 



Log.Exs. I> Lg. Vers. D 



9.25328 
25380 
25432 
25484 
25536 



25588 
25640 
25692 
25743 
25795 



25847 
25899 
25950 
26002 
26054 



26105 
26157 
26209 
26260 
26312 



26364 
26415 
26467 
26518 
26570 



26621 
26673 
26724 
26776 
26827 



26878 
26930 
26981 
27032 
27084 



27135 
27186 
27238 
27289 
27340 



27391 
27443 
27494 
27545 
27596 

27647 
27698 
27749 
27800 
27852 



27903 
27954 
28005 
28056 
28107 



28157 
28208 
28259 
28310 
28361 



9-28412 
Log. Exs. 



20771 
20814 
20856 
20899 
20942 

20984 
21027 
21069 
21112 
21154 



21196 
21239 
21281 
21324 
21366 



21408 
21451 
21493 
21535 
21577 



21620 
21662 
21704 
21746 
21788 



21830 
21872 
21914 
21956 
21998 



22040 
22082 
22124 
22166 
22208 



22250 
22292 
22334 
22376 
22417 



22459 
22501 
22543 
22584 
22626 



22668 
22709 
22751 
22792 
22834 
22876 
22917 
22959 
23000 
23042 



23083 
23124 
23166 
23207 
23248 



23990 
Lg.Vers. 



Log.Exs. 



28412 
28463 
28514 
28564 
28615 



28666 
28717 
28768 
28818 
28869 



28920 
28970 
29021 
29072 
29122 



29173 
29223 
29274 
29324 
20375 



29426 
29476 
29527 
29577 
29627 



29678 
29728 
29779 
29829 
29879 



29930 
29980 
30030 
30081 
30131 



30181 
30231 
30282 
30332 
30382 



30432 
30482 
30533 
30583 
30633 



30683 
30733 
30783 
30833 
30883 



30933 
30983 
31033 
31083 
31133 



9.31183 
31233 
31283 
31333 
31383 
3143? 

Log.Exs. 



I) 



51 
51 
50 
51 

51 
50 
51 
50 
50 

51 
50 
50 
51 
50 

51 
50 
50 
50 
51 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

50 
50 
50 
50 
50 

49 
50 
50 
50 
50 

49 



O 

1 
2 
3 
4 

5 
6 
7 
8 
9 

10 

11 
12 
13 
-li 
15 
16 
17 
18 
39 

20 

21 
22 
23 
^ 
25 
26 
27 
28 
-29 
30 
31 
32 
33 
_34 
35 
36 
37 
38 
li 
40 
41 
42 
43 
44 

45 
46 
47 
48 
49 



50 

51 

52 

53 

-54 

55 
56 
57 
38 
ii 
60 



P. P. 





5' 


3 


51 


51 


6 


5-2 


5.1 


5.1 


7 


6 





6.0 


5.9 


8 


6 


9 


6 8 


6.8 


9 


7 


8 


7.7 


7.6 


10 


8 


6 


86 


8.5 


20 


17 


3 


17.1 


17.0 


30 


26 





25.7 


25.5 


40 


34 


5 


34-3 


34.0 


50 


43 


3 


42.9 


42.5 





50 


50 


4S 


6 


5.0 


5.0 


4. 


7 


5 


9 


5 


8 


5. 


8 


6 


7 


6 


6 


6. 


9 


7 


6 


7 


5 


7. 


10 


8 




8 


3 


8. 


20 


16 


g 


16 


6 


16. 


30 


25 


2 


25 





24. 


40 


33 


5 


33 


3 


33. 


50 


42 


1 


41 


6 


41. 





44 


43 


6 


4.4 


4.31 


7 


5 


1 


5 


1 


8 


5 


8 


5 


8 


9 


6 


6 


6 


5 


10 


7 


3 


7 


2 


20 


14 


6 


14 


5 


30 


22 





21 


7 


40 


29 


3 


29 





50 


36 


6 


36 


2 





42 


4:2 


6 


4.2 


4.2 


7 


4.9 


4.9 


8 


5.6 


5.6 


9 


6.4 


63 


10 


7.1 


7.0 


20 


14.1 


14.0, 


30 


21.2 


21.0 


40 


28.3 


28.0 


50 


35.4 


35.0 



43 

4.3 

5.0 

5.7 

6.4 

71 

14.3 

21.5 

28.6 

35.8 



41. 

41 
4.8 
5.5 



8.2 
b 9 

13. i^ 

20.7 
27.6 
34.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



P. P. 



708 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
34° 35° 



Lg. Vers, 



23290 
23331 
23372 
23414 
23455 



23496 
23537 
23579 
23620 
23661 



23702 
23743 
23784 
23825 
23866 



23907 
23948 
23989 
24030 
24071 



24112 
24153 
24194 
24235 
24275 



24316 
24357 
24398 
24438 
24479 



24520 
24561 
24601 
24642 
24382 



24723 
24764 
24804 
24845 
24885 



24926 
24966 
25007 
25047 
25087 



25128 
25168 
25209 
25249 
25289 



25329 
25370 
25410 
25450 
25490 



25531 
25571 
25611 
25651 
25691 



9. 25731 
Lg. Vers. 



2> 



Log.Exs. 



9-31432 
•31482 
.31532 
.31582 
•31632 



9. 31681 
•31731 
.31781 
.31831 
.3188 

9.31930 
.31980 
.32029 
•32079 
•32129 



9 •32178 
.32228 
.32277 
•32327 
•32377 



9 •32426 
•32476 
.32525 
.32575 
.32624 



9.32673 
.32723 
.32772 
.32822 
•32871 



9.32920 
.32970 
.33019 
.33069 
•33118 



9.33167 
.33216 
•33266 
•33315 

•33364 



9.33413 
•33463 
•33512 
.33561 
•33610 



9.33659 
.33708 
.33758 
.33807 
•33856 



9.33905 
.33954 
. 34003 
.34052 
.34101 



9.34150 
.34199 
. 34248 
.34297 
.34348 



9- 34-395 
Log.Exs. 



I> Lg. Vers, 



25731 
25771 
25811 
25851 
25891 



25931 
25971 
26011 
26051 
26091 



26131 
26171 
26210 
26250 
26290 



26330 
26370 
26409 
26449 
26489 



26528 
26568 
26608 
26647 
26687 



26726 
26766 
26806 
26845 
26885 



26924 
26964 
27003 
27042 
27082 



27121 
27161 
27200 
27239 
27278 



27318 
27357 
27396 
27435 
27475 



27514 
27553 
27592 
27631 
27670 



27709 
27749 
27788 
27827 
27866 



27905 
27944 
27982 
28021 
28060 



28099 
Lg. Vers. 



Log.Exs. 



9 •34395 
. 34444 
•34492 
•34541 
•34590 



9.34639 
•34688 
•34737 
•34785 
•34834 



9 •34883 
•34932 
.34980 
.34029 
.35078 



9.35127 
.35175 
.35224 
.35273 
•35321 

9.35370 
.35419 
.35467 
.35516 
.35564 



9.35613 
.35661 
.35710 
.35758 
.35807 



9.35855 
.35904 
.35952 
.36001 
•36049 



9-36098 
•36146 
.36194 
.36243 
•36291 

9 •36340 
.36388 
•36436 
.36484 
-36533 



9-36581 
.36629 
.36878 
.36726 
•36774 



9.36822 
.36870 
.36919 
.36967 
.37015 



9.37063 
.37111 
.37159 
.37207 
•37255 



9-37303 
Log.Exs. 



D 



O 

i 
2 
3 
4 

5 

6 
7 
8 
9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 

M. 
25 
26 
27 
28 

31 

30 
31 
32 
33 

M 
35" 
36 
37 
38 

_39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



P.P. 





50 


49 


6 


5^0 


4-91 


7 


58 


5 


8 


8 


6-6 


6 


6 


9 


7-5 


7 


4 


10 


8.3 


8 


2 


20 


16-6 


16 


5 


30 


25-0 


24 


7 


40 


33-3 


33 




50 


41.6 


41 


2 



49 

4.9 

5-7 

6.5 

7.3 

8-1 

16-3 

24-5 

32-6 

40-8 



6 
7 
8 
9 

10 
20 
30 
40 
50 



4 


8 


5 


g 


6 


4 


7 


3 


8 


1 


16 


1 


24 


2 


32 


3 


40 


4 



41 


4-11 


4 


8 


5 


5 


6 


2 


6 


9 


13 


8 


20 


7 


27 


g 


34 


6 



48 

4.8 

5^6 

6-4 

7-2 

80 

16.0 

24.0 

32.0 

40.0 

41 

4-1 

4-8 

5-4 

6-1 

6-8 

13-6 

20.5 

27.3 

34.1 





40 


40 


6 


4-0 


^0 


7 


4-7 


4. 


8 


5^4 


5.g 


9 


6.1 


6-0 


10 


6.7 


6-6 


20 


13^5 


13^3 


30 


20^2 


20^0 


40 


27-0 


26.6 


50 


33.7 


33.3 





39 


39 


6 


3-9 


3-9 


7 


4-6 


4 


5 


8 


5-2 


5 


2 


9 


5-9 


5 


8 


10 


6-6 


6 


5 


20 


13-1 


13 





30 


19-7 


19 


5 


40 


26.3 


28 





50 


32.9 


32 


5 



6 
7 
8 
9 

10 
20 
30 
40 
50 



38. 
3 



4 

5 

5 

6 
12 
19 
25 
32 

P.P. 



70^3 



TA.BLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
36° S?"* 



Lg. Vers, 



O 

1 
2 
3 

5 
6 
7 
8 
_9 

10 

11 
12 
13 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
5£ 

55 
56 
57 
58 
59l 
60 



9-28099 
.28138 
.28177 
.28816 
.28255 



9-28293 
.28332 
.28371 
.28410 
-28448 



9-28487 
-28526 
-28564 
-28603 
.28642 



9-28680 
-28719 
.28757 
.28796 
-28835 



9- 28873 
-28912 
.28950 
.28988 
-29027 



9.29065 
.29104 
.29142 
.29180 
-29219 



Log.Exs, 



9-29257 
.29295 
.29334 
.29372 
.29410 



9 - 29448 
.29487 
.29525 
.29563 
-29601 



9-29639 
.29677 
.29715 
.29754 
-29792 



9-29830 
.29868 
.29906 
.29944 
-29982 



9-30020 
.30057 
•30095 
-30133 
-30171 



9-30209 
•30247 
.30285 
•30322 
.30360 



9-30398 



Lg. Vers, 



39 
38 
39 
39 

38 
39 
38 
39 
38 
39 
38 
38 
39 
38 
38 
38 
38 
38 
39 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
38 
38 
38 
38 

38 
37 
38 
38 
38 

38 
37 
38 
37 
38 

38 



9-37303 
•37352 
.37400 
.37448 
-37^^96 



9-37544 
.37592 
•37640 
.37687 
-37735 



9-37783 
.37831 
.37879 
.37927 
.37975 



9.38023 
.38071 
.38119 
.38166 
-38214 



9-38262 
.38310 
.38357 
.38405 
-38453 



9.38501 
.38548 
.38596 
.38644 
-38692 



9.38739 
.38787 
.38834 
.38882 
-38930 



9-38977 
.39025 
.39072 
-39120 
-39168 

9-39215 
.39263 
.39310 
.39358 
-39405 



9.39453 
.39500 
.39548 
.39595 
-39642 

9.39690 
.39737 
.39785 
•39832 
-39879 



9.39927 
.39974 
.40021 
.40069 
.40116 



9.40163 



Log.Exs 



2> 

48 
48 
48 
48 
48 
48 
48 
47 
48 

48 
48 
48 
48 
47 
48 
48 
48 
47 
48 

47 
48 
47 
48 
47 
48 
47 
48 
47 
48 

47 
47 
47 
48 
47 

47 
47 
47 
48 
47 

47 
48 

47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 



Lg. Vers, 



9-30398 
.30436 
.30474 
.30511 
-30549 



9-30587 
.30624 
.30662 
.30700 
-30737 



D Log.Exs. I> 



9.30775 
.30812 
.30850 
.30887 
.30925 



9.30962 
.31000 
.31037 
.31075 
-31112 



9-31150 
.31187 
.31224 
.31262 
.31299 



9.31336 
.31374 
.31411 
.31448 
.31485 



9.31523 
•31560 
.31597 
.31634 
-31671 



9-31708 
•31746 
.31783 
.31820 
-31857 



9.31894 
.31931 
.31968 
.32005 
. 32042 



9.32079 
.32116 
.32153 
.32190 
.32227 



.32263 
.32300 
.32337 
.32374 
.32411 



9.32447 
.32484 
.32521 
.32558 
•32594 



9. 32633 



Lg. Vers, 



37 
38 
37 
37 

38 
37 
37 
38 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 

37 
37 

3Z 
37 
37 

37 
37 
37 
37 
37 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 

37 
37 
37 
37 
37 
37 
37 
37 
37 
37 

36 
37 
37 
36 
37 
36 
37 
36 
37 
36 
37 



9-40163 
•40210 
•40258 
.4C305 
-40352 



9-40399 
.40447 
. 40494 
.40541 
-40588 



9-40635 
.40682 
.40730 
.40777 
•4C824 

9-40871 
-4C918 
-40965 
-41012 
•41059 



9.411C6 
.41153 
-41200 
-41247 
-41294 



9-41341 
.41388 
.41435 
.41482 
.41529 



9.41576 
.41623 
.41670 
.41717 
.41763 



9.41810 

.41857 

.41904 

.41951 

4] 898 



9 . 42044 
•42091 
.42138 
.42185 
-42231 



9.42278 
.42325 
.42372 
.42418 
-42465 



9.42512 
.42558 
.42605 
.42652 
-42688 



9.42745 
.42792 
.42838 
.42885 
•42931 



9.42978 



Log.Exs. 



47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

46 
47 
47 
47 
46 

47 
47 
46 
47 
47 
46 
47 
46 
47 
46 

47 
46 
47 
46 
47 

46 
46 
47 
46 
46 

46 
47 
46 
46 
46 

46 





1 

2 

3 

__4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 



35 
36 
37 
38 
_39 

40 

41 
42 
43 
44 



50 

51 
52 
53 
54 



60 



P.P. 



6 
7 
8 
9 

10 
20 
30 
40 
50 



48_ 

4-8 

5.6 

6.4 

7-3 

8.1 

16.1 

24-2 

32^3 



48 

48 

5.6 

6-4 

7-2 

8.0 

16. 

24.0 

32-0 





47 


6 


4.7 


7 


5.5 


8 


6.3 


9 


7.1 


10 


7.9 


20 


15-8 


30 


23-7 


40 


31-6 


50 


39.6 



6 

7 
8 

9 
10 
20 
30 
40 
50 



6 
7 
8 

9 
10 
20 
30 
40 
50 



6 
7 
8 

9 
10 
20 
30 
40 
PO 



40.4 40.0 



47 

4.7 

5.5 

6.2 

7.0 

7.8 

15.6 

23.5 

31.3 

39.1 



46_ 

4^6 

5^4 

6.2 

7^0 

7-7 

15-5 

23-2 

31^0 

38-7 



38 

3-8 

4^5 

5^1 

5-8 

6-4 

12-8 

19-2 

25.6 

32.1 

37_ 

3.7 

4.4 

5.0 

5-6 

6.2 

12.5 

18.7 

25-0 

31.2 

36_ 

3.6 

4.2 

4-8 

5.5 

6.1 

12.1 

18.2 

24.3 

30.4 



6 


39 


7 


4-5 


8 


5-2 


9 


5-8 


10 


6-5 


20 


13.0 


30 


10.5 


40 


26.0 


50 


32.5 



38 

3-8 
4.4 
5.0 
5-7 



6^3 
12^6 
19.0 
25.3 
31.6 



37 

3-7 

4-3 

4-9 

5-5 

6.1 

12-3 

18.5 

24.6 

30-8 



P.P. 



710 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
38** 39° 



Lg. Vers, 



32631 
32668 
32704 
32741 
32778 



32814 
32851 
32888 
32924 
32961 



32997 
33034 
33070 
33107 
33143 



33180 
33216 
33252 
33289 
333'?5 



33381 
33398 

33434 
33470 
33507 



33543 
33579 
33815 
33652 
33688 

33724 
33760 
33796 
33833 
33869 



33905 
33941 
33977 
34013 
34049 



34085 
34121 
34157 
34193 
34229 



34265 
34301 
34337 
34373 
34408 



34444 
34480 
34516 
34i52 
34587 



34623 
34659 
34695 
34730 
347fiR 



P. 34802 



Lg. Vers* 



Log.Exs, 



42978 
43024 
43071 
43118 
43164 



43211 
43257 
43304 
43350 
43396 



43443 
43489 
43536 
43582 
43629 



43675 
43721 
43768 
43814 
.43861 



43907 
43953 
43999 
. 44046 
44092 



44138 
44185 
44231 
44277 
44323 



44370 
44416 
44462 
44508 
44554 



44601 
44647 
44693 
44739 
44785 



44831 
44877 
44924 
44970 
.45016 



45062 
45108 
45154 
.45200 
45246 

45292 
45338 
45384 
45430 
45476 



45522 
45568 
45614 
45660 
45706 



Q. 45759 



X> Log.Exs, 



D 



46 
47 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 
46 



Lg. Vers, 



34802 
34837 
34873 
34909 
34944 



34980 
.^5016 
.35051 

35087 
.35122 



35158 
35193 
35229 
.35264 
J 5^00 

35335 
•35370 
.35406 
.35441 
.35477 



35512 
35547 
35583 
35618 
35653 



35689 
35724 
35759 
35794 
35829 



35865 
35900 
35935 
35970 
36005 



.16040 
36076 

-36111 
36146 
36181 



36216 
36251 
36286 
36321 
36356 



36391 
36426 
36461 
36495 
36530 



36565 
36600 
36635 
36670 
3R705 



36739 
36774 
36809 
3684-1 
3687P 



Q.SRQl?. 



35 
36 
35 
35 

35 
36 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
35 
35 

35 
35 
35 
34 
35 

35 
35 
34 
35 
35 

34 
35 
34 
35 
34 
35 



Log.Exs, 



45752 
45797 
45843 
45889 
45935 



45981 
46027 
46073 
46118 
46164 



46210 
46256 
46302 
46347 
46393 



46439 
46485 
46530 
46576 
46622 



46668 
46713 
46759 
46805 
46850 



46896 
46942 
46987 
47033 
4707 8 
47124 
47170 
47215 
47261 
47306 



47352 
47398 
47443 
47489 
47534 



47580 
47625 
47671 
47716 
47762 



47807 
47852 
47898 
47943 
47989 



48034 
48080 
48125 
48170 
48216 



48261 
48306 
48352 
48397 
4844-2 



Q /1PA9P 



45 
46 
46 
46 

45 
46 
46 
45 
46 

46 
45 
46 
45 
46 
45 
46 
45 
46 
45 

46 
45 
45 
46 
45 

45 
46 
45 
45 
45 

46 
45 
45 
45 
45 
46 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 



2> Lg.Vers.) D Log.Exs. D 
711 



o 

1 

2 
3 
j4 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

30 

21 
22 
23 
^4 

25 
26 
27 
28 
_29 
30 
31 
32 
33 
li 
35 
36 
37 
38 
li 
40 
41 
42 
43 
44 

45 
46 
47 
48 

M. 

50 
51 
52 
53 
54 

55 
56 
57 
58 
59. 
BO 



P.P. 





47 


6 


4.7 


7 


5.5 


8 


6.2 


9 


7.0 


10 


7.8 


20 


15.6 


30 


23.5 


40 


31.3 


50 


39.1 



46_ 

4.6 

5.4 

6.2 

7.0 

7.7 

15.5 

23.2 

31.0 

38-7 





46 


45 


6 


4.6 


4.5 


7 


5 


3 


5.3 


8 


6 


\ 


6 


9 


6 


9 


6 8 


10 


7 


6 


7.6 


20 


15 


3 


15.1 


30 


23 


22.7 


40 


30 


6,30.3 


50 


38 


3137.9 



45 

4.5 
5-2 



60 
6.7 
7.5 
15. 
22.5 
30.0 
37-5 



37 


3.71 


4 


3 


4 


9 


5 


5 


6 


1 


12 


3 


18 


5 


24 


6 


30 


8 





36 


6 


3.6 


7 


4.2 


8 


4.8 


9 


5.4 


10 


6.0 


20 


12.0 


30 


18.0 


40 


24.0 


50 


30. Ol 





35 


e 


3.5 


7 


4.1 


8 


4.6 


9 


5.2 


10 


5.8 


20 


11.6 


30 


17.5 


40 


23.3 


50 


29. T 



36 

3-6 

4.2 

4.8 

5.5 

6.1 

12.1 

18.2 

24.3 

30.4 

f^ 

%■} 

5.3 
5.9 
11.8 
17.7 
23.6 
29.6 



31. 

3.4 
4.0 
4.6 

i;l 

11.5 
17.2 
23-0 
28.7 



P.P. 



TABLE VIII— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
40° 41° 



O 

1 
2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 



Lg.Vers, 



9.36913 
.36948 
.36982 
.37017 
•37052 



9.37086 
.37121 
.37156 
.37190 
.37225 



9.37259 
.37294 
.37328 
.37363 
•37397 



9-37432 
•37466 
•37501 
.37535 
.37570 



20 

21 
22 
23 
24 

25 

26 

27 

28 

2i 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 



D 



Log.Exs. 



9-37604 
37639 
37673 
37707 
37742 

9-37776 
.37810 
.37845 
.37879 
.37913 

9-37947 
-37982 
-38016 
-38050 
-38084 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9-38118 
-38153 
-38187 
.38221 
.38255 



9.38289 
.38323 
.38357 
.38391 
-38425 



9.38459 
.38493 
.38527 
•38561 
.38595 



9.38629 
•38663 
•38697 
•38731 
•38765 



9 •38799 
.38833 
•38866 
•38900 
•3893 4 

9.3R9R8 



34 
34 
35 
34 

34 
35 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
34 
34 
34 
34 
34 
34 
34 

34 
34 
34 
34 
. 34 
34 
34 
34 
34 
34 

34 
34 
34 
34 
34 

34 
34 
34 
33 
34 

34 
34 
33 
34 
33 
34 



9 •48488 
.48533 
•48578 
.48624 
•48669 

9.48714 
.48759 
.48805 
.48850 
.48895 



9.48940 
.48986 
.49031 
.49076 
.491 21 

9.49166 
.49211 
.49257 
.49302 
-49347 



9.49392 
.49437 
.49482 
.49527 
.49572 



Lg.Vers, 



9.49618 
.49663 
.49708 
.49753 
.49798 



9.49843 
.49888 
.49933 
.49978 
.50023 



9.50068 
.50113 
.50158 
.50203 
-50248 



Lg.Vers.l-2> 



9.50293 
.50338 
.50383 
.50427 
.50472 



9.50517 
.50562 
.50607 
.50652 
-50697 



9-50742 
.50787 
-50831 
.50876 
.50921 



9-50966 
.51011 
.51055 
.51100 
.51145 



9.51190 



45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 

a 

45 

45 
45 
45 
44 
45 

45 
45 
44 
45 
45 

44 
45 
44 
45 
45 
44 



9-38968 
-39002 
.39035 
.39069 
.39103 



9.39137 
.39170 
.39204 
•39238 
.39271 



9.39305 
.39339 
.39372 
.39406 
.39439 



Log.Exs, 



9.39473 
-39507 
-39540 
-39574 
•39607 



9-39641 
-39674 
-39708 
.39741 
-39774 

9-39808 
.39841 
.39875 
-39908 
•39941 



9 39975 
•40008 
•40041 
40075 
40108 



Log.Exs 



9 - 40141 
-40175 
.40208 
.40241 
-40274 



9-40307 
-40341 
.40374 
-40407 
-40440 



9 . 40473 
.40506 
.40540 
.40573 
-40606 



9.40639 
.40672 
.40705 
.40738 
•40771 



9-40804 
-40837 
-40870 
.40903 
-40936 

P.409P9 



34 
33 
34 
33 
34 
33 
33 
34 
33 

33 
34 
33 
33 
33 
33 
34 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
83 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
33 
33 

33 
33 
33 
83 
33 

33 
33 
33 
33 
33 
33 



9-51190 
-51235 
.51279 
.51324 
-51369 



9-51414 
.51458 
.51503 
.51548 
-51592 



9-51637 
-51682 
-51726 
-51771 
-51816 



9-51860 
51905 
51950 
51994 
52039 



Lg. Vers 



9-52084 
52128 
52173 
52217 
52262 



9-52306 
-52351 
-52396 
- 52440 
-52485 



9-52529 
-52574 
-52618 
.52663 
-52707 



9-52752 
-52796 
-52841 
.52885 
-52930 



9.52974 
.53018 
.53063 
.53107 
.53152 



9.53196 
.53240 
.53285 
.53329 
.53374 



9-53418 
-53462 
.53507 
•53551 
.53595 



9 • 53640 
•53684 
.53728 
•53773 
.53817 



P. f^99f^^ 



Log.Exs 



45 
44 
45 
44 

45 
44 
45 
44 
44 

45 
44 
44 
45 
44 

44 
45 
44 
44 
44 

45 

44 

44 

44 

44 

441 

45' 

44 

44 

44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 



10 

11 
12 
13 
14 



20 

21 
22 
23 
24 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

41 
42 
43 
-44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 

55 
56 
57 
58 
31 
60 



P.P. 





45 


6 


4^5 


7 


5-3 


8 


6.0 


9 


6.8 


10 


7.6 


20 


15-1 


30 


22-7 


4C 


30-3 


5C 


37.9 





4^ 


6 


4-4 


7 


5-2 


8 


5-9 


9 


6-7 


10 


7-4 


20 


14-8 


30 


22-2 


40 


29-6 


50 


37.1 





35' 


6 


3.5 


7 


4.1 


8 


4.6 


9 


5.2 


10 


5-8 


20 


11-6 


30 


17-5 


40 


23-3 


50 


29.1 



6 
7 
8 

9 
10 
20 
30 
40 
50 



34 

3-4 



6 

7 
8 

9 
10 
20 
30 
40 
50 



45 

4.5 

52 

6.0 

6.7 

7.5 

15.0 

22.5 

30.0 

37.5 



44 

4.4 
5.1 
58 
6.6 

7.3 
14.6 
22.0 
29-3 
36-6 



32. 

3^4 

4^0 

4.6 

5.2 

5-7 

11.5 

17.2 

23. 

28.7 



33 

3.3 
39 

4.4 
5.0 
5.6 
ll.I 
16.7 
22.3 
127.9 



33 

3.3 
3.8 

4.4 

4.9 

5.5 

11.0 

16.5 
22.0 
27.5 



P. P. 



712 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



42' 



43' 



Lg. Vers, 



40969 
41001 
41034 
41067 
41100 



41133 
41166 
41199 
41231 
41264 



41297 
41330 
41362 
41395 
41428 



41461 
41493 
41526 
41559 
41591 



41624 
41657 
41689 
41722 
41754 



41787 
41819 
41852 
41885 
41917 



41950 
41982 
42014 
42047 
42079 



42112 
42144 
42177 
42209 
42241 



42274 
42306 
42338 
42371 
42403 



42435 
42467 
42500 
42532 
42584 



42596 
42629 
42661 
42693 
42725 



42757 
42789 
42822 
42854 
42886 



9.42918 



Lg. Vers. 



n 



Log.Exs, 



53861 
53906 
53950 
53994 
54038 



54083 
54127 
54171 
54215 
54259 



54304 
54348 
54392 
54436 
54480 



54525 
54569 
54613 
54657 
54701 



54745 
54790 
54834 
54878 
54922 



54966 
55010 
55054 
55098 
55142 



55186 
55230 
55275 
55319 
55363 



55407 
55451 
55495 
55539 
55583 



55627 
55671 
55715 
55759 
55803 



55847 
55890 
55934 
55978 
56022 



56066 
56110 
56154 
56198 
56242 



56286 
56330 
56374 
56417 
56461 



9.56505 



Log.Exs 



D 



4:4: 

44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 

44 
44 
44 
44 
44 
43 
44 
44 
44 

44 
44 
44 
43 
44 

44 
44 
44 
43 
44 

43 



Lg. Vers 



42918 
42950 
42982 
43014 
43046 



43078 
43110 
43142 
43174 
43206 



43238 
43270 
43302 
43334 
43365 



43397 
43429 
43461 
43493 
43525 



43557 
43588 
43620 
43652 
43684 



43715 
43747 
43779 
43810 
43842 



43874 
43906 
43937 
43969 
44000 



44032 
44064 
44095 
44127 
44158 



44190 
44221 
44253 
44284 
44316 



44347 
44379 
44410 
44442 
44473 



44504 
44536 
44567 
44599 
44630 



44661 
44693 
44724 
44755 
44787 



D 



44818 



Log.Exs, 



56505 
56549 
56593 
56637 
56680 



56724 
56ro8 
56812 
56856 



56943 
56987 
57031 
57075 
57118 



57162 
57206 
57250 
57293 
57337 



57381 
57424 
57468 
57512 
57556 



57599 
57643 
57687 
57730 
57774 



57818 
57861 
57905 
57949 
57992 



58036 
58079 
58123 
58167 
58210 



58254 
58297 
58341 
58385 
58428 



58472 
58515 
58559 
58602 
58646 



58689 
58733 
58776 
58820 
58864 



58907 
58951 
58994 
59037 
59081 



59124 



|Lg. Vers. -D Log.Exs 
713 



2> 



10 

11 
12 
13 
li 
15 
16 
17 
18 
ii 
30 
21 
22 
23 
24 

25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39. 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 

60 



P.P. 



41 

4.4i 

5.2 
5-9 
6.7 
7-4 



44 

4.4 
5.1 
5.8 
6.6 
7. '3 





43 


6 


4.31 


7 


5 


1 


8 


5 


8 


9 


6 


5 


10 


7 


2 


20 


14 


5 


30 


21 


7 


40 


29 





50 


36 


2 



14.8 14.6 
22.2 22.0 
29.6,29.3 
37.1136.6 



43 

4.3 

5.0 

5.7 

6-4 

7.1 

14.3 

21.5 

28.6 

35.8 



32 

3.2 
3.8 

4.3 
4.9 
5.4 
10-8 
16.2 
21-6 
27.1 



31. 

31 
3.7 

4.2 
4.7 
5.2 
10.5 
15.7 
21.0 
26.2 



6 


•3 


3 


7 


3 


8 


8 


4 


4 


9 


4 


9 


10 


5 


5 


20 


11 





30 


16 


5 


40 


22 





50 


27 


5 



33 



3 


2 


3 


7 


4 


2 


4 


8 


5 


3 


10 


6 


16 




21 


3 


26 


6 



31 

3.1 

3-6 

4-1 

4.6 

5.1 

10.3 

15.5 

20-6 

25.8 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
44° 45° 



O 

1 
2 
3 

5 
6 
7 
8 
JL 

10 

11 

12 

13 

14 

15 

16 

17 

18 

il 

30 

21 

22 

23 

24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
M 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 

46 

47 

48 

49. 

50 

51 

52 

53 

51 

55 

56 

57 

58 

59. 

60 



Lg.Vers, 



44818 
44849 
44880 
44912 
44943 



44974 
45005 
45036 
45068 
45099 



45130 
45161 
45192 
45223 
45254 



45285 
45316 
45348 
45379 
45410 



45441 
45472 
45503 
45534 
45565 



45595 
45626 
45657 
45688 
45719 



45750 
45781 
45812 
45843 
45873 



45904 
45935 
45966 
45997 
.46027 



46058 
46089 
46120 
46150 
46181 



9.46212 
.46242 
.46273 
.46304 
•46334 



9.46365 
.46396 
•46426 
.46457 
.46487 



9.46518 
•46549 
•46579 
•46610 
.46640 



9.46671 
Lg. Vers, 



3l 
31 
31 
31 
3l 
31 
31 
31 
31 
31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 

30 
31 
31 
31 
31 

31 
30 
31 
31 
30 

31 
31 
30 
31 
30 

31 
30 
31 
30 
31 

30 
30 
31 
30 
30 

31 
30 
30 
30 
30 

31 
30 
30 
30 
30 

30 



Log.Exs. 



59124 
59168 
59211 
59255 
59298 



59342 
59385 
59429 
59472 
59515 



59559 
59602 
59646 
59689 
59732 



59776 
59819 
59863 
59906 
59949 



59993 
60036 
60079 
60123 
60166 



60209 
60253 
60296 
60339 
60383 



60426 
60469 
60512 
60556 
60599 



60642 

60685 

60729 

6077 

60815 



60858 
60902 
60945 
60988 
61031 



61075 
61118 
61161 
61204 
61247 



61291 
61334 
61377 
61420 
61463 



61506 
61550 
61593 
61636 
61679 



9-61722 
Log.Exs. 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 

i 

43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 



Lg.Vers, 



9.46671 
.46701 
•46732 
•46762 
•46793 



9.46823 
.46853 
.46884 
•46914 
.46945 



9.46975 
47005 
47036 
47066 
47096 



9.47127 
.47157 
.47187 
.47218 
.47248 



.47278 
•47308 
.47339 
.47369 
.47399 



47429 
.47459 

47490 
.47520 
.47550 



4758C 
47610 
47640 
.4767C 
4770C 



9.47731 
.47761 
•47791 
.47821 
./5 7851 



9.47881 
.47911 
.47941 
.47971 
.48001 



9.48031 
•48061 
•48090 
•48120 
.48150 



9.4818C 
•48210 
•48240 
.48270 
.48300 



9.48329 
.48359 
•48389 
•48419 
. 48449 



9.4R47P 
Lg. Vers. 



30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

30 
30 
29 
30 
30 

30 
30 
29 
30 
30 

29 
30 
30 
29 
30 

29 

15 



Log.Exs. 



61722 
61765 
61808 
61852 
61895 



61938 
61981 
62024 
62067 
62110 



62153 
62196 
62239 
62282 
62326 



62369 
62412 
62455 
62498 
62541 



62584 
62627 
62670 
62713 
62756 



62799 
62842 
62885 
62928 
62971 



63014 
63057 
63100 
63143 
63186 



63229 
63272 
63315 
63358 
63401 



63443 
63486 
63529 
63572 
63615 



63658 
63701 
63744 
63787 
6383C 



63873 
63915 
63958 
64001 
64044 



64087 
64130 
64173 
64216 
64258 
64301 



Log.Exs 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
42 
43 
43 
43 
43 

43 
42 
43 
43 
43 

43 
42 
43 
43 
43 

42 
43 
43 
43 
42 

43 



10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



30 

21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 
34 

35 
36 
37 
38 
-39 
40 
41 
42 
43 
j44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 

60 



P. P. 





43 


6 


4.8 


7 


5.1 


8 


5.8 


9 


6.5 


10 


7.2 


20 


14.5 


30 


21.7 


40 


29-0 


50 


36.2 



43 

4.3 

50 

5.7 

6.4 

71 

14.3 

21.5 

28.6 

35-8 



43 

4.2 

4.9 

5.6 

6.4 

7.1 

14.1 

21.2 

28.3 

35.4 



50J26 



31 

3.1 

7 
2 
7 
2 
5 
7 

2 



31 

31 



30 


3( 


3.0 


3. 


3 


5 


3. 


4 





4- 


4 


6 


4. 


5 


1 


5 


10 


1 


10. 


15 


2 


15. 


20 


3 


20. 


25 


4 


25. 



39 

2.9 

3.4 

3.9 

4.4 

4-9 

9-8 

14-7 

19-6 

24.6 



P.P. 



714 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

46° 47° 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_1 

10 

11 
12 
13 
14 

15 
16 
17 
18 

11 
30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49_ 

50 

51 
52 
53 

54 

55 

56 
57 
58 
59 

60 



Lg. Vers, 



•48478 
.48508 
48538 
48568 
48597 



J} 



•48627 
48657 
48686 
48716 
43746 



43775 
48805 
48335 
48864 
48394 



48923 
48953 
43983 
49012 
490A? 



49071 
49101 
49130 
49160 
49189 



49219 
49248 
49278 
49307 
•49333 



.49300 
49395 
49425 
49454 
49483 



49513 
49542 
49571 
49601 
49S30 



49059 
49689 
49718 
49747 
49776 



9.49806 
49835 
49864 
49893 
49922 



9-49952 
49981 
50010 
50039 
50068 



9.50097 
.50126 
•50155 
.50185 
.50214 



9.50^4-3 



Lg. Vers 



30 
29 
30 
29 

30 
29 
29 
30 
29 
29 
30 
29 
29 
29 

29 
30 

29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 
29 
29 

29 
29 
29 



29 



D 



Log.Exs, 



9.64301 
. 64344 
•64387 
. 64430 
. 64473 



9.64515 
•64558 
.64601 
.64644 
.64687 



9. 64729 
.64772 
.64815 
.64858 
.64901 



9 . 64943 
•64986 
.65029 
.65072 
.651.14 



9-65157 
.65200 
.65243 
.65285 
.65328 



9.65371 
•65414 
•65456 
•65499 
•65542 



9.65585 
•65627 
•65670 
.65713 
.65755 



9.65798 
•65841 
•65884 
•65926 
.65969 



9-66012 
66054 
66097 
68140 
66182 



66225 
66268 
66310 
66353 
66396 



9.68438 
.66481 
.66523 
.66566 
.66609 



9-66651 
.66694 
.66737 
•66779 
•66822 



9.6686^ 



Log.Exs 



D 

43 
42 
^3 
43 
42 
43 
43 
42 
43 

42 
43 
43 
42 
43 
42 
43 
42 
43 
42 

43 
42 
43 
42 
43 

42 
43 
42 
43 
42 

43 
42 
4:3 
42 
42 

43 

42 
43 
42 
42 

43 
42 
42 
43 
42 

42 
43 
42 
42 
43 

42 
42 
42 
43 
42 

42 
42 
43 
42 
42 

42 



Z> 



Lg. Vers. 



50243 
50272 
50301 
50330 
50359 



50388 
50417 
50446 
50475 
50504 



50533 
50562 
50591 
50619 
50648 



50677 
50706 
50735 
50764 
50793 



50821 
50850 
50879 
50908 
50937 



50965 
50994 
51023 
51052 
51080 



51109 
51138 
51167 
51195 
51224 



51253 
51281 
51310 
51338 
513R7 



51396 
51424 
51453 
51481 
51510 



51539 
51567 
51596 
51624 
51653 



51681 
51710 
51738 
51767 
51795 



51823 
51852 
51880 
51909 
51937 
519R5 



Lg. Vers, 



29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
28 
29 

29 
29 
28 
29 
29 

28 
29 
29 
28 
29 

28 
29 
28 
29 
28 
29 
28 
29 
28 
28 
29 
28 
28 
28 
29 

28 
28 
28 
28 
28 

29 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 



n 



Log.Exs. I> 



9-66864 
66907 
66950 
66992 
67035 



67077 
67120 
67162 
67205 
67248 



67290 
67333 
67375 
67418 
67460 



67503 
67546 
67588 
67631 
67673 



67716 
67758 
67801 
67843 
67886 



67928 
67971 
68013 
68056 
68098 



68141 
68183 
68226 
68268 
68311 



68353 
68396 
68438 
68481 
68523 



68566 
68608 
68651 
68693 
68735 



68778 
68820 
68863 
68905 
68948 



68990 
69033 
69075 
69117 
69160 



69202 
69245 
69287 
69330 
89372 



9-6941^ 



Log.Exs. I> 



42 
43 
42 
42 

42 
42 
42 
43 
42 

42 
42 
42 
42 
42 
42 
43 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 





1 
2 
3 
4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
_29. 

30 

31 
32 
33 
31 
35 
36 
37 
38 
3i 
40 
41 
42 
43 
44 

45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



43 



4 


3 


5 





5 


7 


6 


4 


7 


1 


14 


3 


21 


5 


28 


6 


35 


8 



42 

4-2 

4.9 

5-6 

6-4 

7-1 

14-1 

21-2 

28-3 

35-4 



43 

4-2 

4-9 

5-6 

6-3 

7^0 

14-0 

21-0 

28-0 

35-0 



6 
7 
8 
9 

10 
20 
30 
40 
50 



30 



3 





2- 


3 


5 


3- 


4 





3- 


4 


5 


4- 


5 





4- 


10 





9- 


15 





14- 


20 





19- 


25 





24- 



29 

2-9 
3-4 
3-8 
4-3 
4-8 
9-6 
14-5 
19-3 



6 
7 
8 

9 
10 
20 
30 
40, 
50!24-l 



29 

9 
4 
9 
4 
9 
8 
7 
6 
6 



28 
2-8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



28 
2-8 



P. P. 



715 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTJ 
48° 49° 



f 


Lg.Vers. 





9.51965 


1 


.51994 


2 


.52022 


3 


•52050 


4 


• 52079 


5 


9.52107 


6 


.52135 


7 


.52164 


8 


.52192 


9 


.52220 


10 


9.52249 


11 


.52277 


12 


.52305 


13 


.52333 


14 


•52362 


15 


3.52390 


16 


.52418 


17 


.52446 


18 


.52474 


19 


.52503 






30 


3-52531 


21 


.52559 


22 


.52587 


23 


.52615 


24 


.52643 


25 


). 52671 


26 


.52699 


27 


.52727 


28 


.52756 


29 


.52784 


80 


3-52812 


31 


.52840 


32 


.52868 


33 


.52896 


34 


.52924 


35 


3-52952 


36 


.52980 


37 


-53008 


38 


-53036 


39 


-53064 


40 


9-53092 


41 


.53120 


42 


-53147 


43 


-53175 


44 


•53203 


45 


9.53231 


46 


-53259 


47 


-53287 


48 


-53315 


49 


-53343 


60 


9. 53370 


51 


-53398 


52 


.53426 


53 


•53454 


64 


.53482 


55 


9.5350P 


56 


.53537 


57 


.53565 


58 


.5359?^ 


5i) 


-53620 


60 


^•5.^RAR 


t 


Lg.Vers.) 



28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
2§ 
28 

28 

28 
23 
28 
28 

28 
28 
28 
28 
28 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
27 
28 
28 

28 

27 
28 
28 
28 

27 
28 
28 
27 
28 

27 
28 
27 
28 
27 
28 



Log.Exs. I> 



9.69414 
.69457 
-69499 
.69542 
-69584 



9.69626 
.69669 
-69711 
.69753 
-69796 



9.69838 
.69881 
.69923 
.69965 
-70008 



9.700D0 
-70092 
.70135 
•70177 
-70220 



.70262 
.70304 
.70347 
.70389 
.70431 

.70474 
•70516 
.70558 
.70601 
.70643 



9.70685 
.70728 
.70770 
-70812 
-70854 



9-70897 
•70939 
•70981 
.71024 
.71066 



.71108 
.71151 
.71193 
.71235 
.7ir78 



9.71320 
.71362 
•71404 
. 71447 
.71489 



9.71531 
.71573 
.7161P 
.71658 
71700 



9.71743 
.71785 
•71827 
.71869 
.71912 



9-71954 



Log.Exs, 



D 



Lg. Vers. 



9-53648 
53676 
53704 
53731 
53759 



53787 
53814 
53842 
53870 
53897 



53925 
53952 
53980 
54008 
54035 



54063 
54090 
54118 
54145 
54173 



54200 
54228 
54251; 
54283 
54310 



54338 
54365 
54393 
54420 
54448 



54475 
54502 
54530 
54557 
54585 



54612 
54639 
54667 
54694 
54721 



54748 
54776 
54803 
54830 
54858 



54885 
54912 
54939 
54967 
54994 

55021 
55048 
55075 
55103 
55130 



55157 
55184 
55211 
55238 
55265 



9 55292 
Lg.Vers. 



Log.Exs. 



71954 
71996 
7203b 
72081 
72123 



72165 
72207 
72250 
72292 
72334 



72376 
72419 
72461 
72503 
72545 



72587 
72630 
72672 
72714 
72756 



72799 
72841 
72883 
72925 
72967 



73010 
73052 
73094 
73136 
73178 



73221 
73263 
73305 
73347 
73389 



73431 
73474 
73516 
73558 
73600 



73642 
73685 
73727 
73769 
73811 



73858 
73895 
73938 
7398C 
74022 



74064 
74106 
74148 
74191 
74233 

74275 
74317 
74359 
7440T 
74444 



9 . 74 4. R ft 



Log.Exs, 



o 
1 

2 
3 

_4_ 

5 

6 

7 

8 

_i 

10 

11 

12 

13 

14 

15 
16 
17 
18 

n 

30 

21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 
li 
35 
36 
37 
38 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 

55 
56 
57 
58 
59 

60 



P. P. 





43 


43 


6 


4-2 


4.2 


7 


4 


9 


4.9 


8 


5 


5 


5.6 


9 


6 


4 


6.3 


10 


7 


1 


7.0 


20 


14 


1 


14.0 


30 


21 


2 


21.0 


40 


28 


3 


28.0 


50 


35 


4 


35.0 



6 
7 
8 

9 
10 
20 
30 
40 
50, 



28 



2-81 


3 


3 


3 


8 


4 


3 


4 


7 


9 


5 


14 


2 


19 





23 


7 



38 

2-8 

3-2 

3.7 

4.2 

4.6 

9.3 

14.0 

18. 6 

23.3 





37 


6 


2.7 


7 


32 


8 


3-6 


9 


4-1 


10 


4-6 


20 


9-1 


30 


13-7 


40 


18-3 


50 


22.9 



37 

2.7 

31 

3-6 

4-0 

4.5 

9.0 

13-5 

18. 

22.5 



P. P> 



716 



TABLE YIII-— LOGARITHMIC VEHSED SINES AND EXTERNAL SECANXa 
50** SI** 





1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
M 
15 
16 
17 
18 

li 
30 

21 
22 
23 
2i. 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



Lg. Vers 



40 

41 
42 
43 
il 
45 
46 
47 
48 

50 

51 
52 
53 
54 

■55 
56 
57 
58 
19 



.55292 
.55319 
.55847 
.55374 
.55401 

55428 
55455 
55482 
55509 
_55538 

55563 
55590 
55617 
55644 
55871 



55698 
55725 
55751 
55778 
55805 



9.55832 
.55859 
•55886 
.55913 
.55940 



9-55906 
•55993 
.56020 
.56047 
.56074 



9-56101 
.56127 
.56154 
.56181 
.56208 

9.56234 
.56261 
.56288 
.56315 
•56311 



9.56368 
.56395 
.56421 
.56448 
.56475 



9-56501 
.56528 
.56554 
.56581 
•58608 



9 •56634 
.56661 
.56687 
.56714 
•56741 



9-56767 
•56794 
•56820 
-56847 
•56873 



9-56900 
Lg. Vers, 



27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
2§ 
27 
27 
27 
27 
26 
27 
27 
26 
27 
27 
26 
27 

27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
26 
27 

26 
26 
26 
27 
26 

26 
26 
26 
26 
27 

26 
26 
26 
26 
26 

26 



Log.Exs, 



9-74486 
.74528 
.74570 
.74612 
.74654 



9.74896 
.74739 
.74781 
.74823 
.74865 



9.74907 
.74949 
.74991 
.75033 
.75076 



9.75118 
.75160 
.75202 
.75244 
.75286 



9.75328 
.75370 
.75413 
.75455 
.75497 

9.75539 
.75581 
.75623 
.75660 
^5707 

9.75750 
.75792 
.75834 
.75876 
iZ5918 

9.75960 
.76002 
.76044 
.76086 
.76128 



9.76171 
.76213 
.76255 
.76297 
.76339 



9.76381 
.76423 
.76485 
.76507 
.76549 



9.76592 
.76634 
.76676 
.76718 
.76760 



9.76802 
.76341 
.768^6 
.76928 
.76970 



9-77012 
Log.Exs 



42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 



Lg, Vers 



9-56900 
.56926 
.56953 
.56979 
.57005 



9.57032 
.57058 
.57085 
.57111 
.57138 



9.57164 
.57190 
.57217 
.57243 
:_572_69 

9.57296 
.57322 
.57348 
.57375 
.57401 



9.57427 
.57454 
.57480 
.57506 
J7532 

9.57559 
.57585 
.57611 
.57637 
.57664 



9.57690 
57716 
57742 
57768 
57794 



9.57821 
.57847 
.57873 
.57899 
.5792'-i 



9.57951 
.57977 
.58003 
.58029 
J8p55 

9.58082 
.58108 
.58134 
.58160 

.58212 
.58238 
.58264 
.58290 
.58316 



9-58342 
.58367 
.58393 
.584±9 
-58445 



9 • 5SA71 
Lor. Vers. 



26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 

26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 



Log.Exs 



9-77012 
.77055 
.77097 
.77139 
•77181 



9-77223 
.77265 
.77307 
.77349 
.77391 



D 



9.77433 
.77475 
.77517 
.77560 
l77602 

9 . 77644 
.77686 
•77728 
.77770 
•77812 

9.77854 
77896 
77938 
.77980 
.78 022 

9 . 78064 
.78107 
.78149 
.78191 
.78233 

9.78275 
.78317 
.78359 
.78401 
.J78443 

9.78485 
.78527 
.78569 
.78611 
.7865^ 



9.78696 
.78738 
.78780 
.78822 

->l78864 

9 . 78906 
.78948 
.78990 
.79032 

_^P7| 

9.79116 
.79158 
.79200 
.79242 
.79285 



9-79327 
.79369 
-79411 
-79453 
•79495 



Q. 79537 
Log.Exs. 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



o 

1 

2 
3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

30 

21 
22 
23 
24 
25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39_ 

40 

41 
42 
43 
M 
45 
46 
47 
48 
49 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59 
60 



P.P. 



6 


4S 
4.2 


7 


4.9 


8 


5.6 


9 


6.4 


10 


7-1 


20 


14-1 


30 


21-2 


40 


28-3 


50 


35^4 



43 

4.2 

4.9 

5.3 

6.3 

7.0 

14-0 

21.0 

28.0 

35. 





2^ 


37 


6 


2.7 


2-7 


7 


3-2 


3.1 


8 


3.6 


3-6 


9 


4.1 


4.0 


10 


4.6 


4-5 


20 


9.1 


9.0 


30 


13.7 


13-5 


40 


18.3 


18-0 


50 


22.9 


22.5 



3g 



2-6 


2. 


3 


1 


3- 


3 


5 


3- 


4 





3 


4 


4 


4. 


8 


8 


8 


13 


2 


13- 


17 


5 


17. 


22 


1 


21. 



36 

a 

4 
9 
3 
6 

3 
6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



35. 

2-5 

30 

3.^ 

3.8 

4.2 

8.5 

12.7 

17-0 

21.2 



P.P. 



717 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 



52' 



53' 



Lg, Vers. 



58471 
58497 
58523 
58549 
58575 



58601 
58626 
58652 
58678 
58704 



58^30 
58755 
58781 
58807 
58833 



58859 
58884 
58910 
58936 
58962 



58987 
59013 
59039 
59064 
59090 



59116 
59141 
59167 
59193 
59218 



59244 
59270 
59295 
59321 
59346 



59372 
59397 
59423 
59449 
59474 



59500 
59525 
59551 
595'76 
59602 



59627 
59653 
59678 
59704 
59729 



59754 
59780 
59805 
59831 
59856 



59881 
59907 
59932 
59958 
59983 



6000P 



Lg. Vers. 



Log.Exs. 



79537 
79579 
79621 
79663 
79705 



79747 
79789 
79831 
79874 
79916 



79958 
80000 
80042 
80084 
80126 



80168 
80210 
80252 
80294 
80S36 



80378 
80420 
80463 
80505 
80547 



80589 
80631 
80673 
80715 
80757 
80799 
80841 
80883 
80925 
80968 



81010 
81052 
81094 
81136 
81178 



81220 
81262 
81304 
81346 
81388 



81430 
81473 
81515 
81557 
81599 



81641 
81683 
81725 
81767 
81809 



81851 
81894 
81936 
81978 
82020 



9-82062 
Log.Exs. 



n 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 

42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 



Lg.Vers, 



60008 
60034 
60059 
60084 
60110 



60135 
60160 
60185 
60211 
60236 



60261 
60286 
60312 
60337 
60362 



60387 
60412 
60438 
60463 
60488 



60513 
60538 
60563 
60589 
60614 



60639 
60664 
60689 
60714 
60739 



60764 
60789 
60834 
60839 
6C864 



60889 
60914 
60939 
60964 
60989 



61014 
61039 
61064 
61089 
61114 



61139 
61164 
61189 
61214 
61239 



61264 
61289 
61313 
61338 
61363 



61388 
61413 
61438 
61462 
61487 



61512 



Lg. Vers. 



i> 

25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
25 
2fc 

25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 
25 
25 
24 
25 

25 
25 
24 
25 
25 

25 
24 
25 
24 
25 
25 



Log.Exs. 



9.82062 
.82104 
.82146 
•82188 
•82230 



9-82272 
•82315 
.82357 
•82399 
.82441 

9.82^83 

.82525 
.82567 
•82609 
•82651 

9.82694 
.82736 
•82778 
•82820 
.82862 



9.82904 
.82946 
.82988 
.83031 
.83073 



9.83115 
.83157 
•83199 
•83241 
•83283 



9 •83325 
•83368 
•83410 
•83452 
•83494 



9-83536 
•83578 
.83620 
.83663 
•83705 



9 •83747 
.83789 
.83831 
.83873 
•83916 



9^83958 
.84000 
. 84042 
.84084 
•84.26 



9-84168 
.84211 
.84253 
•84295 
.84337 



9.84379 
• 84422 
.84464 
.84506 
•84548 



9 • 845 90 
Log.Exs, 



2> 



42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 





1 
2 
3 

-A 

5 

6 

7 

8 

_9- 

10 

11 

12 

13 

14 

15 
36 
17 
18 

21 

20 

21 
22 
23 

M. 
25 
26 
27 
28 

-29 

30 
31 
32 
33 

M. 
35 
36 

r37 
38 

19 

40 
41 
42 
43 

J4 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
-54 

55 
56 
57 
58 
59 

60 



P.P. 





42 


42 


6 


4.2 


4 2 


7 


4-9 


4-9 


8 


5.6 


5-6 


9 


6-4 


6-3 


10 


7-1 


7.0 


20 


14-1 


14-0 


30 


21-2 


21-0 


40 


28-3 


28.0 


50 


35.4 


35.0 





26 


25^ 


6 


2.6 


2.5 


7 


3.0 


3 





8 


3.4 


3 


4 


9 


3.9 


3 


i 


10 


4.3 


4 


2 


20 


8.6 


8 


5 


30 


13-C 


12 


7 


40 


17.3 


17 


a 


60 


21.6 


21 





25 


2^ 


6 


2.5 


2- 


7 


2.9 


2- 


8 


3.3 


3- 


9 


3.7 


3- 


10 


4.1 


4- 


20 


8-3 


8. 


30 


12-5 


12- 


40 


16-6 


16- 


50 


20.8 


20. 



P.P. 



718 



TABLE VIIX.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
54° 55° 



Lg. Vers 



61512 
61537 
61562 
61586 
61611 
61636 
61661 
61685 
61710 
61735 



61760 
61784 
61809 
61834 
61858 



61883 
61908 
61932 
61957 
61982 



62006 
62031 
62055 
62080 
62105 



62129 
62154 
62178 
62203 
62227 



62252 
62276 
62301 
62325 
62350 



62374 
62399 
62423 
62448 
62472 



62497 
62521 
62546 
62570 
62594 



62619 
62643 
62668 
62692 
62716 



62741 
62765 
62789 
62814 
62838 



62862 
62887 
62911 
62935 
62960 



9-62984- 



Lg. Vers 



2> 



Log.Exs. 



9.84590 
.84632 
.84675 
.84717 
.84759 



9.84801 

.84843 

•84886 

84928 

.84970 



9.85012 
•85054 
.85097 
•85139 
•85181 



9 •85223 
•85265 
•85308 
•85350 
.85392 



9.85434 
.85476 
.85519 
•85561 
•85603 



9.85645 
.85688 
•85730 
.85772 
•85814 



9^85857 
•85899 
•85941 
•85983 
•86026 



9^86068 
•86110 
•86152 
.86195 
•86237 



9 •86279 
•88321 
•86364 
•86406 
•86448 



9.86490 
•86533 
•86575 
•86617 
.86659 



9.86702 
.86744 
.86786 
•86829 
.86871 



9.86913 
.86956 
.86998 
.87040 
.87082 



9.87125 



Log.Exs. 



Lg. Vers 



62984 
63008 
63032 
63057 
63081 



63105 
63129 
63154 
63178 
63202 



63226 
63250 
63274 
63209 
63323 



63347 
63371 
63395 
63419 
63443 



63468 
63492 
63516 
63540 
63564 



63588 
63612 
63636 
63660 
63684 



63708 
63732 
63756 
63780 
63804 



63828 
63852 
63876 
63900 
63924 



63948 
63972 
63996 
64019 
64048 



64067 
64091 
64115 
64139 
64163 



64187 
64210 
64234 
64258 
64282 



6430? 
64330 
64353 
64377 
64401 



9-64425 



D Lg.Vers 



D 



I> 



9.87971 
.88014 
•88056 
•88099 
•88141 



Log.Exs. 



9.87125 
•87167 
•87209 
.87252 
•87294 



9.87336 
.87379 
•87421 
•87463 
.87506 



9.87548 
.87590 
•87633 
•87675 
•87717 



9.87760 
•87802 
•87844 
•87887 
.87929 



9.88183 
.88226 
.88268 
•88310 
•88353 



9.88395 
•88438 
•88480 
•88522 
•88565 



9.88607 
•88650 
.88692 
.88734 
.88777 



9.88819 
•88862 
.88904 
.88947 
•88989 



9.89031 
•89074 
.89116 
.89159 
■89201 



9.89244 
•89286 
•89329 
•89371 
•89414 



9.89456 
•89499 
•89541 
.89583 
•89626 



9.89668 



Log.Exs, 



D 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



50 

*51 
52 
53 
54 
55 
56 
57 
58 
5i 
60 



P.P. 





4^ 


43 


6 


4.2 


4.2 


7 


4.9 


4.9 


8 


5.6 


5.6 


9 


6.4 


6.3 


10 


7^1 


7.0 


20 


14U 


14.0 


30 


21^2 


21.0 


40 


28-3 


28.0 


50 


35.4 


35.0 





35 


3? 


6 


2.5 


2-4 


7 


2.9 


2.8 


8 


3.3 


3.2 


9 


3.7 


3.7 


10 


4.1 


4-1 


20 


8^3 


8.1 


30 


12^5 


12.2 


40 


16^6 


16.3 


50 


20.8 


20.4 





34 


s5 


6 


2.4 


2.3 


7 


2^8 


3*1 


8 


3.2 


9 


3-6 


3-5 


10 


4.0 


3.9 


20 


8.0 


7.5 


30 


12^0 


11^7 


40 


16^0 


15.6 


50 


20.0 


19.6 



P.P. 



719 



^ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

57° 



56*^ 



O 

1 
2 
3 
_i 
5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
3i 
35 
36 
37 
38 
39 



Iff. Vers, 



9 . 64425 
. 64448 
.64472 
•64496 
.64520 



9.64543 
.64567 
•64591 
•64614 
.64638 



9-64662 
.64685 
.64709 
•64733 
.64756 



9.64780 
• 64804 
•64827 
•64851 
.64875 



9.65016 
.65040 
.65063 
•65087 
•65110 



•64898 
•64922 
.64945 
.64969 
•64992 



•65134 
•65157 
•65181 
•65204 
.65228 



9.65251 
•65275 
•65298 
.65321 
•653A5 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
5i 
55 
56 
57 
58 
59 

60 



•65368 
•65392 
.65415 
.65439 
•65482 

9^65480 
.65509 
.65532 
.65556 
•65579 



9.65602 
.65626 
.65649 
.65672 
J569J6 

9.65719 
.65742 
.65765 
.65789 
.65812 



q. 65835 



23 
24 
23 
24 

23 
24 
23 
23 
24 

23 
23 
24 
23 
23 

24. 
23 
23 
23 
24 

23 
23 
23 
23 
23 

24 
23 
23 
23 
23 

23 
23 
23 

23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 



-og.Exs. 



9-89668 
.89711 
.89753 
.89796 
.8983'8 



9.89881 
.89923 
.89966 
.90008 
.90051 



9.90094 
.90136 
.90179 
.90221 
.90264 



•90306 
•90349 
.90391 
.90434 
.90476 



9-90519 
.90561 
.90604 
.90647 
•90689 



D Lg.Vers 



9-90732 
.90774 
.90817 
.90860 

_l90902 

9.90945 
.90987 
.91030 
.91073 
91115 



9.91158 
.91200 
.91243 
.91286 
.91328 



9.9-1371 
.91414 
.91456 
.91499 
.91541 



9.91584 
.91627 
.91669 
.91712 
.91755 



9-91797 
.91840 
.91883 
.91926 
.91968 



Lg. Vers 



D 



9.92011 
.92054 
.92096 
.92139 

.•921_82 

9 . 92222 



42 
42 
42 
42 

42 
42 
42 
42 
42 

43 

42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
43 
42 
42 

42 
42 
42 
43 
42 

42 
42 
42 
43 
42 

42 
42 
42 
43 
42 

42 
43 
42 
42 
42 

43 
42 
42 
43 
42 

42 
43 
42 
43 
42 

42 
43 
42 
43 
42 

42 



9.65835 
.65859 
.65882 
.65905 
.65928 



Log.Exs, 



9.65952 
.65975 
.65998 
.66021 
.66044 



9.66068 
.66091 
.66114 
.66137 
.66160 



9.66183 
.66207 
.66230 
.66253 
.66276 



9.66299 
•66322 
.66345 
.66368 
•663 91 

9.66415 
.66438 
.66461 
.66484 
•66507 

9 . 66530 
.66553 
.66576 
.66599 
.66622 



Lcg.Exs, 



9.66645 
.66668 
.66691 
.66714 
•_66737 

9 . 66760 
.66783 
.66805 
.66828 
•66851 

9.66874 
.66897 
.66920 
. 66943 
.6696P 

9.66989 
.67012 
.67034 
.67057 
.67080 

9.67103 
.67126 
.67149 
.67171 
.67194 



9-67217 



23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 
23 
23 
23 
23 
23 
23 
23 
23 
23 

23 
23 
23 
23 
23 

23 

23 
23 
22 
23 

23 
23 
22 
23 
23 

22 
23 
23 
22 
23 
22 



9^9222| 



.92267 
•92310 
.92353 
.92395 



9.92438 
•92481 
.92524 
.92566 
•92609 



9^92652 
•92695 
•92737 
•92780 
•92823 

9-92866 
.92909 
•92951 
.92994 
•93037 

9^93C80 
.93123 
•93165 
.93208 
•93251 

9^93294 
.93337 
.93380 
.93422 
•93465 



9 •93508 
•93551 
.93594 
•93637 
•9368C 

9.93722 

.93765 

.93808 

.93851 

93894 



Lg. Vers 



9.93937 
•93980 
.94023 
.94066 
^109 

9.94151 
.94194 
.94237 
.94280 
.94323 

9.94366 
.94409 
.94452 
•94495 
.94538 



9.94581 
.94624 
•94667 
.94710 
-94753 



P. 94796 



D 

43 
42 
43 
42 

43 
42 
43 
42 
43 
42 
43 
42 
43 
42 

43 
43 
42 
43 
42 

43 
43 
42 
43 
43 

42 
43 
43 
42 
43 

43 
42 
43 
43 
43 
42 
43 
43 
43 
42 

43 
4.3 
43 
43 
43 
42 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 

43 
43 
43 
43 

43 



D jLog.Exs 
720 



5 
6 
7 
3 
__9 

10 

11 

12 
13 
ii 

15 
16 
17 
18 
2i 
30 
21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
3i 
40 
41 
42 
43 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
_54 

55 
56 
57 
58 

31 
60 



7> 



P.P. 



43 


42 


4.3 


4 2 


5.0 


4 


9 


57 


5 


g 


6^4 


6 


4 


7^1 


7 


1 


14.3 


14 


\ 


21.5 


21 


2 


28.6 


28 


3 


35^8 


3b 


• 4 



24 



2 


4 


2. 


2 


8 


2- 


3 


2 


3. 


3 


6 


3. 


4 





3. 


8 





7- 


12 





11^ 


16 





15^ 


20 





19. 



25 

3 

\ 

5 
9 
8 
7 
6 
6 



6 
7 
8 

9 
10 
20 
30 
40 
50 



23 


2'2 


2-3 


2.2 


2 


7 


2 


6 


3 





3 





3 


4 


3 


4 


3 


8 


3 


7 


7 


6 


7 


5 


11 


5 


11 


2 


15 


3 


15 





19 


1 


18 


.7 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECAN-ra, 



58 



59' 



10 

11 
12 
13 
14 



Lg. Vers. 



Log. Exs. 



9.67217 
.67240 
.67263 
.67285 
.67308 



1.67331 
•67354 
.67376 
.67399 
.67422 



23 
23 
22 
23 
22 
23 
22 
23 
22 



9.94796 
.94839 
.94882 
.94925 
.94968 



15 
16 
17 
18 

19 



30 

21 
22 
23 
24 



25 
26 
27 
28 
29 



9.67445 
674671 ^§ 
67490 ii 
675131 2| 
67535! "^^ 



30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



9.681221 i^ 



45 
46 
47 
48 
49 



50 

51 
52 
53 

54 



55 
56 
57 
58 
59 



60 



^ 



9.67558 22 
.675811 2| 
.67603 2| 
.67626! ii 
•67649 : 23 

• 676711 22 
•67694! 22 
.677171 i% 
•67739 ii 

• 677621 "^i 
-T 22 



9.95011 
.95,054 
.95097 
.95140 
.95183 



D Lg. Vers. 



9.95226 
.95269 
.95313 
.95356 
.95399 



1.95442 
.95485 
.95528 
.95571 
^.95614 



9.67784 
.67807 
.67830 
.67852 
.67875 



9.67897 
•67920 
•67942 
.67965 
.679871 



9-68010 
•68032 
.68055 
•68077 
•68100 



22 
23 
22 

22 
22 
22 
22 
22 
22 

i 22 
22 
22 
22 
22 

22 
22 



9.96089 
.96132 
.96175 
.96218! 
.96261 



9.96305 
.96348 
.96391 
.96.434 
.96478 



•68145 ii 
.68167 '^- 
.68190 
.68212 



9.68235 
.68257 
.68280 
•68302 
•68324 



9.68347 
.68369 
.68392 
.68414 
•68436 



9.68459 
.68481 
.68503 
.68526 
•68548 



9.68571 



Lg. Vers. 



22 

22 
22 
22 
22 
22 
22 

22 
22 
22 
22 
22 
22 
22 
22 
22 
22 

22 



9-95657 
.95705 
.95744 
.95787 
.95830 



9.95873 
.95916 
.95959 
.96002 
.96046 



9.96521 
.96564 
.96607 
.96650 
.96694 



9.96737 
•96780 
.96824 
.96867 
.96910 



9.96953 
.96997 
.97040 
.97083 
.97127 



9.97170 
.97213 
.97257 
.97300 
.97343 



9.97387 



Log. Exs, 



43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 

1 ^2 

43 

43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 

43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 



9.68571 
.68593 
.68615 
.68637 
•68660 



9.68682 
.68704 
.68727 
.68749 
.68771 



9-68793 
•68816 
.68838 
.68860 
.68882 



9.68905 
.68927 
.68949 
.68971 
•68993 



9.69016 
.69038 
.69060 
.69082 
•69104 



9.69126 
.69149 
.69171 
.69193 
•69215 



9.69237 
.69259 
.69281 
.69303 
.69325 



9.69347 
.69369 
.69392 
.69414 
.69436 



9.69458 
.69480 
.69502 
.69524 
•69546 



9.69568 
.69590 
.69612 
.69634 
.69656 



9.69673 
.69700 
.69721 
.69743 
.69765 



9.69787 
.69809 
.69831 
.69853 
.6-9875 



•69897 



D Lg 



Vers. 



22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
22 
22 
22 
22 
22 
22 
22 
22 

22 
22 
22 
22 
22 

22 
22 
2T 
22 
22 

22 
22 
22 
2T 
22 

22 



Log. Exs. D 



9.97387 
.97430 
.97473 
.97517 
•97560 



9.97603 
.97647 
.97690 
.97734 
.97777 



9-97820 
.97864 
.97907 
.97951 
.97994 



9.98038 
.98081 
.98125 
.98168 
.98211 



9 .98255 
.98298 
.98342 
.98385 
.98429 



9 .98472 
.98516 
.98559 
.98603 
.98647 



9.98690 
.98734 
.98777 
.98821 
=98864 



9.98908 
.98952 
.98995 
.99039 

.99082 



9.99126 
.99170 
.99213 
.99257 
.99300 



9.99344 
•99388 
.99431 
.99475 
.99519 



9.99562 
.99606 
.99650 
.99694 
•99737 



9.99781 
.99825 
.99868 
.99912 

9.99956 



10.00000 



log, Exs. 



D 



60 



P.P. 





44 


6 


4.4 


7 


5.1 


8 


5.8 


9 


6.6 


10 


7.3 


20 


14.6 


30 


22.0 


40 


29.3 


50 


36.6 



45 

4.3 

5.1 

5.8 

6.5 

7.2 

14.5 

21.7 

29.0 

36.2 






23 


6 


2.3 


7 


2.7 


8 


3.0 


9 


3.4 


10 


3-8 


20 


7.6 


30 


11.5 


40 


15.3 


50 


19.1 



2!2L 

2-2 
2.6 
3.0 

1:1 

iL'i 

15j 



18 



6 

7 

8 

9 

10 

20 

30 

40 

50 



22 

2-2 

2.5 

2.9 

3.3 

3.6 

7.3 
11.0 

14.6 

18. 31X7.9 



P.P. 



721 



aCA3LEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
60° 61° 



9.69897 
.69919 
.69940 
.69962 
.69984 



9.70006 
.70028 
.70050 
.70072 
.70093 



10 

11 
12 
13 
14 



9.70115 
.70137 
.70159 
.70181 
•70202 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



Lg. Vers. 



9.70224 
.70246 
.70268 
.70289 
.70311 



9.70333 
.70355 
.70376 
.70398 
.70420 



9 . 70441 
70463 
70485 
70507 
70528 



9.70550 
.70572 
•70593 
.70615 
.70636 



9.70658 
.70680 
.70701 
.70723 
.70745 



9.70766 
.70788 
.70809 
.70831 
.70852 



45 
46 
47 
48 
49 



9.70874 
.70896 
.70917 
.70939 
.70960 



50 

.51 
52 
53 
54 



55 
56 
57 
58 
59 



9.70982 
.71003 
.71025 
.71046 
.71068 



9.71089 
.71111 
.71132 
.71154 
.71175 



60 



9.71197 



22 
21 
22 
22 

22 
21 
22 
22 
21 
22 
21 
22 
22 
21 

22 
21 
22 
21 
22 

2T 
22 
21 
22 
21 

2T 
22 
21 
22 
21 

21 
22 
21 
21 
21 

22 
21 
21 
21 
22 

21 
21 
21 
21 
21 

22 
21 
21 
21 
21 

21 
21 
21 
21 
21 

2T 
21 
21 
21 
21 

21 



10 



10. 



Log. Exs. 



00000 
00044 
00087 
00131 
00175 



00219 
00262 
00306 
00350 
00394 



10. 



00438 
00482 
00525 
00569 
00613 



10. 



00657 
00701 
00745 
00789 
00833 



10 



00876 
00920 
00964 
01008 
01052 



10. 



01096 
01140 
01184 
01228 
01272 



10 



01316 
01360 
01404 
01448 
01492 



10 



01536 
01580 
01624 
01668 
01712 



10 



01756 
01800 
01844 
01889 
01933 



10 



01977 
02021 
02065 
02109 
02153 



10 



Lg. Vers. D Log. Exs 



02197 
02242 
02286 
02330 
02374 



10 



02418 
02463 
02507 
02551 
02595 



Lg. Vers. 



10.02639 



44 
43 
44 
43 
44 
43 
44 
44 
43 

44 
44 
43 
44 
44 

44 
43 
44 
44 
44 

43 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 



71197 
71218 
71239 
71261 
71282 



71304 
71325 
71346 
71368 
71389 



71411 
71432 
71453 
71475 
71496 



D Log. Exs. 



71517 
71539 
71560 
71581 
71603 



71624 
71645 
71667 
71688 
71709 



71730 
71752 
71773 
71794 
71815 



9. 



71837 
71858 
71879 
71900 
71922 



71943 
71964 
71985 
72006 
72028 



72049 
72070 
72091 
72112 
72133 



72154 
72176 
72197 
72218 
72239 



72260 
72281 
72302 
72323 
72344 



723,65 
72386 
72408 
72429 
72450 
72471 



21 
21 
21 
21 

2T 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 
21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 



10. 



DjLg. 



Vers. 



02639 
02684 
02728 
02772 
02816 



10 



02861 
02905 
02949 
02994 
03038 



10 



03082 
03127 
03171 
03215 
03260 



10 



03304 
03348 
03393 
03437 
03481 



10 



10. 



03526 
03570 
03615 
03659 
03704 



03748 
03793 
03837 
03881 
03926 



10 



03970 
04015 
04059 
04104 
04149 



10. 



04193 
04238 
04282 
04327 
04371 



10 



04416 
04461 
04505 
04550 
04594 



10 



04639 
.04684 
.04728 
.04773 

04818 



10 



04862 
04907 
04952 
04996 
05041 



10 



05086 
05131 
05175 
05220 
05265 



10-05310 



D log. Exs. 
722 



44 
44 
44 
44 

44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
45 

44 
44 
44 
44 
44 

44 
45 
44 
44 
44 

45 
44 
44 
44 
45 
44 
45 
44 
44 
45 

44 
45 
44 
45 
44 

45 



50 

51 
52 
53 
5^ 

55 
56 
57 
58 
59 



60 



P.P. 





45 


4l_ 


6 


4.5 


4.4 


7 


5.2 


5-2 


8 


6.0 


5.9 


9 


6.7 


6.7 


10 


7.5 


7.4 


20 


15.0 


14.8 


30 


22.5 


22.2 


40 


30.0 


29.6 


50 


37.5137.1 



44 



4.4 


4. 


5.1 


5. 


5.8 


5. 


6.6 


6. 


7.3 


7. 


14.6 


14. 


22.0 


21. 


29-3 


29. 


36.6 


36. 



43 

3 
1 
8 
5 
2 
5 
7 





22 


21 


6 


2.2 


'J.1 


7 


2.5 


2.5 


8 


2.9 


2.8 


9 


3.3 


3.2 


10 


3.6 


3.? 


20 


7.3 


7.1 


30 


11.0 


10.7 


40 


14.6 


14.3 


50 


18.3 


17.9 



21 



6 
7 


ii 


8 

9 


U 


10 


3.5 


20 


7.0 


30 


10.5 


40 


14.0 


50 


17.5 



P.P. 



OiABLEVIII— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
62° 63° 



7 

8 

_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

1 45 

I 46 

47 

1 49 
60 

! 52 
i 53 
I 54 

55 

56 

1 57 

' 58 

59 

1 60 



Lg, Vers. 



9.72471 
.72492 
.72513 
.72534 
.72555 



D 



9.72576 
.72597 
.72618 
.72639 
•72660 



9.72681 
.72701 
.72722 
.72743 
•72764 



9^72785 
.72806 
.72827 
.72848 
•72869 



9^72890 
.72911 
.72931 
.72952 
•72973 



9.72994 
•73015 
•73036 
.73057 
.73077 



9.73098 
.73119 
.73140 
.73161 
.73181 



9-73202 
•73223 
•73244 
•73265 
•73285 



9.73306 
.73327 
. 73348 
•73368 
.73389 



9 •73410 
.7343Q 
•73451 
•73472 
•73493 



9^73513 
•73534 
•73555 
•73575 
•73596 



9.73617 
•73637 
.73658 
•73679 
•73699 



9.73720 
Lg.Vers. 



21 
21 
21 
21 

21 
21 
21 
21 
21 

21 
20 
21 
21 
21 

2] 
21 
21 
20 
21 

21 
21 
20 
21 
21 

21 
20 
21 
21 
20 
21 
21 
20 
21 
20 
21 
21 
20 
21 
20 

21 
20 
21 
20 
21 

2Q 
20 
21 
20 
21 

20 
20 
21 
20 
20 

21 
20 
20 
21 
20 

20 



Log. Exs. 



10.05310 
.05354 
.05399 
.05444 
•05489 



10^05534 
.05579 
.05623 
.05668 
•05713 



10^05758 
.05803 
.05848 
.05893 
•05938 



10.05983 
.06028 
.06072 
.06117 
•06162 



10.06207 
.06252 
.06297 
.06342 
.06387 



10.06432 
.06477 
.06522 
.06568 
•06613 



10^06658 
.06703 
.06748 
.06793 
•06838 



10^06883 
.06928 
.06974 
.07019 
.07064 



10.07109 
.07154 
.07200 
.07245 
•07290 



10.07335 
.07380 
.07426 
.07471 
.07516 



10.07562 
.07607 
.07652 
.07697 
•07743 



10^07788 
.07834 
.07879 
.07924 
•07970 



10.08015 
Log. Exs, 



44 
45 
45 
44 

45 
45 
44 
45 
45 

45 
44 
45 
45 
45 
45 
45 
44 
45 
45 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 
45 
45 
45 
45 

45 



Lg.Vers. 



9-73720 
.73740 
•73761 
.73782 
•73802 



9 •73823 
•73843 
•73864 
•73884 
•73905 



9.73926 
.73946 
.73967 
.73987 
.74008 



9.74028 
. 74049 
.74069 
. 74090 
.74110 



9.74131 
.74151 
.74172 
.74192 
.74213 



9.74233 
.71254 
.74274 
.74294 
.74315 



9-74335 
.74356 
.74376 
•74396 
.74417 



9.74437 
.74458 
.74478 
.74498 
.74519 



9.74539 
.74559 
•74580 
.74600 
•74620 



9 . 74641 
•74661 
•74681 
•74702 
•74722 



9 • 74742 
•74762 
•74783 
•74803 
•74823 



9 - 74844 
74864 
74881 
74901 
74924 



9-74945 
Lg. Vers. 



20 
20 
21 
20 
20 
20 
20 
20 
21 
20 
20 
20 
20 
20 

20 
20 
20 
20 
20 
20 
20 
20 
20 
20 

20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 



Log. Exs. 



10-08015 
.08061 
.08106 
.08151 
.08197 



10.08242 
.08288 
.08333 
.08379 
.08424 



10.08470 
.08515 
.08561 
.08606 
.08652 



10.08697 
•08743 
.08789 
.08834 
•08880 



10 •08926 
.08971 
.09017 
.09062 
•09108 



10.09154 
.09200 
•09245 
.09291 
•09337 



D 



10.09382 
•09428 
.09474 
.09520 
.09566 



10.09011 
.09657 
.09703 
.09749 
-09795 



10-09841 
.09886 
.09932 
.09978 
.10024 



10.10070 
.10116 
.10162 
.10208 
.10254 



10.10300 
.10346 
.10392 
•10438 
-10484 



10-10530 
•10576 
.10622 
.10668 
.10714 



10-10760 
Log. Exs. 



45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

45 
46 
45 
45 
45 
46 
45 
45 
45 
46 
45 
46 
45 
45 
46 
45 
46 
46 
45 
46 

45 
46 
46 
45 
46 

46 
45 
46 
46 
46 

46 
46 
46 
45 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 

JA 
15 
16 
17 
18 

-ii 

20 
21 
22 
23 

_24 

25 
26 
27 
28 
20^ 

30 

31 

32 

33 

J4 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
Jli 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 

60 





P. P. 


46 46 


6 


4.6 


4.6 


7 


5.4 


5^3 


8 


6.2 


6-1 


9 


7.0 


6.9 


10 


7.7 


7.6 


20 


15.5 


15.3 


30 


23.2 


23^0 


40 


31.0 


30^6 


50 


38.7 


38.3 



45 



6 

7 

8 

9 

10 

20 

30 

40 

50 



4.-5 


4. 


5^3 


5. 


6.0 


6. 


6^8 


6. 


7.6 


7. 


15.1 


15. 


22.7 


22. 


30.3 


30. 


37.9 


37. 



45 
5 
2 

7 
5 

5 

5 



6 

7 

8 

9 

10 

20 

30 

40 

50 



44 

4.4 

5.2 

5^9 

6^7 

7-4 

14-8 

22.2 

29.6 

37.1 



31 



2^1 


2. 


2.4 


2- 


2.8 


2- 


3.1 


3. 


3.5 


3- 


7.0 


6- 


10.5 


10. 


14.0 


13. 


17-5 


17. 



20. 


4 
7 
1 
4 
8 
2 
6 
1 



20 

2.0 

2.3 

2.6 

3.0 

3.3 

6.6 

10.0 

13-3 

16.6 



P. P. 



723 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
64° 65° 



Lg.Vers. 



45 9 

46 

47 

48 

49_ 

50 

51 

52 

53 

54 



74945 
74965 
74985 
75005 
75026 



75046 
75066 
75086 
75106 
75126 



75147 
75167 
75187 
75207 
75227 



75247 
75267 
75287 
75308 
75328 



75348 
75368 
75388 
75408 
75428 



75448 
75468 
75488 
75508 
75528 



75548 
75568 
75588 
75608 
75628 



75648 
75668 
75688 
75708 
75728 



75748 
75768 
75788 
75808 
75828 



75848 
75868 
75888 
75908 
75928 



75947 
75967 
75987 
76007 
76027 



76047 
76067 
76087 
76106 
76126 



9.76146 



20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
20 

20 
20 
20 
20 
19 

20 
20 
20 
20 
20 

19 
20 
20 
20 
19 

20 
20 
20 
19 
20 
20 



Log.Exs. 



10 



10760 
10807 
10853 
10899 
10945 



10 



10991 
11037 
11084 
11130 
11176 



10 



11222 
11269 
11315 
11361 
11407 



10 



11454 
11500 
11546 
11593 
11639 



10 



11685 
11732 
11778 
11825 
11871 



10 



11917 
11964 
12010 
12057 
12103 



10 



12150 
12196 
12243 
12289 
12336 



10 



12383 
12429 
12476 
12522 
12569 



10. 



12616 
12662 
12709 
12756 
12802 



10 



12849 
12896 
12942 
12989 
13036 



' Lg. Vers-|l> Log.Exs 



10 



13083 
13130 
.13176 
13223 
.13270 



10 



10 



13317 
13364 
13411 
13457 
13504 

T355T 



46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

46 
46 
46 
46 
46 

47 
46 
46 
46 
46 

47 
46 
46 
47 
46 

46 
47 
40 
47 
46 

47 
47 
46 

46 

47 
47 
47 
46 
47 

47 



Lg. Vers, 



76146 
76166 
76186 
76206 
76225 



76245 
76265 
76285 
76304 
76324 



76344 
76364 
76384 
76403 
76423 



76443 
76463 
76482 
76502 
76522 



76541 
76561 
76581 
76600 
76620 



76640 
76659 
76679 
76699 
76718 



76738 
76758 
76777 
76797 
76817 



76836 
76856 
76875 
76895 
76915 



76934 
76954 
76973 
76993 
77012 



77032 
77052 
77071 
77091 
77110 



77130 
77149 
77169 
77188 
77208 



77227 
77247 
77266 
77286 
77305 



9.77325 



l> Lg.Vers 



D 



19 
20 
20 
19 
20 
19 
20 
19 
20 
20 
19 
20 
19 
20 

19 
20 
19 
19 
20 

19 
20 
19 
19 
20 

19 
19 
20 
19 
19 

20 
19 
19 
19 
20 

19 
19 
19 
20 
19 
19 
19 
19 
19 
19 

20 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 
19 



Log.Exs. 



10-13551 
.13598 
.13645 
.13692 
.13739 



D 



10.13786 
.13833 
.13880 
.13927 

. .13974 



10.14021 
.14068 
.14115 
.14162 
.14210 



10.14257 
.14304 
.14351 
.14398 
.14445 



10.14493 
.14540 
.14587 
.14634 
.14682 



10.14729 
.14776 
.14823 
.14871 
.14918 



10.14965 
.15013 
.15060 
.15108 
.15155 



10.15202 
.15250 
.15297 
.15345 
.15392 



10.15440 
.15487 
.15535 
.15582 
.15630 



10.15678 
.15725 
.15773 
.15820 
.15868 



10.15916 
.15963 
.16011 
.16059 
.16106 



10.16154 
.16202 
.16250 
.16298 
.16345 



10.16393 



Log. Exs. 



47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 

^1 
45 

47 
47 
48 

47 
47 
48 
47 
47 

48 
47 
48 
48 
47 
48 



10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
34 
35 
36 
37 
38 
_39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
_59 

60 



P.P. 



48 



4 


8 


4. 


5 


6 


5. 


6 


4 


6. 


7 


2 


7. 


8 





7. 


16 





15- 


24 





23- 


32 





31. 


40 





39. 



47 



4 


7 


4. 


5 


5 


5. 


6 


2 


6- 


7 





7. 


7 


3 


7. 


15 


6 


15. 


23 


5 23- 


31 


3 31 


39 


138. 



47 

7 
5 
3 
1 
9 
8 
7 
6 
6 



46 

6 
4 
2 

7 
5 
2 

7 



46 

4.6 



20 15 
30 23 
40 30 
50 38 





20 


2( 


6 2-0 


2. 


7' 


4 


2. 


8 


' 


7 


2- 


9 


3 


1 


3. 


10 


3 


4 


3- 


20 


6 


3 


6 


30 


10 


2 


10. 


40 


13 


6 


13- 


50 


17 


1 


16- 



19 



6 


1- 


7 


2. 


8 


2. 


9 


2- 


10 


3. 


20 


6. 


30 


9 


40 


13- 


50 


16. 



P.P. 



72^ 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
66'' 67° 



Lg. Vers, 



77325 
77344 
77363 
77383 
77402 



77422 
77441 
77461 
77480 
77499 



77519 
77538 
77557 
77577 
77596 



77616 
77635 
77654 
77674 
77693 



77712 
77732 
77751 
77770 
77790 
77809 
77828 
77847 
77867 
77886 



77905 
77925 
77944 
77963 
77982 



78002 
78021 
78040 
78059 
78078 



78098 
78117 
78136 
78155 
78174 



78194 
78213 
78232 
78251 
78270 



78289 
78309 
78328 
78347 
78386 



78385 
78404 
78423 
78442 
78462 



9-78481 



Lg. Vers, 



2> 

19 
19 
19 
19 

19 
19 
19 
i9 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

1? 
19 
19 
19 
19 
19 
19 

19 
19 
19 

18 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 
19 
19 
19 
19 
19 

19 
19 
19 
19 
19 
19 



Log.Exs. 



10.16393 
.16441 
.16489 
.16537 
.16585 



10.16633 
.16680 
.16728 
.16776 
.16824 



10.16872 
.16920 
.16968 
.17016 
.17064 



10.17112 
.17160 
.17209 
.17257 
.17305 



10.17353 
.17401 
.17449 
.17498 
.17546 



10.17594 
.17642 
.17690 
.17739 
.17787 

10.17835 
.17884 
.17932 
.17980 
^1802| 

10.18077 
.18126 
.18174 
.18222 
.18271 



10.18319 
.18368 
.18416 
.18465 
.18514 



10.18562 
.18611 
.18659 
.18708 
.18757 



10.18805 
.18854 
.18903 
.18951 
.19000 



10.19049 
.19098 
.19146 
.19195 
.19244 

10.19293 



Log.Exs. 



D 

48 
47 
48 
48 
48 
47 
48 
48 
48 

48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

48 
48 
48 
49 
48 

48 
48 
48 
48 
49 

48 
48 
49 
48 
49 

48 
49 
48 
49 
49 

48 



Lg.Vers, 



9.78481 
.78500 
•78519 
.78538 
.78557 



9.78576 
.78595 
.78614 
.78633 
■78652 

9-78671 
.78690 
.78709 
.78728 
•78747 



9.78766 
.78785 
.78804 
.78823 
.78842 



9.78861 
.7888C 
.7889 
.78918 
=78937 



9.78956 
78975 
78994 
79013 
79032 



9.79051 
.79069 
.79088 
.79107 
.79126 



9.79145 
.79164 
.79183 
.79202 
•79220 



9.79239 
.79258 
.79277 
.79296 
l7_9315 

9.79333 
.79352 
.79371 
.79390 
.79409 



9.79427 
.79446 
.79465 
.79484 
.79503 



9.79521 
•79540 
.79559 
.79578 
.79596 
9-79615 
Lg. Vers. 



19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
19 

19 
19 
19 
19 
18 

19 
19 
19 
19 
19 
19 
18 
19 
19 
19 
19 
18 
19 
19 
18 
19 
19 
18 
19 
19 
18 
19 
19 
18 
19 

18 
19 
19 
18 
19 

18 
19 
18 
19 
18 

19 



Log, Exs, 



10-19293 
.19342 
.19391 
.19439 
.19488 



10.19537 

.19586 

.19635 

.19684 

19733 



10.19782 
.19831 
.19880 
.19929 
.19979 



10-20028 
.20077 
.20126 
.20175 
.20224 



10.20273 
.20323 
.20372 
.20421 
.20470 



10.20520 
.20569 
.20618 
.20668 
.20717 



10-20767 
.20816 
.20865 
.20915 
^964 

10^21014 
.21063 
.21113 
.21162 
•21212 



10^21262 
.21311 
.21361 
.21410 
•21460 



10-21510 
.21560 
.21609 
.21659 
.21709 



10-21759 
.21808 
.21858 
.21908 
•21958 



10-22008 
.22058 
.22108 
.22158 
-22208 



10-22258 
Log.Exs. 



49 
49 
48 
49 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 

49 
49 
49 
49 
50 

49 
49 
49 
49 
50 

49 
50 
49 
50 
49 
50 
49 
50 
50 
49 

50 
50 
50 
50 
50 
50 



10 

11 

12 
13 
li 

15' 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
2b 
27 
28 
29 

30 

31 
32 
33 
34 

35 

36 
37 
38 
39, 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P.P. 



6 

7 
8 
9 
10 
20 
30 
40 
50 



50 

5-0 

5-8 

6-6 

7-5 

8-3 

16-6 

25-0 

33-3 

41 • 6 



49 

4-9 
5-7 
6-5 
7-3 
8-1 
16-3 



3024-5 
40132-6 
50i40.8 





48 


6 


4-8 


7 


5-6 


8 


6-4 


9 


7-2 


10 


8-0 


20 


16-0 


30 


24-0 


40 


32-0 


50 


40^0 



49 

4^9 

5.8 

6.6 

7.4 

8.2 

16.5 

24.7 

33.0 

41.2 



48 

4^8 

^:l 

7^3 

ili 

24^3 
32-3 
40. 4 



4.2 
5.5 
6^3 
7^1 
7> 

23^7 
31^6 
39^S 



6 

7 

8 

9 

10 

20 

30 

40 

50 



19 


19 


1.9 


1.9 


2.3 


2.? 


2.6 


2.5 


2.9 


2.8 


3.2 


31 


6.5 


6-3 


9.7 


9.5 


13-0 


12.6 
15.8 


16.2 



V 
8 
9 
10 
20 
SO 
40 
50 



15 

1.' 

2. 
2. 
2.8 

3-1 

6^I 



12.3 
15.4 



P.P. 



725 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
es" 69° 



60 



f 


Lg.Vers. 





9.79615 


1 


.79634 


2 


.79653 


3 


.79671 


4 


.79690 


5 


9.79709 


6 


.79727 


7 


.79746 


8 


.79765 


9 


.79783 


10 


9.79802 


11 


.79821 


12 


.79839 


13 


.79858 


14 


.79877 


15 


9.79895 


16 


.79914 


17 


.79933 


18 


.79951 


19 


.79970 


20 


9.799«8 


21 


.80007 


22 


.80026 


23 


.80044 


24 


.80063 


25 


9.80081 


26 


.80100 


27 


.80119 


28 


.80137 


29 


.80156 


30 


9.80174 


31 


.80193 


32 


.80211 


33 


.80230 


34 


.80248 


35 


9.80267 


36 


.80286 


37 


.80304 


38 


.80323 


39 


.80341 


40 


9-80360 


41 


.80378 


42 


.80397 


43 


.80415 


44 


.80434 


45 


9.80452 


46 


.80470 


47 


.80489 


48 


.80507 


49 


.80526 


50 


9.80544 


51 


.80563 


52 


.80587 


53 


.80600 


54 


.80618 


55 


9.80636 


56 


.80655 


57 


.80673 


58 


.80692 


59 


.80710 



9.80728 



2> 

18 
19 
18 
18 

19 
18 
19 
18 
18 

19 
18 
18 
19 
18 

18 
18 
19 
18 
18 

18 
19 
18 
18 
18 

18 
19 
18 
18 
18 

18 
18 
18 
18 
18 

19 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

1§ 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 



Log.Exs, 



10-22258 
.22308 
.22358 
.22408 
.22458 



10.22508 
.22558 
.22608 
.22658 
.22708 



10.22759 
.22809 
.22859 
.22909 
.22960 



10.23010 
.23060 
.23110 
.23161 
.23211 



10.23262 
.23312 
.23362 
.23413 
.23463 



10.23514 
.23564 
.23615 
.23666 
.23716 



10.23767 
.23817 
.23868 
.23919 
.23969 



10.24020 
.24071 
.24122 
.24172 

^4223 



10.24274 
.24325 
.24376 
.24427 
.24478 



10.24529 
.24580 
.24631 
.24682 
.24733 



Lg. Vers. l> 



10.24784 
.24835 
.24886 
.24937 

^ ,24988 

10.^5039 
.25090 
.25142 
.25193 
.25244 



10.25295 



Log.E 



xs. 



Lg.Vers, 



9.80728 
80747 
80765 
80783 
80802 



80820 
80839 
80857 
80875 
80894 



80912 
80930 
80949 
80967 
80985 



81003 
81022 
81040 
81058 
81077 



81095 
81113 
81131 
81150 
81168 



81186 
81204 
81223 
81241 
81259 



81277 
81295 
81314 
81332 
81350 



81368 
81386 
81405 
81423 
81441 



81459 
81477 
81495 
81513 
81532 



81550 
81568 
81586 
81604 
81622 



81640 
81658 
81676 
81695 
81713 



81731 
81749 
81767 
81785 
81803 



9-81821 
Lg. Vers. 



2> 



18 
18 
18 
18 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
^8 
18 
18 
18 
18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 
18 
18 
18 
18 

18 



D 



Log.Exs. 



10.25295 
.25347 
.25398 
.25449 
.25501 



10-25552 
.25604 
.25655 
.25707 
-25758 



10-25810 
.25861 
.25913 
.25964 
-2S016 



10-26067 
.26119 
.26171 
.26222 
.26274 



10.26326 
.26378 
.26429 
.26481 

^6533 



10.26585 
.26637 
.26689 
.26741 
.26793 



10.26845 
.26897 
.26949 
-27001 
-27053 



10.27105 
.27157 
.27209 
.27261 

,27314 



10.27366 
.27418 
-27470 
.27523 
.27575 



10-22627 
.27680 
.27732 
.27785 
.27837 



10-2789C 
.27942 
.27995 
-28047 
.28100 



10.28152 
.28205 
.28258 
.28310 
.28363 



10.28416 



Log.Exs. 





1 

2 
3 

4 

5 
6 
7 
8 
_1 

10 

11 
12 
13 
14 

15 
16 
17 
18 

21 
30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 

34 



40 

41 
42 
43 
44 



50 

51 
52 
53 
54 



55 
56 
57 
58 
59, 
60 



n' 



p.p. 





53 


6 


5.31 


7 


6 


2 


8 


7 





9 


7 


9 


10 


8 


g 


20 


17 


5 


30 


26 


5 


40 


35 


3 


50 


44 


1 



52 

5.2 
6.1 
7-0 
7-9 
8-7 



53 

5-2 



61 

5-1 



51 


50 


5.1 


5.0 


5 


9 


5 


9 


6 


8 


6 


7 


7 


g 


7 


6 


8 


5 


8 


4 


17 





16 


8 


25 


5 


25 


2 


34 





33 


g 


42 


5 


42 


1 



50 

5-0 

5-8 

6-6 

7-5 

8.3 

16-6 

25.0 

33.3 

41-6 



6 

7 

8 

9 

10 

20 
80 
40 
50 



19 


It 


1.9 


1- 


2 


2 


2. 


2 


5 


2. 


2 


g 


2. 


3 


1 


3. 


6 


3 


6. 


9 


5 


9. 


12 


6 


12. 


15 


8 


15. 



18 

18 

2.1 
2.4 
27 
30 
60 
90 
12.0 
15 



P.P. 



726 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 



70*= 



11' 



Lg.Vers. 



81821 
81839 
81857 
81875 
81893 



81911 
81929 
81947 
81965 
81983 



82001 
82019 
82037 
82055 
82073 



82091 
82109 
82127 
82145 
82163 



82181 
82199 
82217 
82235 
82252 



82270 
82288 
82306 
82324 
82342 



82360 
82378 
82396 
82413 
82431 



82449 
82467 
82485 
82503 
82520 



82538 
82556 
82574 
82592 
82609 



82627 
82645 
82663 
82681 
82698 



82716 
82734 
82752 
82769 
82787 



82805 
82823 
82840 
82858 
82876 



9-82894 
Lg. Vers 



D 



Log. Exs, 



10.28416 
.28469 
.28521 
.28574 
.28627 



10.28680 
.28733 
.28786 
.28839 
.28892 



10.28945 
.28998 
.29051 
.29104 
.29157 



10.29210 
.29263 
.29316 
.29370 
.29423 



10.29476 
.29529 
.29583 
.29636 
•29689 



10.29743 
.29796 
.29850 
.29903 
.29957 



10.30010 
.30064 
.30117 
.30171 
.30225 



10.30278 
.30332 
.30386 
.30440 
.30493 



10.30547 
.30601 
.30655 
.30709 
.30763 



10.30817 
.30871 
.30925 
.30979 
.31033 



10.31087 
.31141 
.31195 
.31249 
.31303 



10-31358 

.31412 
.31466 
.31521 
.31575 



10-31629 
Log. Exs, 



2> 



53 
52 
53 
53 
52 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
53 
53 
53 
53 
53 
53 
53 

53 
53 
53 
53 
53 

53 
53 
53 
54 
53 

53 
54 
53 
54 
53 
54 
53 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 
54 
54 
54 
54 

54 



Lg. Vers. 



9.82894 
.82911 
.829'>9 
.82947 
.82964 



9.82982 
.83000 
.83017 
.83035 
i83^_3 

9.83070 
.83088 
.83106 
.83123 
.83141 



9.83159 
.83176 
.83194 
.83211 
.83229 



9.83247 
.83264 
.83282 
.83299 
.83317 



9.83335 
.83352 
.83370 
.83387 
.83405 



9.83422 
.83440 
.83458 
.83475 
.83493 



9.83510 
.83528 
.83545 
.83563 
.83580 



9.83598 
.83615 
.83633 
.83650 
.83668 



.83685 
.83703 
.83720 
.83737 
.83755 



9.83772 
.83790 
.83807 
.83825 
.83842 



9.83859 
•83877 
.83894 
.83912 
.83929 



9-83946 
Lg. Vers, 



Log. Exs. 



10-31629 
.31684 
.31738 
.31793 
-31847 



10-31902 
.31956 
.32011 
.32066 
.32120 



10.32175 
.32230 
.32284 
.32339 
.32394 



10-32449 
.32504 
.32558 
.32613 
-32668 



10-32723 
.32778 
.32833 
.32888 
.32944 



10-32999 
.33054 
.33109 
.33164 
.33220 



10.33275 
.33330 
.33385 
.33441 
.33496 



10.33552 
.33607 
.33663 
.33718 
.33774 



10.33829 
.33885 
.33941 
.33996 
.34052 



10.34108 
.34164 
.34220 
.34275 
.34331 



10.34387 
.34443 
-34499 
-34555 
.34611 

10.34667 
.34723 
.34780 
.34836 
.34892 



10-34948 



Log. Exst 



10 

11 

12 

13 

J4 

15 
16 
17 
18 
19 

20 

21 
22 
23 
_24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
_34 

35 
36 
37 
38 
li 
40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

55 
56 
57 
58 
^ 
60 



P.P. 





56 


6 


5.6 


7 


6.6 


8 


7.5 


9 


8.5 


10 


9.4 


20 


18.8 


30 


28.2 


40 


37.6 


50 


47.1 





55 


6 


5-5 


7 


6.5 


8 


7.4 


9 


8-3 


10 


9-2 


20 


18-5 


30 


27.7 


40 


37.0 


50 


46.2 



51 



5 


4 


6 


-3 


7 


2 


8 


2 


9 


1 


18 


1 


27 


2 


36 


3 


45 


4 



56 

5.6 

6.5 

7.4 

8.4 

9.3 

18.6 

28.0 

37.3 

46.6 

55 

5.5 

6.4 

7.3 

8.2 

91 

18.3 

27.5 

36-6 

45.8 

54 

5.4 
6.3 
7.2 
8.1 
9.0 





53 


6 


5-3 


7 


6-2 


8 


7.1 


9 


8.0 


10 


8.9 


20 


17.8 


30 


26-7 


40 


35-6 


50 


44.6 



53 

5.3 

6-2 

7.0 

7.9 

8.8 

17.6 

26.5 

35-3 

44.1 



53 

5.2 

6-1 

7-0 

7-9 

8-7 

17-5 

26-2 

35-0 

43.7 





18 


17 


6 


1-8 


1-7 


7 


2.1 


2.0 


8 


2-4 


2-3 


9 


2-7 


2-6 


10 


3-0 


2-9 


20 


6-0 


5-8 


30 


9-0 


8-7 


40 


12-0 


11-6 


50 


15.0 


14-6 



17 

1-7 
2.0 

2.2 
2.5 
2.8 
5.6 
8.5 
11.3 
14-1 



P.P. 



727 



TABLE VIII.-LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
T«>° 73° 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



Lg.Versi 



9-83946 
.83964 
.83981 
.83999 
.84016 



9-84033 
.84051 
. 84068 
.84085 
-84103 



9-84120 
-84137 
-84155 
.84172 

_^84189 

9-84207 

.84224 
.84241 
.84259 
-84276 



J> 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

81 
32 
33 
34 

35 
36 
37 
38 
39 



Log.Exs. 



9-84293 
.84310 
.84328 
.84345 
-84362 



9-84380 
-84397 
-84414 
.84431 
- 84449 



9-84466 

-84483 

-84500 

84517 

.84535 

9-84552 
-84569 
-84586 
-84603 
-84620 



40 

41 
42 
43 
44 

45 

46 

47 

48 

49 

60 

51 

52 

53 

5£ 

55 

56 

57 

58 

59 

60 



9-84638 
-84655 
-84672 
-84689 
-84706 




9-84809 
-84826 
-84844 
-84861 
-84878 



9-84895 
.84912 
-84929 
-84946 
-84963 



17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

17 



10-34948 
.35005 
.35061 
.35117 
.35174 



10-35230 
.35286 
.35343 
.35399 
•35456 



10.35513 
.35569 
.35626 
.35683 
-35739 



2> 



10-35796 
.35853 
.35910 
.35967 
.36023 



Lg.Vers, 



10-36080 
.36137 
.36194 
.36251 
-36308 



10-36366 
.36423 
.36480 
.36537 
.36594 



10-36652 
.36709 
.36766 
.36824 
-36881 



10-36938 
.36996 
.37054 
.37111 
-37169 



10-37226 
-37284 
.37342 
.37399 
.37457 



10-37515 
.37573 
.37631 
.37689 
.37747 



10-37805 
.37863 
.37921 
.37979 
-38037 



10-38095 
-38153 
.38212 
.38270 
.38328 



10-38387 
Log.Exs. 



56 
56 
56 
56 

56 
58 
56 
56 
57 
56 
56 
56 
57 
56 

57 
56 
57 
57 
56 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
57 
57 
57 

57 
57 
58 
57 
57 

57 
57 
58 
57 
58 

58 
57 
58 
58 
58 

58 
58 
58 
58 
58 

58 
58 
58 
58 
58 

58 



9-84980 
.84997 
.85014 
.85031 
.85049 



-85066 
-85083 
.85100 
-85117 
-85134 



9-85151 
-85168 
-85185 
-85202 
.85219 



D 



Log. Exs. 



9. 85236 
-85253 
-8.5270 
-85287 
.85304 



9-85321 
-85338 
.85355 
-85372 
-85389 



9-85405 
-85422 
-85439 
-85456 
.85473 



9-85490 
-85507 
.85524 
.85541 
-85558 



-85575 
.85592 
.85608 
.85625 
-85642 



-85659 
.85676 
.85693 
-85710 
.85726 



9-85743 
.85760 
.85777 
-85794 
-85811 



9.8582 
-85844 
-85861 
.85878 
-85895 



9-85911 
-85928 
-85945 
-85962 
.85979 



Q.Rpi995 
Lg' Vers. 



17 
17 
17 
17 

17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

17 
17 
17 
17 
17 

16 
17 
17 
17 
17 

17 
17 
16 
17 
17 

17 
17 
16 
17 
17 

17 
16 
17 
17 
16 

17 
17 
16 
17 
17 

16 
17 
17 
16 
17 

16 
17 

iZ 

16 
17 
16 



10-38387 
•38445 
.38504 
.38562 
.38621 



10-38679 
.38738 
.38796 
.38855 
.38914 



10-38973 
.39031 
.39090 
.39149 
.39208 



10-39267 
.39326 
.39385 
. 39444 
.39503 



10.39562 
-39621 
.39681 
.39740 
-39799 



n 

58 
58 
58 
58 

58 
58 
58 
59 
58 

59 
58 
59 
59 
58 

59 
59 
59 
59 
59 

59 
59 
59 
59 
59 



10 

11 
12 
13 
14 



10-39859 
-39918 
.39977 
-40037 
.40096 



10.40156 
-40216 
-40275 
-40335 
-40395 



10-40454 
.40514 
.40574 
.40634 
-40694 

10-40754 
.40814 
.40874 
.40934 
.40994 



30 

21 
22 
23 
24 



10-41054 
.41114 
.41174 
.41235 
-41295 



10-41355 
-41416 
.41476 
.41537 
-41597 



10-41658 
-41719 
.41779 
-41840 
-41901 



10-41962 
Log. Exs. 



59 
59 
59 
59 

59 
60 
59 
59 
60 

59 
60 
59 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
60 
60 
60 
60 

60 
61 
60 
60 
61 
61 



25 
26 
27 
28 
_29 

30 

31 
32 
33 
M 
35 
36 
37 
38 
Ji 
40 
41 
42 
43 
44 



45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P. P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



61 

6-1 

7-1 

8-1 

9-1 

10-1 

20-3 

30-5 

40-6 

50.8 



60 


6 





7 





8 





9 





10 





20 





30 


-0 


40 


.0 


50 


• 





59 


6 


5-9 


7 


6-9 


8 


7-8 


9 


8.8 


10 


9.8 


20 


19.6 


30 


29.5 


40 


39-3 


50 


49-1 



58 

5-8 

6-7 

7-7 

8.7 

9.6 

19.3 

29-0 

38-6 

48-3 

57 



60 

6-0 

7 

8-0 

9-1 

10-1 

20-1 

30-2 

40-3 

50-4 

59 

5-9 

6-9 

7-^ 

8-9 

9-9 

19-8 

29-Z 

39-6 

49.6 

58 

5-8 

6-8 

78 

8-8 

9-7 

19-5 

29-2 

39-0 

48-7 

57 
5-7 



6 


5-7 


7 


6-6 


9 


7-6 


9 


8-5 


10 


9-5 


20 


19-0 


3028-5 


40!38.0 


50 


47.51 



56, 

5-6 
6.6 
7-5 
8-5 
9-4 

18-8 

28 

37 



2 
6 
47.1 





17 


6 


1-7 


7 


20 


8 


2-3 


C 


2-6 


10 


2-9 


20 


5-8 


30 


8-7 


40 


11-6 


50 


14-6 



17 

1-7 



2 

2 

2 

2 

5 

8 
11 _ 
14. 1 



16 

1-6 
1-9 
2-2 
2-5 
2-7 
5-5 
8-2 
11-0 
13.7 



P.P. 



728 



TABLE VIII.--LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 



74^ 



75' 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9_ 

10 

11 
12 
13 
ii 
15 
16 
17 
18 
19 



20 

21 
22 
23 
24 

25 
26 
27 
28 
21 
30 
31 
32 
33 
34 

35 
36. 
37 
38 
39 



Lg. Vers, 



9.85995 
.86012 
.86029 
.86046 
.86062 



9.86079 
.86096 
.86113 
.86129 
.86146 



9-86163 
.86179 
.86196 
.86213 
.86230 



9.86246 
.86263 
.86280 
.86296 
.86313 



9.86330 
•86346 
.86363 
.86380 
.86396 



9.86413 
.86430 
.86446 
.86463 
.86479 



9.86496 
.86513 
.86529 
.86546 
.86562 



9.86579 
.86596 
.86612 
.86629 
-86645 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

60 

51 
52 
53 
54 
55 
56 
57 
58 
59 

60 

"7 



9.86662 
.86678 
.86695 
.86712 
.86728 



9.86745 
.86761 
.86778 
• 86794 
.86811 



9.86827 

•86844 

-86860 

.86877 

86893 



9-86910 
.86926 
.86943 
-86959 
-86976 



9.86992 



Lg. Vers, 



2) 

17 
16 
17 
16 
17 
16 
17 
16 
17 

16 
16 
17 
16 
17 

16 
16 
17 
16 
16 

17 
16 
16 
17 
16 

16 
17 
16 
16 
16 

17 
16 
16 
16 
16 

17 
16 
16 
16 
16 

16 
16 
17 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 



Log. Exs. 



10-41962 
.42022 
.42083 
.42144 
.42205 



10.42266 
.42327 
.42388 
.42450 
.42511 



10-42572 
.42633 
.42695 
.42756 
.42817 



10.42879 
.42940 
.48002 
.43063 
.43125 



10.43187 
.43249 
.43310 
.43372 
.43434 



10.43496 
•43558 
.43620 
.43682 
.43744 



10.43806 
.43868 
.43931 
.43993 
.44055 



10.44118 
.44180 
.44242 
.44305 
.44368 



10.44430 
.44493 
.44556 
.44618 
.44681 



J> 



10.44744 
.44807 
.44870 
.44933 
.44996 



10-45059 
.45122 
.45185 
.45248 
.45312 



10.45375 
.45439 
.45502 
.45565 
.45629 



10-45693 



Log. Exs. 



60 
61 
61 
61 
61 
61 
61 
61 
61 

61 
61 
61 
61 
61 

61 
61 
61 
61 
62 

61 
62 
61 
62 
6l 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
62 
63 
62 
62 
63 
62 
63 
62 
63 
63 
63 
63 

63 
63 
63 
63 
63 

63 
63 
63 
63 
63 
64 



Lg. Vers. 



9.86992 
.87009 
.87025 
.87042 
.87058 



9.87074 
.87091 
.87107 
.87124 
.87140 



9.87157 
.87173 
.87189 
.87206 
.87222 



9.87239 
.87255 
.87271 
.87288 
•87304 



9.87320 
.87337 
.87353 
.87370 
.87386 



9.87402 
.87419 
.87435 
.87451 
.87468 



9.87484 
.87500 
.87516 
.87533 

.87549 



9.87565 
.87582 
.87598 
.87614 
.87631 



9.87647 
.87653 
.87679 
.87696 
.87712 



9-87728 

.87744 
.87761 
.87777 
.87793 



D 



9.87809 
.87825 
.87842 
.87858 
-87874 



9-87890 
.87906 
.87923 
.87939 
-87955 



9.87971 



Lg. Vers 



Log. Exs. 



10.45693 
.45756 
.45820 
•45884 
.45947 



10.46011 
.46075 
•46139 
•46203 
.46267 



10.46331 
•46395 
•46460 
.46524 
•46588 



10.46652 
.46717 
.46781 
•46846 
.46910 



10.46975 
•47040 
•47104 
•47169 
.47234 



10.47299 
•47364 
•47429 
•47494 
.47559 



10-47624 
.47689 
•47754 
•47820 
.47885 



10.47950 
•48016 
•48081 
•48147 
-48213 



10-48278 
•48344 
•48410 
•48476 
.48542 



10.48607 
•48674 
•48740 
.48806 
.48872 



10.48938 
.49004 
•49071 
.49137 
-49204 



10-49270 
.49337 
•49403 
•49470 
-49537 



10.49604 
Log. Exs. 



I) 

63 
63 
64 
63 

64 
64 
64 
64 
64 
64 
64 
64 
64 
64 

64 
64 
64 
64 
64 

64 
65 
64 
65 
64 

65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
65 
66 

65 
65 
66 
66 
66 

65 
66 
66 
66 
66 

66 
66 
66 
66 
66 
66 
66 
66 
67 
66 

67 
1) 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
-29 
30 
31 
32 
33 
34 



35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 

60 



P.P. 





67 


66 


6 


6.7 


6.6 


7 


7.8 


7.7 


8 


8.9 


8.8 


9 


10.0 


10.0 


10 


11.1 


11.: 


20 


22.3 


22.;. 


30 


33.5 


33.2 


40 


44.6 


44.3 


50 


55.8 


55.4 





65 


65 


6 


6.5 


6-5 


7 


7.6 


7-6 


8 


8.7 


8-6 


9 


9-8 


9-7 


10 


10-9 


10.8 


20 


21-8 


21.6 


30 


32.7 


32.5 


40 


43.6 


43.3 


50 


54.6 


54.1 





64 


63 


6 


6-4 


6.3 


7 


7.4 


7-4 


8 


8.5 


8-4 


9 


9-6 


9.5 


10 


10.6 


10.6 


20 


21-3 


21.1 


30 


32-0 


31.7 


40 


42-6 


42.3 


50 


53.3 


52.9 





62 


62 


6 


6-2 


6-21 


7 


7-3 


7 


2 


8 


8-3 


8 


2 


9 


9-4 


9 


3 


10 


10-4 


10 


3 


20 


20-8 


20 


6 


30 


31.2 


31 





40 


41.6 


41 


3 


50 


52.1 


51 


6 



66 

6-6 
7-7 
8^8 
9-9 

110 
22.0 
33-0 
44.0 
55-0 

61 

6.4 

7.5 

86 

9.7 

10.7 

21.5 

32.2 

43. 

53.7 

63 

6-3 

7.3 

8.4 

9-4 

10.5 

21.0 

31.5 

42 

52.5 

61. 

6.1 

7-2 

8.2 

9.2 

10.2 

20.5 

30.7 

41.0 

51.2 



6 

7 

8 

9 

10 

20 

30 

40 

50 



61 

6-1 

7.1 

8-1 

9-1 

10. 1 

20.3 

30.5 

40.6 

50.8 



17 


16 


1.7 


1-6 


2.0 


1-9 


2-2 


2-2 


2-5 


2-5 


2-8 


2.7 


5.6 


5.5 


8.5 


8.2 


11-3 


11-0 


14-1 


13-7 



16 

1.6 

1. 

2. 

2.4 

2. 

5. 

8. 
10. 
13-3 



P. P. 



729 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
76° 77° 



O 

1 
2 
3 

_4 

5 
6 
7 
8 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19 



20 

21 
22 
23 
24 
25 
26 
27 
28 
29. 
80 
31 
32 
33 
34 

35 
36 
37 
38 
39 



Lg. Vers. 



40 

41 

42 

43 

44 

45 

46 

47 

48 

4i 

50 

51 

52 

53 

54 

55 
56 
57 
58 
51 
60 



87971 
87987 
88003 
88020 
88036 



88052 
88068 
88084 
88100 
88116 



88133 
88149 
88165 
88181 
88197 



88213 
88229 
88245 
88261 
88277 



88294 
88310 
88326 
88342 
88358 



88374 
88390 
88406 
88422 
88438 



88454 
88470 
88486 
88502 
88518 



88534 
88550 
88566 
88582 
88598 



88614 
88630 
88646 
88662 
88678 



88694 
88710 
88726 
88742 
88758 



88774 
88790 
88805 
88821 
88837 



88853 
88869 
88885 
8890] 
88917 



88933 



Lg. Vers. -D 



16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
16 
16 

16 
16 
16 
15 
16 

16 
16 
16 
16 
16 

16 
16 
15 
16 
16 
16 
16 
15 
16 
16 
16 



Log.Exs. 



10.49604 
.49670 
.49737 
.49804 
.49871 



10.49939 
.50006 
.5007^ 
.50140 
.50208 



10.50275 
.50342 
.50410 
.50477 
.50545 



10.50613 
.50681 
.50748 
.50816 
.50884 



10.50952 
.51020 
.51088 
.51157 
.51225 



10.51293 
.51361 
.51430 
.51498 
•51567 



10.51636 
.51704 
.51773 
.51842 
.51911 



10.51980 
.52049 
.52118 
.52187 
.52256 



10.52325 
.52394 
.52464 
•52533 
.52603 



10.52672 
.52742 
.52812 
.52881 
.52951 



10.53021 
.53091 
.53161 
.53231 
.53301 



10.53372 
. 53442 
.53512 
.53583 
.53653 



10 53724 



Log.Exs. 



66 
67 
67 
67 

67 
67 
67 
67 
67 
67 
67 
67 
67 
68 

67 
68 
67 
68 
68 

68 
68 
68 
68 
68 

68 
68 
68 
68 
68 
69 
68 
68 
69 
69 

69 
69 
69 
69 
69 
69 
69 
69 
69 
69 

69 
70 
69 
69 
70 

70 
70 
70 
70 
70 

70 
70 
70 
70 
70 

70 



Lg. Vers, 



8893^ 
88949 
88964 
88980 
88996 



89012 
89028 
89044 
89060 
89075 
89091 
89107 
89123 
89139 
89155 



89170 
89186 
89202 
89218 
89234 



89249 
89265 
89281 
89297 
89312 



89328 
89344 
89360 
89376 
89391 



89407 
89423 
89438 
89454 
89470 



9.89486 
.89501 
.89517 
.89533 
•89548 



9.89564 
.89580 
.89596 
.89611 
.89627 



9 . 89643 
.89658 
.89674 
.89690 
.89705 



9.89721 
.89737 
.89752 
.89768 
.89783 



9.89799 
.89815 
.89830 
.89846 
•89862 



n 



9-89877 



Lg. Vers, 



16 
15 
16 
16 

16 
15 
16 
16 
15 

16 
16 
15 
16 
16 
15 
16 
15 
16 
16 
15 
16 
15 
16 
15 

16 
15 
16 
16 
15 

15 
16 
15 
16 
15 

16 
15 
16 
15 
15 

16 
15 
16 
15 
15 

16 
15 
15 
16 
15 

15 
16 
15 
15 
15 

16 
15 
15 
15 
16 
15 

1^ 



Log.Exs. 



10.53724 
.53794 
.53865 
.53936 
.54007 



10.54078 
.54149 
.54220 
.54291 
.54362 



10.54433 
.54505 
.54576 
. 54647 
.54719 



10.54791 
.54862 
.54934 
.55006 
.55078 



10.55150 
.55222 
.55294 
.55366 
.55438 



10.55511 
.55583 
.55655 
.55728 
.55801 



10.55873 
.55946 
.56019 
.56092 
.56165 



10.56238 
.56311 
.56384 
.56457 
.56531 



10.56604 
.56678 
.56751 
.56825 
.56899 



10.56973 
.57047 
.57120 
•57195 
.57269 



10 . 57343 
.57417 
.57491 
.57566 
.57640 



10.57715 
.57790 
.57864 
.57939 
.58014 



10.58089 



Log.Exs. 



D 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
_19 

30 

21 
22 
23 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
34 

35 
36 
37 
38 
_39 

40 

41 

42 

43 

j44 

45 

46 

47 

48 

19 

50 

51 

52 

53 

55 
56 
57 
58 

60 



P.P. 





75 


74 


6 


7.5 


7-4 


7 


8^7 


8.6 


8 


10^0 


9.8 


9 


11^2 


11.1 


10 


12^5 


12.3 


20 


25.0 


24.6 


30 


37.5 


37.0 


40 


50.0 


49.3 


50 


62.5 


61.6 





73 


71 


6 


7.2 


7.1 


7 


8 


4 


8.3 


8 


9 


6 


9^4 


9 


10 


8 


10.6 


10 


12 





11^8 


20 


24 





23^6 


30 


36 





35^5 


40 


48 





47^3 


50 


60 





59.1 



69 

6 



68 



6 

71 8 
8! 9 
9 10 
10 11 
20 23 
30 34 
40 46 
50 57 



• 9 


6.8 


• 


7.9 


• 2 


9^0 


• 3 


10.2 


• 5 


11.3 


• 


22.6 


• 5 


34.0 


.0 


45.3 


.5 


56^6 



73 

73 
8.5 
9^7 
10^9 
12^I 
24-3 
36.5 
48.6 
60.8 



70 

7-0 
8.2 
9-4 
10^6 
11^7 
23^3 
35^2 
47.0 
58.7 



67 

6.7 
7.8 
8.9 
10.0 
11.1 
22.3 
33-5 
44.6 
55.5 





66 





6 


6.6 


0.0 


7 


7 


7 








8 


8 


8 








9 


9 


9 





1 


10 


11 








X 


20 


22 








1 


30 


33 








2 


40 


44 








3 


50 


55 








4 





16 


16 


1 


6 


1.6 


1.6 


1. 


7 


1.9 


1-8 


1 


8 


2-2 


2.1 


2. 


9 


2.5 


2-4 


2. 


10 


2.7 


2 6 


2. 


20 


5.5 


5.3 


5 


30 


8.2 


8 


7. 


40 


11.0 


10.6 


10. 


50 


13.7 


13-3 


12. 



P. p. 



730 



TABLE VIII —LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 



78' 



79° 



Lg. Vers 



89877 
89893 
89908 
89924 
89939 



89955 
89971 
89986 
90002 
90017 



90033 
90048 
90064 
90080 
90095 



90111 
90126 
90142 
90157 
90173 



90188 
90204 
90219 
90235 
90250 



90266 
90281 
90297 
90312 
90328 



90343 
90359 
90374 
90389 
90405 



90420 
90436 
90451 
90467 
90482 



90497 
90513 
90528 
90544 
90559 



90574 
90590 
90605 
90621 
90636 



90651 
90667 
90682 
90697 
90713 



90728 
90744 
90759 
90774 
90790 



9-90805 
Lg. Vers. 



Log.Exs. 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



10 



58089 
58164 
58239 
58315 
58390 



58465 
58541 
58616 
58692 
58768 



58844 
58920 
58995 
59072 
59148 



59224 
59300 
59377 
59453 
59530 



59606 
59883 
59760 
59837 
59914 



59991 
60088 
60145 
60223 
60300 



60378 
60455 
60533 
60611 
60688 



60766 
60844 
60923 
61001 
61079 



61158 
61236 
61315 
61393 
61472 



61551 
61630 
61709 
61788 
61867 



61947 
62026 
62105 
62185 
62265 



62345 
62424 
62504 
62585 
62665 



10-62745 
Log.Exs 



2> 

75 
75 
75 
75 

75 
75 
75 
76 
75 
76 
76 
75 
75 
76 

76 
76 
76 
76 
76 

76 
77 
76 
77 
77 
77 
77 
77 
77 
77 
77 
77 
77 
78 
77 

78 
78 
78 
78 
78 
78 
78 
78 
78 
79 

78 
79 
79 
79 
79 

79 
79 
79 
80 

79 

80 
79 
80 
80 
80 

80 



Lg. Vers 



90805 
90820 
90835 
90851 
90866 



90881 
90897 
90912 
90927 
90943 

90958 
90973 
90988 
91004 
91019 



91034 
91049 
91065 
91080 
91095 



91110 
91126 
91141 
91156 
91171 



91187 
91202 
91217 
91232 
91247 



91263 
91278 
91293 
91308 
91323 



91338 
91354 
91369 
91384 
91399 



91414 
91429 
91445 
91460 
91475 



91490 
91505 
91520 
91535 
91550 



91565 
91581 
91596 
91611 
91626 



91641 
91656 
91671 
91686 
91701 



91716 



Lg. Vers, 



15 
15 
15 
15 

15 
15 
15 
15 
15 

15 

15 
15 
15 
15 
15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 

15 
15 
15 
15 
15 
15 



Log. Exs. 



10-62745 
.62825 
.62906 
.62986 
.63067 



10-63148 
.63229 
.63310 
.63391 

.63472 



10-63553 
.63634 
.63716 
.63797 
-63879 



10-63961 
. 64043 
.64125 
.64207 
-64289 



10-64371 
.64453 
.64536 
.64618 
-64701 



10-64784 
.64867 
.64950 
.65033 
.65116 



10.65199 
.65283 
.65366 
.65450 
.65534 



10.65617 
.65701 
.65785 
.65870 
.65954 



10-66038 
.66123 
.66207 
.66292 
-66377 



10-66462 
-66547 
-66632 
-66717 
-66803 



10-66888 
.66974 
.67059 
.67145 
-67231 



10-67317 
.67403 
.67490 
.67576 
-67663 



10-67749 



Log.Exs. 



80 

80 

80 

81 

80 

8 

8 

8 

8 

8 
8 
8 
8 
8 

82 
82 
82 
82 
82 
82 
82 
82 
82 
83 
82 
83 
83 
83 
83 

33 
83 
83 
83 
84 

83 
84 
84 
84 
84 

84 
84 
84 
84 
85 

85 
85 
85 
85 
85 

85 
85 
85 
86 
86 

86 
86 
86 
86 
86 

86 



10 

11 
12 
13 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29^ 

30 

31 
32 
33 
3i 
35 
36 
37 
38 

39- 
40 

41 
42 
43 

M. 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



P. P. 





86 


85 


84 


6 


8.6 


8.5 


8. 


7 


10.0 


9 


g 


9. 


8 


11-4 


11 


3 


11. 


9 


12-9 


12 


7 


12. 


10 


14-3 


14 


1 


14. 


20 


28-6 


28 


3 


28- 


30 


43-0 


42 


5 


42- 


40 


57-3 


56 


6 


56. 


50 


71.6 


70 


8 


70. 



83 



10 13 
20i27 
3041 
40 55 
50169 



• 3! 

•7i 9 

• 10 
.4 12 
-8 13 
-6 27 
-541 
-3,54 

• lies 



83 
8-2 



81 
8-1 



80 79 



8 


0| 7 


9 


7. 


9 


3! 9 


2 


9. 


10 


6 10 


5 


10- 


12 


Oil 


8 


11- 


13 


3 13 


1 


13- 


26 


6 26 


3 


26- 


40 


39 


5 


39- 


53 


3 52 


6 


52- 


66 


6 65 


8 


65. 





77 


76 


7^ 


6 


7-7 


7-6 


7- 


7 


9 





8 


8 


8- 


8 


10 


2 


10 


1 


10- 


9 


11 




11 


4 


11- 


10 


12 


3 


12 


6 


12- 


20 


25 


6 


25 


3 


25- 


30 


38 


5 


38 





37- 


40 


51 


3 


50 


6 


50- 


50 


64 


1 


63 


3 


62. 



78 
8 
1 
4 
7 








6 

7 

8 

9 

10 

20 

30 

40 



O 

0.0 
0.0 
0-0 
0.1 
0.1 
0.1 
0-2 
0-3 



500-4 



16 

1 

1 

2 

2 

2. 

5 

8 
10 
13 



15_ 

5 
8 

3 
6 
I 
7 
3 

9 

P. P. 



"-§ 


X • 
1- 


.1 


2- 


-4 


2- 


-6 


2- 


.3 


5- 


-0 


7- 


-6 


10 


-3 


12 



15 

1-5 
1-7 



731 



TABLE VIII.— J.OGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



80° 



81^ 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19_ 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49_ 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



Lg. Vers. 



91716 
91731 
91746 
91761 
91776 



91791 
91807 
91822 
91837 
91852 



91867 
91882 
91897 
91912 
91927 



91942 
91957 
91972 
91987 
92002 



92016 
92031 
92046 
^2061 
92076 



92091 
92106 
92121 
92136 
92151 



92166 
92181 
92196 
92211 
92226 



92240 
92255 
92270 
92285 
92300 



92315 
92330 
92345 
92360 
92374 



92389 
92404 
92419 
92434 
92449 

92463 
92478 
92493 
92508 
92523 



92538 
92552 
92567 
92582 
92597 



9-92612 



Lg. Vers, 



J> 



Log. Exs. 



10.67749 
.67836 
.67923 
.68010 
.68097 



10.68184 
.68272 
.68359 
. 68447 
.68534 



10-68622 
.68710 
.68798 
.68886 
•68975 



10.69063 
.69152 
.69240 
.69329 
.69418 



10.69507 
•69596 
•69686 
.69775 
.69865 



10.69955 
• 70044 
•70134 
.70224 
•70315 



10.70405 
.70495 
•70586 
•70677 
.70768 



10.70859 
•70950 
•71041 
.71133 
.71224 



10.71316 
.71408 
.71500 
.71592 
.71684 



10.71776 
.71869 
.71961 
.72054 
.72147 



10.72240 
•72333 
.72427 
.72520 
.72614 



10-72707 
.72801 
•72895 
.72990 
•73084 



10-73178 



Log. Exs. 



n 

86 
87 
87 
87 
87 
87 
87 
87 
87 
88 
88 
88 
88 
88 

88 
88 
88 
89 
89 
89 
89 
89 
89 
89 
90 
89 
90 
90 
90 
90 
90 
91 
90 
91 
91 
91 
91 
91 
91 

91 

92 
92 
92 
92 

92 
92 
92 
93 
92 

93 
93 
93 
93 
93 

93 
94 
94 
94 
94 
94 

17 



Lg. Vers. 



9.92612 
.92626 
.92641 
•92656 
.92671 



9-92686 
.92700 
.S2715 
.92730 
•92745 



9.92759 
.92774 
.92789 
.92804 
.92818 



9.92833 
•92848 
.92862 
•92877 
.92892 



9.92907 
.92921 
•92936 
•92951 
.92965 

9.92980 
.92995 
•93009 
•93024 
.93039 

9-9~3053 
.93068 
.93083 
.93097 
.93112 



9-93127 
.93141 
.93156 
.93171 
.93185 



9.93200 
•93214 
•93229 
•93244 
.93258 



9.93273 
•93287 
•93302 
•93317 
.93331 



9.93346 
.93360 
•93375 
•93389 
•93404 



9.93419 
•93433 
•93448 
•93462 
.93477 



9-93491 
Lg. Vers. 



14 
15 
14 
15 

15 
14 
15 
14 
15 
14 
15 
14 
15 
14 

15 
14 
14 
15 
14 

15 
14 
14 
15 
14 

15 
14 
14 
15 
14 

14 
15 
14 
14 
15 

14 
14 
14 
15 
14 

14 
14 
15 
^4 
14 

14 
14 
15 
14 
14 

14 
14 
14 
14 
15 

14 
14 
14 
14 
14 

14 



Log. Exs. 



10.73178 
•73273 
•73368 
•73463 
•73558 



10-73653 
•73748 
•73844 
•73940 
-74035 



10.74131 
.74227 
.74324 
. 74420 
•74517 



10.74613 
.74710 
.74807 
.74905 
•75002 



10^75099 
•75197 
.75295 
.75393 
•75491 



10-75589 
.75688 
.75786 
.75885 
.75984 



10.76083 
•76182 
•76282 
•76382 
.76481 



10-76581 
•76681 
•76782 
•76882 
.76983 



10.77083 
.77184 
.77286 
.77387 
.77488 



10.77590 
.77692 
.77794 
.77896 
•77998 



10 •78101 
.78203 
.78306 
.78409 
•78513 



10.78616 
.78720 
.78823 
.78927 
.79031 



10-79138 



Log. txs. 



D 

95 
94 
95 
95 

95 
95 
95 
96 
95 
96 
96 
96 
96 
96 

96 
97 
97 
97 
97 

97 
98 
97 
98 
98 

98 

98 

98 

99 

99 

99 

99 

99 

100 

99 

100 

100 

100 

100 

100 

100 
101 
101 
101 
101 

lOl 
102 
102 
102 
102 

102 
102 
103 
103 
103 

103 
104 
103 
104 
104 
104 



10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 



30 

31 
32 
33 

Ik 
35 
36 
37 
38 

J9 

40 

41 

42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
14 

55 
56 
57 
58 
3^ 
60 



P. P. 





90 


6 


9.0 


7 


10.5 


8 


12.0 


9 


13.5 


10 


15.0 


20 


30.0 


30 


45-0 


40 


60-0 


50 


75.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



9 

0.9 
1.0 
1.2 
1.3 
1.5 
3.0 
4.5 
6-0 
7-5 

7 
0.7 
0.8 
0.0 
1.0 
1.1 
2-3 
3.5 
4.6 
5.8 





5 


6 


0-5 


7 


0-6 


8 


0-6 


9 


0-7 


10 


0-8 


20 


1-6 


30 


2-5 


40 


3-3 


50 


4.1 





15 


6 


1.5 


7 


1.8 


8 


2.0 


9 


2.3 


10 


2.6 


20 


5^1 


30 


7.7 


40 


10.3 


50 


12.9 



80 
8.0 
9.3 
10^6 
12^0 
13^3 
26-6 
40.0 
53.3 
66.6 

8 
0.8 
0^9 
1.0 
1-2 
1.3 
2.6 
4.0 
5^3 
6.6 

6 

0.6 
0^7 
0^8 
0.9 
1.0 
2.0 
3^0 
4^0 
5.0 

4 

0.4 
0^4 
0^5 
0^6 
0.6 
1.3 
2^0 
2.6 
3^3 

15 

1^5 
1.7 
2^0 
2^2 
2^5 
5^0 
7^5 
10^0 
12.5 



12_ 



6 


l.i 


7 


1.7 


8 


1^9 


9 


2^2 


10 


2^4: 


20 


4.8 


30 


7.2 


40 


9^6 


50 


12.1 



P.P. 



732 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

SS"* 83° 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
14 
15 
16 
17 
18 

11 
20 

21 
22 
23 
24 

25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 

35 
36 
37 
38 

39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
M 
55 
56 
57 
58 
59 
60 



Lg. Vers 



93491 
93506 
93520 
93535 
93549 



93564 
93578 
93593 
93607 
93622 



93636 
93651 
93665 
93680 
93694 



93709 
93723 
93738 
93752 
93767 



93781 
93796 
93810 
93824 
93839 



93853 
93868 
93882 
93897 
93911 



93925 
93940 
93954 
93969 
93983 



93997 
94012 
94026 
94041 
94055 



94069 
94084 
94098 
94112 
94127 

94141 
94155 
94170 
94184 
94198 



94213 
94227 
94241 
94256 
94270 



94284 
94299 
94313 
94327 
94341 



9-94356 
Lg. Vers, 



Log. Exs. 



10.79136 
.79240 
.79345 
•79450 
•79555 



10.79660 
.79766 
.79871 
.79977 
•80083 



10.80189 
.80296 
.80402 
.80509 
.80616 



10.80723 
.80831 
.80938 
.81046 
.81154 



10-81262 
.81371 
.81479 
.81588 
.81697 



10.81806 
.81916 
.82025 
.82135 
.82245 



10.82356 
.82466 
.82577 
.82688 
.82799 



10.82910 
.83022 
.83133 
.83245 
.83358 



10.83470 
.83583 
.83695 
.83809 
.83922 



10.84035 

.84149 
.84263 
.84377 
.84492 



10.84607 
.84721 
.84837 
.84952 
.85068 



10.85183 

.85299 
.85416 
.85532 
•85649 



10.85766 



Log, ExS: 



104 
105 
104 
105 
105 
105 
105 
106 
106 

106 
106 
106 
107 
107 
107 
107 
107 
108 
108 

108 
108 
108 
109 
109 

109 
109 
109 
110 
110 

110 

110 

110 

11 

11 

11 

11 

11 

112 

112 

112 
112 
112 
113 
113 

113 
114 
114 
114 
114 

115 
114 
115 
115 
116 
115 
116 
116 
116 
117 

117 



Lg. Vers 



9-94356 
.94370 
•94384 
•94398 
.94413 



9 . 94427 
.94441 
•94456 
.94470 
.94484 



9 . 94498 
94512 
94527 
94541 
94555 



9.94569 
•94584 
.94598 
.94612 
-94626 



9.94640 
•94655 
.94669 
.94683 
.94697 



9.94711 
.94726 
.94740 
.94751 
.94768 



9 .94782 
.94796 
.94810 
.94825 
•94839 



9 .94853 
.94867 
.94881 
.94895 
-94909 



9 .94923 
.94938 
.94952 
.94966 
.94980 



9 . 94994 
•95008 
•95022 
•95036 
•95050 



9.95064 
.95078 
.95093 
•95107 
•95121 



9^95135 
.95149 
.95163 
.95177 
•95191 



9-95205 
Lg. Vers. 



2> 

14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
14 

14 



Log. Exs. 



10-85766 
.85884 
•86001 
•86119 

. .86237 



10.86355 
.86474 
.86592 
•86711 
•86831 



10.86950 
•87070 
.87190 
.87310 
.87431 



10-87552 
•87673 
•87794 
.87916 
.88038 



10.88160 
.88282 
.88405 
.88528 
•88651 



10.88775 
•88898 
.89022 
.89147 
.89271 



10.89396 
.89521 
.89647 
.89773 
.89899 



10 



-90025 
•90152 
.90279 
.90406 
■90533 



10.90661 
.90789 
•90917 
•91046 
•91175 



10^91304 
.91434 
.91564 
•91694 
•91825 



10-91956 
.92087 
.92218 
.92350 
•92482 



10.92614 
.92747 
.92880 
.93014 
.9314 7 

10.93^31 



Log. Exs. 



n 

117 
117 
117 
118 

118 
118 
118 
119 
119 

119 
120 
120 
120 
120 

121 
121 
121 
121 
122 

122 
122 
122 
123 
123 

124 
123 
124 
124 
124 

125 
125 
125 
126 
126 

126 
126 
127 
127 
127 

128 
127 
128 
129 
129 

129 
130 

129 
130 
130 

131 
131 
131 
131 
132 

132 
133 
133 
133 
133 
134 



10 

11 
12 
13 
li 
15 
16 
17 
18 

11 
20 

21 
22 
23 
24 

25 
26 
27 
28 
_29 
30 
31 
32 
33 
-34 

35 
36 
37 
38 

M, 
40 

41 
42 
43 
_4i 
45 
46 
47 
48 
49 



50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

60 



I' 



p.p. 





130 


6 


13.0 


7 


15.1 


8 


17.3 


9 


19.5 


10 


21.6 


20 


43 .3 


30 


'65.0 


40 


86-6 


50 


108.3 



110 

ejii.o 

7112 

8,14 

9ll6 
10 18 
20 36 





3 


6 


U.3 


7 


0.3 


8 


0.4 


9 


-•4 


10 


0.5 


20 


1.0 


30 


1.5 


40 


2.0 


50 


2.5 



6 
7 
8 

9 
10 
20 
30 
40 
50 



0-1 


0. 


0.1 


0. 


0.1 


0. 


0.1 


0. 


0.1 


0. 


0.3 


0. 


0-5 


0. 


0-6 


0. 


0.8 


0. 



6 
7 
8 

9 
10 
20 
30 
40 
50 



15 

1.4 
1.7 
1.9 
2.2 
2.4 
4.8 
7.2 
9.6 
12.1 



130 

12.0 
14.0 
16^0 
18^0 
20.0 
40.0 
60.0 
80.0 
100.0 



100 

10.0 
11-6 
13.3 
15.0 
16.6 
33.3 
50.0 
66-6 
83-3 



3 

0^2 
0.2 
0.2 
0.3 
0.3 
0.6 
l.Q 
1.3 
1.6 



14 

1.4 
1.6 
1.8 
2.1 
2.3 
4^6 
7^0 
9^3 
11.6 



P.P. 



733 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



84^ 



85^ 



5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 
3£ 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



Lg.Vers. -» 



95205 
95219 
95233 
95247 
95261 



95275 
95289 
95303 
95317 
95331 



95345 
95359 
95373 
95387 
95401 



95415 
95429 
95443 
95457 
95471 



9.95485 
.95499 
.95513 
•95527 
•95540 



9^95554 
•95568 
•95582 
.95596 
•95610 



9-95624 
95638 
95652 
95666 
95680 



9-95693 
•95707 
•95721 
•95735 
•95749 



9^95763 
•95777 
.95791 
.95804 
•95818 



9^95832 
.95846 
.95860 
.95874 
•95888 



9 •95901 
.95915 
.95929 
.95943 
.95957 



55 9.95970 



56 
57 
58 
59_ 



.95984 
.95998 
.96012 
.96026 
9.96039 



14 
14 
14 
14 

14 
14 
14 
14 
14 
14 
14 
13 
14 
14 

14 
14 
14 
14 
14 

14 
14 
14 
14 
13 

14 
14 
14 
14 
14 

14 
13 
14 
14 
14 

13 
14 
14 
14 
14 

13 
14 
14 
13 
14 

14 
14 
13 
14 
14 

13 
14 
13 
14 
14 

13 
14 
14 
13 
14 

13 



Log.Exs. 



10.93281 
.93416 
.93551 
.93686 
.93821 



10.93957 
.94093 

.94229 
.94366 
•94503 



10 •94641 
.94778 
.94917 
.95055 
.95194 



10.95333 
.95473 
.95613 
.95753 
.95894 



10.96035 
.96176 
.96318 
.96461 
.96603 



10.96746 
.96889 
.97033 
.97177 
•97322 



10 •97467 
.97612 
.97758 
.97904 
•98050 



10^98197 
.98345 
.98492 
.98640 
.98789 



10.98938 
.99087 
.99237 
.99387 
.99538 



Lg, Vers, 



10.99689 

•99841 

10.99993 

11.00145 

•00298 



11.00451 
.00605 
.00759 
.00914 
•01069 



11-01225 
.01381 
.01537 
.01694 
.01852 



11.02010 



Log.Exs, 



134 
135 
135 
135 

135 
136 
136 
137 
137 
137 
137 
138 
138 
139 

139 
139 
140 
140 
140 

14l 
141 
142 
142 
142 

143 

143 
144 
144 
144 

145 
145 
145 
146 
146 

147 
14.7 
147 
148 
149 

149 
149 
150 
150 
151 

151 
151 
152 
152 
153 

153 
154 
154 
155 
155 

155 
156 
156 
157 
157 
158 



Lg, Vers, 



9-96039 
.96053 
.96067 
.96081 
.96095 



9.96108 
.96122 
.96136 
.96150 
•96163 



9^96177 
.96191 
.96205 
.96218 
•96232 



9 •96246 
.96259 
.96273 
.96287 
•96301 



9 •96314 
.96328 
.96342 
.96355 
.96369 



9-96383 
.96397 
.96410 
.96424 
.96438 



96451 
96465 
96479 
96492 
96506 



9.96519 
.96533 
•96547 
•96560 
•96574 



996588 
•96601 
•96615 
.96629 
•96642 



9 •96656 
•96669 
.96683 
96697 
96710 



9.96724 
•96737 
•96751 
•96764 
•96778 



9^96792 
.96805 
•96819 
•96832 
•96846 



9-9685P 



Lg. Vers 



14 
13 
14 
14 
13 
14 
13 
14 
13 

14 
13 
14 
13 
14 

13 
13 
14 
13 
14 

13 
14 
13 
13 
14 

13 
14 
13 
13 
14 

13 
13 
14 
13 
13 
IS 
14 
13 
13 
14 
IS 
13 
13 
14 
13 

13 
13 
13 
14 
13 

13 
13 
13 
13 
14 

13 
13 
13 
13 
13 

13 



Log.Exs. 



11 



02010 
02168 
02327 
02487 
02646 



jy 



11 



02807 
02968 
03129 
03291 
03453 



11 



03616 
03780 
03944 
04108 
04273 



11 



04438 
04604 
04771 
04938 
05106 



11 



05274 
05443 
05612 
05782 
05952 



11 



06123 
06295 
06467 
06640 
06813 



11 



06987 
07161 
07336 
07512 
07688 



11 



07865 
08043 
08221 
08400 
08579 



11 



08759 
08940 
09121 
09303 
09486 



11 



09669 
09853 
10038 
10223 
10409 



11 



10595 
10783 
10971 
11160 
11349 



11 



11539 
11730 
11922 
12114 
12307 



11^12501 



158 
159 
159 
159 

160 
161 
161 
161 
162 

163 
163 
164 
164 
165 
165 
166 
167 
167 
167 

168 
169 
169 
169 
170 

171 
171 
172 
173 
173 

174 
174 
175 
176 
176 

177 

177 

178 
179 
179 

180 
180 
181 
182 
182 

183 
184 
185 
185 
186 
186 
187 
188 
189 
189 

190 
191 
191 
192 
193 
193 



jy Log.Exs. 





1 

2 

3 

_4 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
jL4 

15 
16 
17 
18 
_19 

20 

21 
22 
23 

25 
26 
27 
28 
2^ 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 

50 

51 
52 
53 
_54 

55 
56 
57 
58 
-59 
60 



n 



P.P. 



190 



6 

7 

8 

9 

10 

20 

30 

40 

50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



19 





22 


1 


25 


3 


28 


5 


31 


6 


63 


3 


95 





126 


6 


158 


3 



170 



17 





16^ 


19 


8 


18 • 


22 


6 


21. 


25 


5 


24. 


28 


3 


26^ 


56 


6 


53 • 


85 





80 • 


113 


3 


106 • 


141 


6 


133 • 



150 


14( 


15.0 


14. 


17 


5 


16. 


20 





18. 


22 


5 


21^ 


25 





23. 


50 





46 • 


75 





70 • 


100 





93- 


125 





116^ 



180 

18^0 
21.0 
24.0 
27.0 
30.0 
60.0 
90.0 
120.0 
150.0 

160 


6 
S 

6 
3 

6 
3 



130 9 8 



13 


C 





g 


0. 


15 


1 


1 





0^ 


17 


3 


1 


2 


1. 


19 


5 


1 


3 


1^ 


21 


6 


1 


5 


1^ 


43 


3 


3 





2. 


65 


C 


4 


5 


^• 


86 


6 


6 





5. 


108 


3 


7 


5 


6- 





7 


6 


5 


6 


0^7 


0^6 


0-5 


7 


0^8 





7 





6 


8 


0^9 





8 





g 


9 







9 





7 


10 


1^1 


1 








g 


20 


2^3 


2 





1 


6 


30 


3^5 


3 





2 


5 


40 


4^6 


4 





3 


3 


50 


5-8 


5 





4 


1 





14 


14 


6 


1-4 


1-4 


7 


1-7 


1-6 


8 


1-9 


1-8 


9 


2-2 


2-1 


10 


2-4 


2-3. 


20 


4-8 


4-6 


30 


7^2 


7-0 


40 


9^6 


9-3 


50 


12-1 


n.6 



13 

1-3 
1.6 
1.8 
2.0 
2.2 
4-5 
6.7 
90 
11^2 



P.P. 



734 



TABLE YIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

86° 87° 



Lg. Vers J D 



10 

11 
12 
13 
14 



9-96859 
96837 
96887 
96900 
96914 



9.96927 
.96941 
•96954 
•96968 
•96981 



30 

21 
22 
23 
24 

25 
26 
27 
28 
29 



30 

31 
32 
33 

34 



35 
36 
37 
38 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



50 
51 
52 
53 
54 



55 

, 56 

' 57 

58 

59 



9^96995 
•97008 
.97022 
•97035 
•97049 



97062 
97076 
97089 
97103 
97116 



97130 
97143 
97157 
97170 
97183 



97197 
97210 
97224' 
972371 
97251 



97264 
97277 
97291 
97304 
97318 



97331 
97345 
97358 
97371 
9788'^ 



973981 
974121 
97425! 
97438 
.97452 



9 •97465 
•97478 
•97492; 
• 975051 
•975191 



9-97532 
•97545 
•97559 
•97572 
•97585 



9^97599 
•97612 
•97625 
•97639 
•97652 



'60 9.97665 
' iLg, Vers 



13 
14 
13 
13 

ll 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 



Log. Exs. 



11 



.12501 
.12696 
.12891 
.13087 
.13284 



11 



13482 
13680 
13879 
14079 
14280 



11 



11 



11 



14482 
14684 
14887 
15092 
15297 



11 



15502 
15709 
15917 
16125 
163^4 



11 



16544 
16755 
16967 
17180 
17394 



17609 
17824 
18041 
18259 
18477 



18697 
18917 
19138 
19361 
19584 



11 



19809 
20034 
20261 
20489 
20717 



11. 



20947 
21178 
21410 
21643 
21877 



11 



22112 
22349 
22586 
22825 
23065 



11 



23306 
23548 
23792 
24037 
24283 



11 



24530 
24778 
25028 
25279 
25531 



11-25785 



Lg. Vers, 



195 
195 
196 
196 

198 
198 
199 
200 
201 

201 
202 
203 
20S 
205 

205 
206 
208 
208 
209 

210 
211 
212 
213 
214 

214 
215 
216 
218 
218 

219 
220 
221 
222 
223 

224 
225 
227 
227 
228 

230 
230 
232 
233 

23i 

235 
236 
237 
239 
239 

241 
242 
243 
245 
246 

247 
248 
250 
251 
252 

254 



Log. Exs. D Lg. Vers 



97665 
97679 
97692 
97705 
9.7718 



97732 
97745 
97758 
97772 
97785 



97798 
97811 
97825 
97838 
97851 



97864 
97978 
97891 
97904 
97917 



97931 
97944 
97957 
97970 
97984 



97997 
98010 
98023 
98036 
98050 



98063 
98076 
98089 
98102 
98116 



98129 
98142 
98155 
98168 
98181 



98195 
98208 
98221 
98234 
98247 



98260 
98273 
98287 
98300 
98313 



98326 
98339 
98352 
98365 
98378 



98392 
98405 
98418 
98431 
98444 



98457 



3 
3 
3 
3 
3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 

3 

3 
3 
3 
3 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 

3 
3 
3 
3 
3 
13 



Log. Exs. 



11-25785 
•26040 
.26297 
.26554 
.26814 



11.27074 
.27336 
.27599 
.27864 
•28131 



11.28398 
.28668 
.28938 
.29211 
-29485 



11-29760 
•30037 
•30316 
.30596 
-30878 



11-31162 
.31447 
.31734 
.32023 
-32313 



11-32606 
.32900 
•33196 
.33494 
•33793 



11-34095 
.34398 
.34704 
.35011 
-35321 



11.35632 
.35946 
.36261 
.36579 



11.37221 
.37546 
.37872 
.38201 
•38532 



11.38866 
.39201 
.39540 
•39880 
.40224 



11.40569 
.40918 
.41269 
.41622 
.41979 



11.42338 
.42699 
•43064 
.43431 
.43802 



11.44175 



Log. Exs. 



255 
256 
257 
259 

260 
262 
263 
265 
266 

267 
269 
270 
272 
274 
275 
277 
278 
280 
282 

283 
285 
287 
288 
290 
292 
294 
296 
298 
299 
301 
303 
305 
307 
309 

311 
313 
315 
318 

320 

322 
324 
326 
328 
331 

333 
335 
338 
340 
343 

345 
348 
351 
353 
356 

359 
361 
364 
367 
370 
373 





1 

2 

3 

__4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

li. 

15 

16 

17 

18 

li 

20 

21 

22 

23 

25 
26 
27 
28 
29_ 
30 
31 
32 
33 
34_ 
35 
36 
37 
38 
IL 
40 
41 
42 
43 
44 



45 
46 
47 
48 
ii- 
50 
51 
52 
53 
54 



60 





P.P. 


*^50 


6 


25.0 


7 


29.1 


8 


33^3 


9 


37.5 


10 


41.6 


20 


83.3 


30 


125.0 


40 


166.6 


50 


208.3 





330 


6 


23.0 


7 


26.8 


8 


30.6 


9 


34.5 


10 


38.3 


20 


76.6 


30 


115^0 


40 


153-3 


50 


191.6 





310 


6 


21.0 


7 


24.5 


8 


28^0 


9 


51^5 


10 


35^0 


20 


70-0 


30 


105.0 


40 


140.0 


50 


175-0 





190 


4 


6 


19.0 


0.41 


7 


22-1 


0.4 


8 


25-3 


O.u 


9 


28.5 


0.6 


10 


31^6 


0.6 


20 


63^8 


1.3 


30 


95.0 


2.0 


40 


126.6 


2-6 


50 


158.3 


3.3 





3 


1 


6 


0^2 


0.1 


7 


0^2 


0.1 


8 


0.2 


0-1 


9 


0.3 


0.1 


10 


0.3 


0.1 


20 


0.6 


0.3 


30 


1.0 


0.5 


40 


1-3 


0.6 


50 


1.6 


0.8 





14 


13 


6 


1.4 


1.3 


7 


1.6 


1.6 


8 


1.8 


1.8 


9 


2.1 


2.0 


10 


2.3 


2.2 


20 


4.6 


4.5 


30 


7.0 


6-7 


40 


9^3 


9.0 


50 


11^6 


11.2 



340 

24.0 

28.0 

32.0 

36.0 

40.0 

80.0 

120.0 

160.0 

200.0 

330 

22.0 

25.6 

29.3 

33.0 

36.6 

73.3 

110.0 

146.6 

183.3 

300 

20.0 

23.3 

26.6 

30.0 

33.3 

66-6 

100.0 

133-3 

166.6 



3 

0.3 
0.3 
0.4 
0.4 
0-5 
1.0 
1.5 
2.0 
2.5 

O 

0.0 
0.0 
0.0 
0.1 
0.1 
0.1 
0.2 
0.3 
0.4 

13 

1.3 
1.5 
1.7 
1.9 
2.1 
4.3 
6.5 
8.6 
10.8 



P.P. 



735 



TABLFj VIII.— logarithmic versed sines and external SECANll 

88** 89° 



O 9 

1 



Lg. Vers. 



98457 
98470 
98483 
98496 
98509 



98522 
98535 
98548 
98562 
98575 



98588 
98601 
98614 
98627 
98840 



98653 
98666 
98679 
98692 
98705 



98718 
98731 
98744 
98757 
98770 



98783 
98796 
98809 
98822 
98835 



98848 
98861 
98874 
98887 
98900 



98913 
98925 
98938 
98951 



98977 
98990 
99003 
99016 
99C29 



99042 
99055 
99068 
99081 
99093 



99106 
99119 
99132 
99145 
99158 



99171 
99184 
99197 
99209 
99222 



99235 



' Lg. Vers. 



13 
13 
13 
13 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
13 
13 
13 
13 

13 
12 
13 
13 
13 

13 
13 
13 
13 
12 

13 
13 
13 
13 
12 

13 
13 
13 
13 
12 

13 
13 
13 
12 
13 
13 



Log.Exs. 



11.44175 
.44551 
.44931 
.45313 
•45699 



11.46088 
.46480 
.46876 
.47275 
.47677 



11.48083 
.48493 
.48906 
.49323 
.49743 



11.50168 
•50597 
.51029 
.51466 
.51906 



11.52351 
.52801 
•53255 
•53713 
.54176 



D 



11 •54643 
.55116 
•55593 
.56076 
.56563 



11-57056 
•57554 
.58058 
.58567 
•59082 



11.59602 
.60129 
•60662 
.61202 
.61747 



11.62300 
.62859 
.63425 
.63998 
.64579 



11.65167 
.65762 
.66366 
.66978 
.67598 



11.68227 
.68865 
.69511 
•70168 
•70834 



11^71509 
.72196 
•72892 
•73600 
.74319 



11.75050 



Log.Exs, 



376 
379 
382 
386 
389 
392 
39'5 
399 
402 

406 
409 
413 
417 
420 
425 
428 
432 
436 
440 

445 
449 
454 
458 
463 

467 
472 
477 
482 
487 

492 
498 
504 
509 
515 
520 
527 
533 
539 
545 
552 
559 
566 
573 
581 

588 
595 
604 
611 
620 

628 
638 
646 
656 
666 

675 
686 
896 
707 
719 
730 



Lg, Vers. 



.99235 
.99248 
.99261 
.99274 
.99287 



•99299 
•99312 
.99325 
.99338 
.99351 



•99363 
.99376 
.99389 
.99402 
•99415 



•99428 
•99440 
•99453 
.99466 
•99479 



•99491 
•99504 
•99517 
.99530 
.99543 



•99555 
•99568 
-99581 
.99594 
•99606 



.99619 
.99632 
.99645 
.99657 
.99670 



99683 
99695 
99708 
99721 
99734 



99746 
99759 
99772 
99784 
99797 



99810 
99823 
99835 
99848 
99861 



99873 
99886 
99899 
99911 
99924 



99937 
99949 
99962 
99974 
99987 



10-00000 
Lg 



Vers. 



12 
13 
13 
13 
12 
13 
13 
12 
13 
12 
13 
13 
12 
13 

13 
12 
13 
12 
13 
12 
13 

1^ 
12 
13 
12 
13 
12 
13 
12 

13 
12 
13 
12 
13 

12 
12 
13 
12 
13 
12 
13 
12 
12 
13 
12 
13 
12 
12 
13 

12 
12 
13 
12 
12 

13 
12 
12 
12 
13 
12 



Log.Exs, 



11.75050 
.75792 
.76547 
.77316 

J8097 

11.78892 
•79702 
•80527 
•81367 

i82223 

11.83095 
•83986 
.84894 
.85821 
•86768 



11.87735 
•88724 
•89735 
•90769 
.91829 



11.92914 
•94026 
•95167 
•96338 
.97541 



11.98777 
12.00048 
•01358 
•02707 
•04098 



12.05535 
•07020 
•08557 
•10149 
.11801 



12.13517 
•15302 
.17163 
.19106 
.21139 



12.23271 
.25511 
.27872 
.30367 
•33013 



12-35828 
.38837 
.42068 
.45557 
.49349 



12.63501 
.58089 
.63217 
.69029 
.75736 



12.83667 
•93371 

13.05877 
•23499 
•53615 



D 



Infinity 



JO J Log.Exs, 

736 



742 
755 
768 
781 

795 
809 
825 
840 
856 

872 
890 
908 
927 
947 

967 

989 

1009 

1034 

1059 

108! 
1112 
1140 
1171 
1203 
1236 
1271 
1309 
1349 
1391 

1436 
1485 
1537 
1592 
1652 

1716 
1785 
1861 
1943 
2033 
2131 
2240 
2361 
2495 
2645 
2815 
3009 
3231 
3489 
3791 

4152 
4588 
5127 
5812 
6707 

7931 

9704 

12506 

17621 

30116 



JD 





1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
ii 
15 
16 
17 
18 

11 
30 

21 
22 
23 
24 

25 
26 
27 
28 
2i 
30 
31 
32 
33 
34 

35 
36 
37 
38 
3i 
40 
41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 



60 



P. P. 





13 


6 


1-3 


7 


1.6 


8 


1.8 


9 


2.0 


10 


2.2 


20 


4.5 


30 


6.7 


40 


9.0 


50 


11.2 



13 

1.3 
1.5 
1.7 
1-2 
2.1 
4.3 
6^5 
8.6 
10.8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



13 

1 
1 
1 
1 
2 
4 
6 
8 
10 



P.P. 



J 



f^Al 


3LEIX. 


-NATURAL SINES, COSINES, TANGENTS. AND COTANGENTa 
0'' !*> 




Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 

.01746 
.01775 
.01804 
.01833 
.01862 


Cot. 


# 


1 

2 
3 

4 


.00000 
.00029 
.00058 
.00087 
.00116 


One 
One 
One 
One 
One 

One 
One 
One 
One 
One 


.00000 
.00029 
.00058 
.00087 
.00116 

.00145 
.00175 
.00204 
.00233 
.00262 


Infinite 
3437.75 
1718.87 
1145.92 
809^436 


.01745 
.01774 
.01803 
.01832 
.01862 


.99985 
.99984 
.99984 
•99983 
.99983 


57.2900 
56.3506 
55-4415 
54-5613 
53-7086 


60 

59 
58 
57 
56 


5 

i 6 
7 
8 

9 


.00145 
.00175 
.00204 
.00233 
.00262 


687.549 
572.957 
491.106 
429.718 
381.971 


.01891 
.01920 
.01949 
.01978 
.02007 


.99982 
.99982 
.99981 
.99980 
-99980 

-99979 
.99979 
.99978 
.99977 
.99977 


.01891 
.01920 
.01949 
.01978 
.02007 


52-8821 
52.0807 
51-3032 
50.5485 
49.8157 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.00291 
.00320 
.00349 
.00378 
.00407 


One 
.99999 
.99999 
.99999 
.99999 


.00291 
.00320 
.00349 
.00378 
.00407 

.00436 
.00465 
.00495 
.00524 
.00553 


343.774 
312.521 
286.478 
264.441 
245.552 


.02036 
.02065 
.02094 
.02123 
.02152 


.02036 
.02066 
.02095 
.02124 
.02153 


49.1039 
48.4121 
47.7395 
47.0853 
46.4489 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.00436 

.00465 
.00495 
.00524 
.00553 


.99999 
.99999 
.99999 
.99999 
.99998 


229.182 
214.858 
202.219 
190.9^4 
180.932 


.02181 
.02211 
.02240 
.02269 
.02298 


.99976 
.99976 
.99975 
.99974 
•99974 


.02182 
.02211 
.02240 
.02269 
-02298 


45.8294 
45.2261 
44:6386 
44.0661 
43-5081 


45 
44 
43 
42 
41 


30 

21 
22 
23 


.00582 
.00611 
.00640 
.00669 
.00698 


.99998 
.99998 
.99998 
.99998 
.99998 

.99997 
.99997 
.99997 
.99997 
.99996 

.99996 
.99993 
.99996 
.99995 
.99995 


.00582 
.00611 
.00640 
.00669 
.00698 


171-885 
163.700 
156.259 
149.465 
143.237 


.02327 
.02356 
.02385 
.02414 
.02443 


.99973 
.99972 
.99972 
.99971 
.99970 

.99969 
.99969 
.99968 
.99967 
.99966 


.02328 
.02357 
.02386 
.02415 
.02444 


42-9641 
42-4335 
41.9158 
41.4106 
40.9174 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.00727 
.00756 
2)0785 
.00814 
.00844 


.00727 
.00756 
.00785 
.00815 
.00844 


137.507 
132.219 
127.321 
122.774 
118.540 


.02472 
.02501 
.02530 
.02560 
.02589 


.02473 
.02502 
.02531 
.02560 
.02589 


40.4358 
39.9655 
39.5059 
39-0568 
38-6177 


35 
34 
33 
32 
31 


30 

Jl 
)2 

H 


.00873 
.00902 
.00931 
.00960 
.00989 


.00873 
.00902 
.00931 
.00960 
.00989 


114.589 
110.892 
107.426 
104.171 
101.107 


.02618 
.02647 
.02676 
.02705 
.02734 


.99966 
.99965 
.99964 
.99963 
.99963 


.02619 
.02648 
.02677 
.02706 
•02735 


38-1885 
37-7686 
37-3579 
36-9560 
36-5627 


30 

29 
28 
27 
26 


J5 
i6 
J7 
)8 


.01018 
.01047 
.01076 
.01105 
.01134 


.99995 
.99995 
.99994 
.99994 
.99994 


.01018 
.01047 
.01076 
.01105 
.01135 


98.2179 
95.4895 
92.9085 
90.4633 
88.1436 


.02763 
.02792 
.02821 
-02850 
.02879 


.99962 
.99961 
.99960 
.99959 
•99959 


•02764 
.02793 
.02822 
.02851 
•02881 


36^1776 
35 •8006 
35-4313 
35-0695 
34.7151 


25 
24 
23 
22 
21 


10 

11 
112 
;i3 
i4 


.01164 

.01193 

.01222. 

.01251 

.01280 


.99993 
.99993 
.99993 
.99992 
.99992 


.01164 
.01193 
.01222 
.01251 
.01280 


859398 
83.8435 
81-8470 
79.9434 
78.1263 


•02908 
•02938 
-02967 
.02996 
.03025 


•99958 
•99957 
.99956 
.99955 
.99954 

.99953 
.99952 
.99952 
.99951 
.99950 


•02910 
.02939 
.02968 
.02997 
.03026 


34-3678 
34.0273 
33.6935 
33.3662 
33-0452 


30 

19 
18 
17 
16 


IS 
16 
17 
|t8 
19 


.01309 
.01338 
.01367 
.01396 
.01425 


.99991 
.99991 
.99991 
.99990 
.99990 


.01309 
.01338 
.01367 
.01396 
.01425 


76.3900 
74.7292 
73.1390 
71.6151 
70-1533 


•03054 
•03083 
•03112 
•03141 
.03170 


.03055 
•03084 
•03114 
.03143 
.03172 


32.7303 
32.4213 
32.1181 
31.8205 
31.5234 


15 
14 
13 
12 
11 


50 

il 

li 


•01454 
.01483 
.01513 
.01542 
.01571 


.99989 
.99989 
.99989 
.99988 
.99988 


.01455 
.01484 
.01513 
.01542 
.01571 


68.7501 
67.4019 
68.1055 
64.8580 
63.6567 


•03199 
•03228 
.03257 
.03286 
.03316 


.99949 
.99948 
.99947 
.99946 
.99945 


.03201 
.03230 
.03259 
.03288 
.03317 


31.2416 
30.9599 
30.6833 
30.4116 
30.1446 


10 

9 
8 
7 
6 


55 
56 
57 
$8 

59 


•01600 
^01629 
•01658 
.01687 
•01716 


.99987 
.99987 
.99986 
.99986 
.99985 


.01600 
.01629 
.01658 
'.01687 
.01716 


62.4992 
61.3829 
60.3058 
59.2659 
58.2612 
57.2900 


.03345 
.03374 
.03403 
.03432 
.03461 


.99944 
.99943 
.99942 
.99941 
.99940 


•03346 
.03376 
.03405 
.03434 
.03463 


29.8823 
29.6245 
.09.3711 
^<).1220 
28.8771 


5 

4 
3 

2 

1 


go 


.01745 


.99985 


.01746 


•03490 


.99939 


-03492 


28.6363 


«o 


^f/ ■ 


Cos. 


Sin. 1 Cot. 


Tan. 


Cos. j Sin. 


Cot. 


Tan. 


/^ 






S 


9^ 


73 


7 


8 


8^ 


-- 





TABLE IX.- 


-NATURAL SINES. COSINES, TANGENTS, AND COTANGENTS. 
2^ 3° 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 1 Cot. ] 


"So 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 



1 
2 
3 

4 


.03490 
.03519 
.03548 
.03577 
.03606 


.99939 
.99938 
.99937 
.99936 
.99935 
.99934 
.99933 
.99932 
.99931 
.99930 


.03492 
.03521 
.03550 
.03579 
.03609 


28.6363 
28.3994 
28.1664 
27.9372 
27.7117 


.05234 
.05263 
.05292 
.0532L- 
.05350 


.99863 
.99861 
.99860 
.99858 
.99857 


.05241 
.05270 
.05299 
.05328 
.05357 


19.0811 
18.9755 
18.8711 
18.7678 
18.6656 


5 
6 
7 
8 
9 


.03635 
.03664 
.03693 
.03723 
.03752 


.03638 
.03667 
.03696 
.03725 
.03754 
.03783 
.03812 
.03842 
.03871 
.03900 


27.4899 
27.2715 
27.0566 
26.8450 
26.6367 


.05379 
.05408 
.05437 
.05466 
.05495 


.99855 
.99854 
.99852 
.99851 
.99849 
.99847 
.99846 
.99844 
.99842 
.99841 


.05387 
.05416 
.05445 
.05474 
.05503 


18.5645 
18.4645 
18.3655 
18.2677 
18.1708 


10 

11 
12 
13 
14 


.03781 
.03810 
.03839 
.03868 
.03897 


.99929 
.99927 
.99926 
.99925 
.99924 


26.4316 
26.2296 
26.0307 
25.8348 
25.6418 


.05524 
.05553 
.05582 
.05611 
.05640 


.05533 
.05562 
.05591 
.05620 
.05649 


18.0750 
17.9802 
17.8863 
17.7934 
17.7015 


15 
16 
17 
18 
19 


.03926 
.03955 
.03984 
.04013 
.04042 


.99923 
.99922 
.99921 
.99919 
.99918 


.03929 
.03958 
.03987 
.04016 
.04046 


25.4517 
25.2644 
25.0798 
24.8978 
24.7185 


.05669 
.05698 
.05727 
.05756 
.05785 


.99839 
.99838 
.99836 
.99834 
.99833 


.05678 
.05708 
.05737 
.05766 
.05795 


17.6106 
17.5205 
17.4314 
17.3432 
17.2558 


30 

21 
22 
23 

24 


.04071 
.04100 
.04129 
.04159 
.04188 


.99917 
.99916 
.99915 
.99913 
.99912 


.04075 
.04104 
.04133 
.04162 
.04191 


24.5418 
24.3675 
24.1957 
24.0263 
23.8593 


.05814 
.05844 
.05873 
.05902 
.05931 
.05960 
.05989 
.06018 
.06047 
.06076 


.99831 
.99829 
.99827 
.99826 
.99824 


.05824 
.05854 
.05883 
.05912 
.05941 


17.1693 
17.0837 
16.9990 
16.9150 
16.8319 


40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

29 
28 
27 
26 
25 
24 
23 
22 

30 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 


25 
26 
27 
28 
29 


.04217 
.04246 
.04275 
.04304 
.04333 


.99911 
.99910 
.99909 
.99907 
.99906 


.04220 
.04250 
.04279 
.04308 
.04337 


23.6945 
23.5321 
23.3718 
23.2137 
23.0577 


.99822 
.99821 
.99819 
.99817 
.99815 


.05970 
.05999 
.06029 
.06058 
.06087 


16.7496 
16.6681 
16.5874 
16.5075 
16.4283 
16.3499 
16.2722 
16.1952 
16.1190 
16.0435 


30 

.31 
32 
33 
34 


.04362 
.04391 
.04420 
.04449 
.04478 


.99905 
•99904 
.99902 
.99901 
.99900 


.04366 
.04395 
.04424 
.04454 
.04483 


22.9038 
22.7519 
22.6020 
22.4541 
22.3081 


.06105 
.06134 
.06163 
.06192 
.06221 


.99813 
.99812 
.99810 
.99808 
.99806 


.06116 
.06145 
.06175 
.06204 
•06233 


35 
36 
37 
38 
39 


.04507 
.04536 
.04565 
.04594 
.04623 


.99898 
.99897 
.99896 
.99894 
.99893 


.04512 
.04541 
.04570 
.04599 
.04628 


22.1640 
22.0217 
21.8813 
21.7426 
21.6056 


.06250 
.06279 
.06308 
.06337 
.06366 


.99804 
.99803 
.99801 
.99799 
.99797 


.06262 
.06291 
.06321 
.06350 
.06379 


15.9687 
15.8945 
15.8211 
15.7483 
15.6762 


40 

41 
42 
43 
44 


.04653 
.04682 
.04711 
.04740 
.04769 
.04798 
.04827 
.04856 
.04885 
.04914 


.99892 
.99890 
.99889 
.99888 
.99888 


.04658 
.04687 
.04716 
.04745 
.04774 


21.4704 
21.3369 
21.2049 
21.0747 
20.9460 


.06395 
.06424 
.06453 
.06482 
.06511 


.99795 
.99793 
.99792 
.99790 
.99788 


.06408 
.06437 
.06467 
•06496 
•06525 


15.6048 
15.5340 
15.4638 
15.3943 
15.3254 


45 
46 
47 
48 
49 


.99885 
.99883 
.99882 
.99881 
.99879 


.04803 
.04833 
■04862 
.04891 
.04920 


20.8188 
20.6932 
20.5691 
20.4465 
20.3253 


.06540 
.06569 
.06598 
.06627 
.06656 


.99786 
.99784 
.99782 
.99780 
.99778 
.99776 
.99774 
.99772 
.99770 
.99768 


•06554 
.06584 
.06613 
.06642 
.06671 
.06700 
.06730 
.06759 
.06788 
.06817 
.06847 
.06876 
.06905 
.06934 
.06963 


15-2571 
15.1893 
15.1222 
15.0557 
14.9898 


50 

51 
52 
53 
54 


.04943 
.04972 
.05001 
.05030 
.05059 


.99878 
.99876 
.99875 
.99873 
.99872 


.04949 
.04978 
.05007 
.05037 
.05066 


20.2056 
20.0872 
19.9702 
19.8546 
19.7403 


.06685 
.06714 
.06743 
.06773 
.06802 


14.9244 
14.8596 
14.7954 
14.7317 
14.6685 


55 
56 
57 

59 


.05088 
.05117 
.05146 
.05175 
.05205 


.99870 
.99869 
.99867 
.99866 
.99864 


.05095 
.05124 
.05153 
.05182 
.05212 


19.6273 
19.5156 
19.4051 
19.2959 
19.1879 
19. 0811 


.06831 
.06860 
.06889 
.06918 
.06947 


.99766 
.99764 
.99762 
.99760 
.99758 


14.6059 
14.5438 
14.4823 
14.4212 
14.3607 


£2- 


.05234 


.99863 


.05241 


.06976 


.99756 


.06993 


14.3007 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


1 


8 


r 738 86*^ 

i 



TABLE IX.- 


-NATURAL SINES. COSINES. TANGENTS. AND COTANGENTa 
4° 5° 


1^ 

Sf 

\JL. 

^ 6 

^ 8 

10 

11 

12 

13 

? 14 

Sl5 
16 
17 
18 

20 

21 

! 22 

; 23 

^ 24 

; 25 

' 26 

27 

' 28 

' 29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 


.06976 
.07005 
.07034 
.07063 
.07092 


.99756 
.99754 
.99752 
.99750 
.99748 


.06993 
.07022 
.07051 
.07080 
.07110 


14.3007 
14.2411 
14.1821 
14.1235 
14.0655 


.08716 
.08745 
.08774 
08803 
.08831 


.99619 
.99617 
.99614 
.99612 
.99609 


.08749 
.08778 
.08807 
.08837 
.08866 


11.4301 
11.3919 
11.3540 
11.3163 
11.2789 


60 

59 
58 
57 
56 


.07121 
.07150 
.07179 
.07208 
.07237 


.99746 
.99744 
.99742 
.99740 
.99738 


.07139 
.07168 
.07197 
.07227 
.07256 


14.0079 
13.9507 
13.8940 
13.8378 
13.7821 


.08860 
.08889 
.08918 
.08947 
.08976 


.99607 
.99604 
.99602 
.99599 
.99596 


.08895 
.08925 
.08954 
.08983 
.09013 


11.2417 
11.2048 
11.1681 
11.1316 
11.0954 


55 
54 
53 
52 
51 


.07266 
.07295 
.07324 
.07353 
.07382 


.99736 
.99734 
.99731 
.99729 
.99727 


.07285 
.07314 
.07344 
.07373 
.07402 


13.7267 
13.6719 
13.6174 
13.5634 
13.5098 


.09005 
.09034 
.09063 
.09092 
.09121 


.99594 
.99591 
.99588 
.99586 
.99583 


.09042 
•09071 
.09101 
•09130 
•09159 


11.0594 
11.0237 
10.9882 
10.9529 
10.9178 


50 

49 
48 
47 
46 


.07411 
.07440 
.07469 
.07498 
.07527 


•99725 
.99723 
.99721 
.99719 
.99716 

.99714 
.99712 
.99710 
.99708 
.99705 


.07431 
.07461 
.07490 
.07519 
.07548 


13.4566 
13.4039 
13.3515 
13.2996 
13.2480 


.09150 
.09179 
.09208 
.09237 
.09266 


.99580 
.99578 
.99575 
.9S572 
.99570 


•09189 
•09218 
•09247 
.09277 
.09306 


10.8829 
10.8483 
10.8139 
10.7797 
10.7457 


45 
44 
43 
42 
41 


.07556 
.07585 
.07614 
•07643 
.07672 


.07578 
.07607 
.07636 
.07665 
.07695 


13.1969 
13.1461 
13.0958 
13.0458 
12.9962 


.09295 
.09324 
.09353 
.09382 
.09411 


.99567 
.99564 
.99562 
.99559 
.99556 

.99553 
.99551 
.99548 
.99545 
.99542 


.09335 
.09365 
.09394 
.09423 
.09453 


10.7119 
10.6783 
10.6450 
10.6118 
10.5789 


40 

39 
38 
37 
36 


.07701 
.07730 
.07759 
.07788 
.07817 


.99703 
.99701 
.99699 
.99696 
.99694 


.07724 
.07753 
.07782 
.07812 
.07841 


12.9469 
12.8981 
12.8493 
12.8014 
12.7536 


c 09440 
.09469 
.09498 
.09527 
.09556 


.09482 
.09511 
.09541 
.09570 
.09600 

•09629 
.09658 
.09688 
.09717 
.09746 


10^5462 
10.5136 
10.4813 
10.4491 
10.4172 


35 
34 
33 
32 
31 


.07846 
.07875 
.07904 
.07933 
.07962 

.07991 
.08020 
.08049 
.08078 
.08107 


.99692 
.99689 
.99687 
.99685 
.99683 


.07870 
.07899 
.07929 
.07958 
.07987 


12.7062 
12.6591 
12.6124 
12.5660 
12.5199 


.09585 
.09614 
.09642 
.09671 
.09700 


.99540 
.99537 
.99534 
.99531 
.99528 


10.3854 
10.3538 
10.3224 
10.2913 
10.2602 


30 

29 
28 

27 
26 


.99680 
.99678 
.99676 
.99673 
.99671 


.08017 
.08046 
.08075 
.08104 
.08134 


12.4742 
12.4288 
12.3838 
12.3390 
12.2946 


.09729 
.09758 
.09787 
.09816 
.09845 


.99526 
.99523 
.99520 
.99517 
99514 


.09776 
.09805 
.09834 
.09864 
.09893 


10.2294 
10.1988 
10.1683 
x0.1381 
10.1080 


25 
24 
23 
22 
_21 


.08136 
.08165 
.08194 
.08223 
.08252 


.99668 
.99666 
.99664 
.99661 
.99659 


.08163 
.08192 
.08221 
.08251 
.08280 


12.2505 
12.2067 
12.1632 
12.1201 
12.0772 


.09874 
.09903 
.09932 
.09961 
.09990 


.99511 
•99508 
.99506 
.99503 
.99500 


.09923 
.09952 
.09981 
.10011 
.10040 

.10069 
.10099 
.10128 
.10158 
.10187 

.10216 
.10246 
.10275 
.10305 
.10334 


10.0780 
10.0483 
10. 0187 
9.98931 
9.96007 


20 

19 
18 
17 
18 


.08281 
.08310 
.08339 
.08368 
.08397 


.99657 
.99654 
.99652 
.99649 
.99647 


.08309 
.08339 
.08368 
.08397 
.08427 


12.0346 
11.9923 
11.9504 
11.9087 
11.8673 


.10019 
.10048 
.10077 
.10106 
.10135 


.99497 
.99494 
.99491 
.99488 
.99485 


9.93101 
9.90211 
9.873S8 
9.84482 
9.81641 


15 
14 
13 
12 
11 


60 

51 
52 
53 

54 

55 
56 
57 
58 
59 

22- 


.08426 
.08455 
.08484 
.08513 
.08542 


.99644 
.99642 
.99639 
.99637 
.99635 


.08456 
.08485 
.08514 
.08544 
.08573 


11.8262 
11.7853 
11.7448 
11.7045 
11.6645 


.10164 
.10192 
.10221 
.10250 
.10279 


.99482 
.99479 
.99476 
.99473 
.99470 


9.78817 
9.76009 
9.73217 
9.70441 
9.67680 


10 

9 
8 
7 
8 


.08571 
.08600 
.08329 
.08658 
.08687 


.99632 
.99630 
.99627 
.99625 
.99622 


.08602 
.08632 
.08661 
.08690 
.08720 


11.6248 
11.5853 
11.5461 
11.5072 
11.4685 


.10308 
.10337 
.10366 
.10395 
.10424 


.99467 
.99464 
.99461 
.99458 
.99455 


.10363 
.10393 
.10422 
.10452 
.10481 


9.64935 
9.62205 
9.59490 
9. 56791 
9.54106 


5 
4 
3 
2 
1 


.08716 


.99619 


.08749 


11.4301 


.10453 


.99452 


.10510 


9.51436 





■"V— 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. Cot. 


Tan. 


/ 



85^ 



739 



84^ 



TABLE tX.- 


-NATURAL SINFR, COSINES, TANGENTS, AND C0TANGENTI3. 1 1 


Sin. 1 


Cos. 


Tan. 


Cot. Sin. 1 


Cos. 


Tan. 1 Cot. \ ' ' 




1 
2 
3 
4 


.10453 
.10482 
.10511 
.10540 
.10569 


.99452 
.99449 
.99446 
.99443 
•99440 


.10510 
.10540 
.10569 
.10599 
.10628 


9.51436 
9.48781 
9.46141 
9.43515 
9.40904 


.12187 
.12216 
.12245 
.12274 
.12302 


.99255 
.99251 
.99248 
.99244 
•99240 


.12278 
.12308 
.12338 
.12367 
.12397 


8-14435 
8.12481 
8-10536 
8^08600 
8^06674 

8 • 04756 
8 •02848 
8 • 00948 
7.99058 
7.97176 


60 

59 
58, 
57 
56 

55 

541 

53 i 
52 
51 

50 1 

49 
48 
47 
46 

45 
44 
43 
42 

40 

39 
38 
37 
36 

35 

34 

33, 

32 

31 

30 

29 1 

28; 

27 

26 


5 
6 
7 
8 
9 


.10597 
.10626 
.10655 
.10684 
.10713 


.99437 
.99434 
.99431 
.99428 
.99424 


.10657 
.10687 
.10716 
.10746 
.10775 


9.38307 
9.35724 
9.33155 
9-30599 
9.28058 


.12331 
.12360 
.12389 
.12418 

.12447 


.99237 
•99233 
•99230 
.99226 
•99222 


.12426 
.12456 
.12485 
.12515 
.12544 


10 

11 
12 
13 
14 


.10742 
.10771 
.10800 
.10829 
.10858 

.10887 
.10916 
.10945 
.10973 
.11002 


.99421 
.99418 
.99415 
.99412 
.99409 

.99406 
.99402 
.99399 
.9l396 
.99393 


.10805 
.10834 
.10863 
.10893 
.10922 

.10952 
.10981 
.11011 
.11040 
.11070 


9.25530 
9 23016 
9.20516 
9.18028 
9.15554 


.12476 
.12504 
.12533 
.12562 
12591 


•99219 
•99215 
.99211 
.99208 
.99204 


.12574 
.12603 
.12633 
.12662 
.12692 


7.95302 
7.93438 
7.91582 
7.89734 
7.87895 


15 
16 
17 
18 
19 


9.13093 
9.10646 
9.08211 
9.05789 
9.03379 


.12620 
.12649 
.12678 
.12706 
.12735 


.99200 
.99197 
.99193 
.99189 
•99186 


.12722 
.12751 
.12781 
.12810 
.12840 


7.86064 
7.84242 
7.82428 
7.80622 
7.78825 


20 

21 
22 
23 

24 


.11031 
.11060 
.11089 
.11118 
.11147 


.99390 
.99386 
.99383 
.99380 
.99377 


.11099 
.11128 
.11158 
.11187 

.11217 


9.00983 
8.98598 
8.96227 
8.93867 
8.91520 


.12764 
.12793 
.12822 
.12851 
.12880 


.99182 
.99178 
.99175 
.99171 
.99167 


.12869 
.12899 
.12929 
.12958 
.12988 


7.77035 
7.75254 
7.73480 
7.71715 
7.69957 


25 
26 
27 
28 
29 


.11176 
.11205 
.11234 
.11263 
.11291 


.99374 
.99370 
.99367 
.99364 
.99360 


.11246 
.11276 
.11305 
.11335 
.11364 


8.89185 
8.86862 
8.84551 

8.82252 
8.79964 


.12908 
.12937 
.12966 
.12995 
.13024 


•99163 
.99160 
.99156 
.99152 
.99148 


.13017 
.13047 
.13076 
.13106 
.13136 


7.68208 
7.66466 
7.64732 
7.63005 
7.61287 

7.59575 
7.57872 
7.56176 
7.54487 
7.52806 


30 

31 
32 
33 
34 


.11320 
.11349 
.11378 
.11407 
.11436 


.99357 
.99354 
.99351 
.99347 
.99344 


.11394 
.11423 
.11452 
.11482 
.11511 


8.77689 
8.75425 
8.73172 
8.70931 
8.68701 


.13053 
.13081 
.13110 
.13139 
.13168 


.99144 
.99141 
.99137 
.99133 
.99129 


.13165 
.13195 
.13224 
.13254 
.13284 


35 
36 
37 
38 
39 


.11465 
.11494 
.11523 
.11552 
.11580 


.99341 
.99337 
.99334 
.99331 
.99327 


.11541 
.11570 
.11600 
.11629 
.11659 


8.66482 
8.64275 
8.62078 
8.59893 
8.57718 


.13197 
.18226 
.13254 
.13283 
.13312 


.99125 
.99122 
.99118 
.99114 
.99110 


.13313 
.13343 
.13372 
.13402 
.13432 


7^51132 
7^49465 
7-47806 
7.46154 
7.44509 


25 
24 
23 
22 
21 


40 

41 
42 
43 

44 


.11609 
.11638 
11667 
.11696 
.11725 


.99324 
.99320 
.99317 
.99314 
.99310 


.11688 
.11718 
.11747 
.11777 
.11806 


8.55555 
8.53402 
8.51259 
8.49128 
8.47007 


.13341 
.13370 
.13399 
.13427 
.13456 


.99106 
.99102 
.99098 
.99094 
.99091 


.13461 
.13491 
.13521 
.13550 
.13580 


7.42871 
7.41240 
7.39616 
7.37999 
7.36389 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.11754 
.11783 
.11812 
.11840 
.11869 


.99307 
.99303 
.09300 
.99297 
.99293 


.11836 
.11865 
.11895 
.11924 
.11954 


8.44896 
8.42795 
8.40705 
8.38625 
8.36555 


.13485 
.13514 
.13543 
.13572 
.13600 


.99087 
.99083 
.99079 
.99075 
.99071 


.13609 
.13639 
.13669 
.13698 
.13728 


7-34786 
7-33190 
7-316C0 
7. 30018 
7.28442 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.11898 
.11927 
.11956 
.11985 
.12014 


.99290 
.99286 
.99283 
.99279 
.99276 


.11983 
.12013 
.12042 
.12072 
.12101 


8.34496 
8.32446 
8-30406 
8.28376 
8.26355 


.13629 
.13658 
.13687 
.13716 
.13744 


.99067 
.99063 
.99059 
.99055 
.99051 


.13758 
.13787 
.13817 
•13846 
a3876_. 

.13906 
.13^^35 
.13965 
•13995 
.14024 


7.26873 
7-25310 
7-23754 
7-22204 
7.20661 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•12043 
.12071 
.12100 
.12129 
.12158 


.99272 
.99269 
.99265 
.99262 
.99258 


.12131 
.12160 
.12190 
.12219 

.12249 


8.24345 
8.22344 
8.20352 
8.18370 
8.16398 

. 8 14435 


.13773 
.13802 
.13831 
.13860 
.13889 

.13917 


.99047 
.99043 
99039 
.99035 
^9903L 
. 99027 


7.19125 
7.17594 
7.16071 
7-14553 
7.13042 

7.11537 


5 

4 
3 
2 
1 


60 


.12187 


.99255 


12278 


•14054 


ft 


/ 


Cos. 


Sin. 


Cot. 1 Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ - 



83" 



740 



83' 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTa 







8° 






' 


9^ 






t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin, 


Cos. 


Tan. 


Cot. 


# 




1 

2 
3 
4 


.13917 
.13946 
.13975 
.14004 
.14033 


.99027 
.99023 
.99019 
.99015 
.99011 


.14054 

.14084 

.14113 

.14143 

• 14173, 

.14202 

.14232 

.14262 

.14291 

.14321 


7.11537 
7.10038 
7.08546 
7.07059 
7.05579 


.15643 
.15672 
.15701 
.15730 
.15758 

.15787 
.15816 
.15845 
.15873 
15902 


.98769 
.98764 
.98760 
.98755 
.98751 

.98746 
.98741 
.98737 
.98732 
.98728 


•15838 
.15868 
.15893 
.15928 
.15958_ 

.15988 
.16017 
.16047 
.16077 
.16107 


6^31375 
6.30189 
6.29007 
6.27829 
6.26655 

6.25486 
6.24321 
6.23160 
6.22003 
6.20851 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.14061 
.14090 
.14119 
.14148 
.14177 


.99006 
.99002 
.98998 
.98994 
.98990 

.98986 
.98932 
.98978 
.98973 
.98969 


7.04105 
7-02637 
7.01174 
6.99718 
6.98268 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.14205 
.14234 
.14263 
.14292 
.14320 

.14349 
.14378 
.14407 
.14436 
.14464 


.14351 
.14381 
.14410 
. 14440 
.14470 


6^96823 
6.95385 
6.93952 
6^92525 
6^91104 


.15931 
.15959 
.15988 
.16017 
.16046 


.98723 
.98718 
.98714 
.98709 
.98704 


.16137 
.16167 
.16196 
.16226 
.16256 


6.19703 
6.18559 
6.17419 
6.16283 
6.15151 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.98965 
.98961 
.98957 
.98953 
.98948 


.14499 
.14529 
.14559 
.14588 
•14618 


6^89688 
6^88278 
6^86874 
6 • 85475 
6 •84082 


.16074 
.16103 
.16132 
.16160 
.16189 


•98700 
.98695 
.98690 
.98686 
.98681 


.16286 
.16316 
.16346 
.16376 
•16405 


6.14023 
6.12899 
6.11779 
6.10664 
6.09552 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.14493 
.14522 
.14551 
.14580 
. 14608 


.98944 
.98940 
.98936 
.98931 
.98927 


.14648 
.14678 
.14707 
.14737 
.14767 


6.82694 
6.81312 
6.79936 
6.78564 
6.77199 


.16218 
.16246 
.16275 
.16304 
.16333 


.98676 
.98671 
.98667 
.98662 
.98657 


.16435 
.16465 
.16495 
.16525 
.16555 


6.08444 
6.07340 
6.06240 
6.05143 
6.04051 


40 

39 
38 
37 
36 


25 
26 
l«7 
28 
29 


.14637 
.14666 
.14695 
.14723 
.14752 


.98923 
•98919 
•98914 
.98910 
.98906 


.14796 
.14826 
.14856 
.14886 
.14915 


6.75838 
6.74483 
6.73133 
6.71789 
6.70450 


.16361 
.16390 
.16419 
.16447 
.16476 

.16505 
.16533 
.16562 
.16591 
-16620 


.98652 
.98648 
.98643 
.98638 
.98633 


.16585 
.16615 
.16645 
.16674 
.16704 


6.02962 
6.01878 
6.00797 
5.99720 
5.98646 

5.97576 
5.96510 
5.95448 
5.94390 
5.93335 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.14781 
.14810 
.14838 
.14867 
.14896 


•98902 
.98897 
.98893 
.98889 
.98884 

.98880 
.98876 
.98871 
.98867 
.98863 


.14945 
.14975 
.15005 
.15034 
.15064 

.15094 
.15124 
.15153 
.15183 

.15213 


6.69116 
6.67787 
6.66463 
6.65144 
6.63831 


.98629 
.98624 
.98619 
.98614 
.98609 


.16734 
.16764 
.16794 
.16824 
.16854 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.14925 
.14954 
.14982 
.15011 
-15040 


6.62523 
6.61219 
6.59921 
6.58627 
6.57339 


.18648 
.16677 
•16706 
.16734 
.16763 


.98604 
.98600 
.98595 
98590 
.9858& 


.16884 
.16914 
.16944 
.16974 
.17004 


5.92283 
5.91236 
5.90191 
5.89151 
5.88114 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.15069 
.15097 
.15126 
.15155 
.15184 


.98858 
.98854 
.98849 
.98845 
.98841 


.15243 
.15272 
.15302 
.15332 
.15362 


6.56055 
6.54777 
6.53503 
6.52234 
6.50970 


.16792 
.16820 
.16849 
.16878 
.16906 


.98580 
.98575 
.98570 
.98565 
.98561 


.17033 
.17063 
.17093 
.17123 
.17153 


5-87080 
5-86051 
5-85024 
5.84001 
5.82982 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.15212 
.15241 
.15270 
.15299 
•15327 


.98836 
.98832 
.98827 
.98823 
.98818 


.15391 
.15421 
.15451 
.15481 
.15511 


6.49710 
6.48456 
6.47206 
6.45961 
6=44720 


.16935 
.16964 
.16992 ! 
.17021 i 
-17050 j 


.98556 
.98551 
.98543 
.98541 
.98536 


.17183 
.17213 
.17243 
.17273 
.17303 


5.81966 
5.80953 
5.79944 
5.78938 
5.77936 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.15356 
.15385 
.15414 
.15442 
.15471 

.15500 
15529 
.15557 
.15586 
.15615 


.98814 
.98809 
.98805 
.98800 
.98796 


.15540 
.15570 
.15600 
.15630 
.15860 


6.43484 
6.42253 
6.41026 
6.39804 
6.38587 
"6.37374 
6.36165 
6.34961 
6.33761 
6-32566 


.17078 
•17107 
•17136 
.17164 
.17193 

.17222 
.17250 
.17279 
.17308 
.17336 


.98531 
.98526 
.98521 
.98516 
.98511 

.98506 
.98501 
.98496 
.98491 
•98486 


.17333 
.17363 
.17393 
.17423 
.17453 


5-76937 
5.75941 
5-74949 
5.73960 
5.72974 

5-71992 
5.71013 
5.70037 
5-69064 
5-68094 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.98791 
.98787 
.98782 
.98778 
.98773 

.98769 


.15689 
.15719 
.15749 
.15779 
.15809 


.17483 
.17513 
.17543 
.17573 
•17603 


5 
4 
3 
2 

1 


60 


.15643 


.15838 


6.31375 


•17365 


.98481 


.17633 


5-67128 


-.2 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 





8i' 



741 



80' 



TA3LE IX.— NATURAL SINES. COSINES, TANGENTS, AND COTANGENTS. 
10° 11° 



Sin. 


Cos. 


Tan. 


.17365 
.17393 
•17422 
.17451 
.17479 


.98481 
.98476 
.98471 
.98466 
.98461 


.17633 
.17663 
.17693 
.17723 
.17753 


.17508 
.17537 
.17565 
.17594 
.17623 


.98455 
.98450 
.98445 
.98440 
•98435 


.17783 
.17813 
.17843 
.17873 
.17903 


.17651 
.17680 
.17708 
.17737 
•17766 


.98430 
.98425 
.98420 
.98414 
.98409 


17933 
.17963 
.17993 
.18023 
.18053 


.17794 
.17823 
•17852 
.17880 
.17909 


.98404 
.98399 
.98394 
.98389 
.98383 


.18083 
.18113 
.18143 
.18173 
•18203 


17937 
.17966 
.17995 
.18023 
.18052 


.98378 
.98373 
.98368 
.98362 
.98357 


.18233 
•18263 
.18293 
•18323 
•18353 


.18081 
.18109 
.18138 
.18166 
.18195 


.98352 
.98347 
.98341 
.98336 
.98331 


•18384 
•18414 
• 18444 
.18474 
.18504 


.18224 
.18252 
.18281 
.18309 
.18338 


.98325 
.98320 
.98315 
•98310 
•98304 


.18534 
.18564 
•18594 
•18624 
•18654 


.18367 
.18395 
.18424 
.18452 
.18481 


•98299 
.98294 
•98288 
.982»3 
.98277 


•18684 
•18714 
.18745 
.18775 
.18805 


.18509 
.18538 
.18567 
.18595 
.18624 


.98272 
•98267 
•98261 
•98256 
•98250 


.18835 
.18865 
.18895 
.18925 
.18955 


.18652 
.18681 
.18710 
•18738 
•18767 


.98245 
•98240 
•98234 
.98229 
•98223 


.18986 
•19016 
•19046 
.19076 
.19106 


•18795 
•18824 
.18852 
.18881 
.18910 

.18938 
.18967 
.18995 
.19024 
•19052 


•98218 
.98212 
.98207 
.98201 
.98196 


.19136 
•19166 
.19197 
•19227 
.19257 


.98190 
.98185 
.98179 
.98174 
.98168 


.19287 
•19317 
•19347 
•19378 
•19408 


•19081 


.98163 


•19438 


Cos. 


' Sin. 


Cot. 



Cot. 



5^67128 
5^66165 
5^65205 
5-64248 
5-63295 



5^62344 
5-61397 
5-60452 
5^59511 
5-58573 



5^57638 
5.56706 
5.65777 
5.54851 

5.53927 



5.53007 
5.52090 
5.51176 
5.50264 
5.49356 



5.48451 
5.47548 
5.46648 
5-45751 
5-44857 



5.43966 
5.43077 
5.42192 
5.41309 
5-40429 



5-39552 
5.38677 
5-37805 
5-36936 
5-36070 



5-35206 
5-34345 
5-33487 
5-32631 
5-31778 



5-30928 
5-30080 
5-29235 
5 •28393 
5-27553 



5-26715 
5-25880 
5-25048 
5.24218 
5-23391 



5-22566 
5-21744 
5.20925 
5-20107 
5-19293 



5-18480 
5-17671 
5-16863 
5-16058 
5-15256 



5-14455 



Tan. 



Sin. 



19081 
19109 
19138 
19167 
19195 



19224 
19252 
19281 
19309 
19338 



19366 
19395 
19423 
19452 
19481 



19509 
19538 
19o66 
19595 
19623 



19652 
19680 
19709 
19737 
19766 



19794 
19823 
19851 
19880 
19908 



19937 
19965 
19994 
20022 
20051 



20079 
20108 
20136 
20165 
20193 



20222 
20250 
20279 
20307 
20336 



20364 
20393 
20421 
20450 
20478 



20507 
20535 
20563 
20592 
20620 



20649 
20677 
20706 
20734 
20763 



20791 



Cos. 



Cos. 



98163 
98157 
98152 
98146 
98140 



98135 
98129 
98124 
98118 
98112 



98107 
98101 
98096 
98090 
98084 



98079 
98073 
98067 
98061 
98056 



98050 
98044 
98039 
98033 
98027 



98021 
98016 
98010 
98004 
97998 



97992 
97987 
97981 
97975 
97969 



97963 
97958 
97952 
97946 
97940 



97934 
97928 
97922 
97916 
97910 



97905 
97899 
97893 
97887 
97881 



97875 
97869 
97863 
97857 
97851 



97845 
97839 
97833 
97827 
97821 



97815 

Sin. 



Tan. 



19438 
19468 
19498 
19529 
19559 



19589 
19619 
19649 
19680 
19710 



19740 
19770 
19801 
19831 
19861 



19891 
19921 
19952 
19982 
20012 



20042 
20073 
20103 
20133 
20164 



20194 
20224 
20254 
20285 
20315 



20345 
20376 
20406 
20436 
20466 



20497 
20527 
20557 
20588 
20618 



20648 
20679 
20709 
20739 
20770 



20800 
20830 
20861 
20891 
20921 



20952 
20982 
21013 
21043 
21073 



21104 
21134 
21164 
21195 
21225 



21256 



Cot. 



Cot. 



5-14455 
5-13658 
5-12862 
5-12069 
5-11279 



5-10490 
5-09704 
5-08921 
5-08139 
5-07360 



5-06584 
5-05809 
5-05037 
5-04267 
5-03499 



5-02734 
5-01971 
5-01210 
5-00451 
4-99695 



4-98940 
4-98188 
4-97438 
4-96690 
4-95945 



4.95201 
4.94460 
4.93721 
4-92984 
4-32249 



4-91516 
4-90785 
4-90056 
4-89330 
4-88605 



4-87882 
4-87162 
4-86444 
4-85727 
4. 85013 



4-84300 
4-83590 
4-82882 
4-82175 
4-81471 



4-80769 
4-80068 
4-79370 
4-78673 
4-77978 



4.77286 
4.76595 
4-75906 
4.75219 
4-74534 



4-73851 
4-73170 
4-72490 
4-71813 
4-71137 



4-70463 



Tan. 



•^9' 



742 



7S' 



TABLE IX. 



-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
12° 13° 



/ 


Sin. Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 1 


/ 


. 

, 1 

2 

'■ 3 
4 


.20791 
•20820 
.20848 
.20877 
•20905 


.97815 
.97809 
.97803 
.97797 
.97791 


.21256 
.21286 
.21316 
.21347 
.21377 
•21408 
•21438 
•21469 
•21499 
.21529 


4.70463 
4.69791 
4.69121 
4^68452 
4^67786 


.22495 
.22523 
.22552 
.22580 
.22608 


.97437 
.97430 
.97424 
.97417 
.97411 


•23087 
•23117 
.23148 
.23179 
.23209 

•23240 
.23271 
.23301 
.23332 
•23363 


4. 

4. 

4 

4 

4 


33148 
32573 
32001 
31430 
30860 


60 

59 
58 
57 
56 


5 

! ? 

8 
1 9 


.20933 
.20962 
.20990 
.21019 
.21047 


.97784 
.97778 
.97772 
•97766 
•97760 


4^67121 
4^ 66458 
4^65797 
4^65138 
4.64480 


.22637 
.22665 
.22693 
.22722 
.22750 
.22778 
.22807 
.22835 
.22863 
.22892 


.97404 
.97398 
•97391 
•97384 
•97378 


4 
4 
4 
4 
4 


30291 
29724 
29159 
28595 
28032 


55 
54 
53 
52 
51 


10 

11 
12 
13 

14 


.21076 
.21104 
.21^32 
.21161 
.21189 


•97754 
•97748 
.97742 
.97735 
.97729 


•21560 
•21590 
.21621 
.21651 
.21682 


4.63825 
4.63171 
4.62518 
4.61868 
4-61219 


•97371 
•97365 
•97358 
•97351 
.97345 


•23393 
•23424 
•23455 
•23485 
-23516 


4 
4 
4 
4 
4 


27471 
26911 
26352 
25795 
25239 


50 

49 
48 
47 
46 


: 15 
16 
17 
18 

1 19 


.21218 
.21246 
.21275 
.21303 
.21331 


.97723 
.97717 
.97711 
•97705 
•97698 


.21712 
.21743 
.21773 
•21804 

.21834 


4.60572 
4.59927 
4.59283 
4.58641 
4-58001 


.22920 
.22948 
.22977 
.23005 
.23033 


.97338 
•97331 
•97325 
•97318 
.97311 


.23547 
.23578 
.23608 
•23639 
-23670 


4 
4 
4 
4 
4 


24685 
24132 
23580 
23030 

22481 


45 
44 
43 
42 
41 


30 

1 21 
22 

; 23 
24 


.21360 
.21388 
.21417 
.21445 
.21474 


.97692 
.97686 
.97680 
.97673 
.97667 


.21864 
•21895 
•21925 
•21956 
•21986 


4-57363 
4.56726 
4.56091 
4.55458 
4-54826 


•23062 ' 
.23090 ! 
.23118 
.23146 
.23175 


-97304 
•97298 
•97291 
•97284 
•97278 


-23700 
-23731 
.23762 
.23793 
.23823 


4 
4 
4 
4 
4 


21933 
21387 
20842 
20298 
19756 


40 

39 
38 
37 
36 


25 
. 26 
,27 
1 28 
I 29 


.21502 
.21530 
.21559 
.21587 
.21616 


.97661 
.97655 
.97648 
.97642 
.97636 


•22017 
.22047 
.22078 
.22108 
•22139 


4.54196 
4.53568 
4-52941 
4.52316 
4.51693 


.23203 
.23231 
.23260 
.23288 
•23316 
•23345 
•23373 
•23401 
•23429 
•23458 


.97271 
•97264 
•97257 
.97251 
.97244 

.97237 
.97230 
.97223 
.97217 
.97210 

.97203 
.97196 
.97189 
.97182 
.97176 


.23854 
.23885 
.23916 
.23946 
.23977 

.24008 
.24039 
.24069 
.24100 
.24131 


4 
4 
4 
4 
4 


19215 
18675 
18137 
17600 
17064 


35 
34 
33 
32 
31 


30 

31 

, 32 

33 

34 


.21644 
.21672 
•21701 
.21729 
.21758 


.97630 
.97623 
.97617 
.97611 
.97604 


•22169 
•22200 
•22231 
•22261 
•22292 


4.51071 
4.50451 
4.49832 
4.49215 
4.48600 


4 
4 
4 
4 
4 


16530 
15997 
15465 
14934 
14405 


30 

29 
28 
27 
26 


35 

36 

. 37 

1 38 

39 


.21786 
.21814 
.21843 
.21871 
.21899 


.97598 
.97592 
.97585 
.97579 
.97573 


•22322 
•22353 
•22383 
•22414 
. 22444 


4.47986 
4.47374 
4^46764 
4.46155 
4.45548 


23486 
•23514 
•23542 
•23571 
•23599 


.24162 
.24193 
.24223 
.24254 
.24285 


4 
4 
4 
4 
4 


13877 
13350 
12825 
12301 
11778 


25 
24 
23 
22 
21 


Uo 

41 

42 

43 

1 44 


.21928 
.21956 
.21985 
.22013 
.22041 

.22070 
.22098 
.22126 
.22155 
.22183 


•97566 
•97560 
.97553 
.97547 
.97541 


•22475 
•22505 
•22536 
•22567 
.22597 


4.44942 
4.44338 
4^43735 
4.43134 
4.42534 

4.41936 
4.41340 
4^40745 
4^40152 
4-39560 


.23627 
.23656 
.23684 
.23712 
•23740 

.23769 
.23797 
.23825 
.23853 
.23882 


.97169 
.97162 
.97155 
.97148 
•97141 


.24316 
.24347 
.24377 
.24408 
.24439 


4 
4 
4 
4 
4 


11256 
10736 
10216 
09699 
09182 


30 

19 
18 
17 
16 


\ 45 
1 46 
i 47 
148 
149 


.97534 
.97528 
.97521 
.97515 
.97508 


.22628 
.22658 
•22689 
•22719 
.22750 


•97134 
.97127 
.97120 
.97113 
.97106 


.24470 
.24501 
.24532 
.24562 
.24593 


4 
4 
4 
4 
4 


08666 
08152 
07639 
07127 
06616 


15 
14 
13 
12 
11 


50 

1 51 
52 
53 

' 64 


.22212 
.22240 
.22268 
.22297 
•22325 


.97502 
.97496 
.97489 
.97483 
.97476 


•22781 
.22811 
•22842 
•22872 
•22903 

•22934 
•22964 
.22995 
.23026 
.23056 


4.38969 
4.38381 
4.37793 
4.37207 
4.36623 


.23910 
.23938 
.23966 
.23995 
•24023 


.97100 
.97093 
.97086 
.97079 
•97072 


.24624 
.24655 
.24686 
.24717 
.24747 


4 
4 
4 
4 
4 


06107 
05599 
05092 
04586 
04081 


10 

9 
8 
7 
6 


55 
56 
57 
58 

1 59 


.22353 
•22382 
.22410 
.22438 
.22467 
.22495 


•97470 
•97463 
•97457 
.97450 
.97444 


4.36040 
4.35459 
4.34879 
4.34300 
4.3^723 


.24051 
.24079 
.24108 
.24136 
.24164 
.24192 


.97065 
.97058 
.97051 
.97044 
.97037 


.24778 
•24809 
. 24840 
.24871 
.24902 


4 
4 
4 
4 
4 


03578 
03076 
02574 
02074 
01576 


5 
4 
3 
2 
1 


60 


.■97437 


.23087 


4-33148 


•97030 


•24933 


4 


01078 





/ 


Cos. 


Sin. 1 Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 1 


/ 



77' 



743 



76' 



TABLE IX. 



-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
14° 15° 



, 


Sin. 


Cos. 


Tan. 

.24933 
.24964 
.24995 
.25026 
.25056 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


.24192 
•24220 
• 24249 
.24277 
•24305 


.97030 
.97023 
.970]y5 
.97008 
.97001 


4-01078 
4^00582 
4^ 00086 
3-99592 
3-99099 


-25882 
.25910 
.25938 
-25966 
-25994 


-96593 
-96585 
-96578 
-96570 
-96562 


-26795 
-26826 
-26857 
-26888 
-26920 


3 
3 
3 
3 
3 


73205 
72771 
72338 
71907 
71476 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.24333 
•24362 
.24390 
.24418 
. 24446 


.96994 
.96987 
.96980 
.96973 
.96966 


.25087 
.25118 
.25149 
.25180 

.25211 


3-98607 
3-98117 
3^97627 
3-97139 
3-96651 


-26022 
-26050 
-26079 
-26107 
-26135 


-96555 
-96547 
-96540 
-96532 
-96524 


-26951 
.26982 
.27013 
.27044 
.27076 


3 
3 
3 
3 
3 


71046 
70616 
70188 
69761 
69335 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


•24474 
.24503 
.24531 
.24559 
.24587 

•24615 
■ 24644 
•24672 
.24700 
.24728 


.96959 
.96952 
.96945 
.96937 
.96930 


.25242 
25273 
.25304 
.25335 
.25366 


3-96165 
3-95680 
3-95196 
3.94713 
3.94232 


-26163 
-26191 
-26219 
-26247 
-26275 


-96517 
-96509 
-96502 
-96494 
-96486 


-27107 
-27138 
-27169 
-27201 
-27232 

-27263 
.27294 
.27326 
.27357 
.27388 


3 
3 
3 
3 
3 


68909 
68485 
68061 
.67638 
67217 


50 

49 
48 
47 

46 


15 
16 
17 
18 
19 


.96923 
.96916 
•96909 
.96902 
.96894 


.25397 
.25428 
.25459 
. 25490 
.25521 


3.93751 
3.93271 
392793 
3.92316 
3. 91839 


.26303 
.26331 
-26359 
-26387 
-26415 


-96479 
-96471 
-96463 
-96456 
-96448 


3 
3 
3 
3 

3 


.66796 
.66376 
.65957 
.65538 
•65121 


45 
44 
43 

-ii 


30 

21 
22 
23 
24 


.24756 
.24784 
.24813 
.24841 
.24869 


.96887 
•96880 
•96873 
•96866 
.96858 


.25552 
.25583 
.25614 
.25645 
.25676 

.25707 
.25738 
.25769 
.25800 
.25831 


3.91364 
3.90890 
3.90417 
3-89945 
3.89474 


-26443 
-26471 
-28500 
-26528 
-26556 

.26584 
.26612 
.26640 
.26668 
-26696 


-96440 
-96433 
-96425 
-96417 
.96410 


.27419 
.27451 
.27482 
.27513 
.27545 


3 
3 
3 
3 
3 


•64705 
. 64289 

63874 
.63461 

63048 


40 

39 
38 
37 
38 


25 
26 
27 
28 
29 


•24897 
.24925 
.24954 
.24982 
.25010 


•96851 
•96844 
•98837 
•98829 
.96822 


3 •83004 
3-88536 
3.88068 
3.87601 
3.87133 


.96402 
.96394 
.96386 
.96379 
.96371 


.27576 
.27607 
.27638 
.27670 
.27701 


3 
3 
3 
3 
3 


62636 
62224 
61814 
61405 
60996 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.25038 
•25066 
•25094 
.25122 
.25151 

.25179 
.25207 
.25235 
.25263 
•25291 


.96815 
.96807 
■96800 
•96793 
•96786 

•96778 
•96771 
.96764 
•96756 
.96749 


.25862 
.25893. 
.25924 
.25955 
.2598R 


3.86871 
3.86208 
3. 85745 
3. 85284 
3.84824 


-26724 
.26752 
.26780 
.26808 
.26836 


.96363 
.96355 
.96347 
.96340 
-96332 


.27732 
.27764 
.27795 
.27826 
.27858 


3 
3 
3 
3 
3 


60588 
60181 
59775 
59370 
58966 


30 

29 
28 
27 
28 


35 
36 
37 
38 
39 


.26017 
.26043 
.26079 
.26110 
.26141 


3. 84384 
3.83908 
3. 83449 
3.82992 
3-82537 


.26864 
.26892 
. 26920' 
.26948 
.26976 


-96324 
-96316 
-96308 
-96301 
.96293 


.27889 
.27921 
.27952 
.27983 
-28015 


3 
3 
3 
3 
3 


58562 
58160 
57758 
57357 
56957 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.25320 
•25348 
.25376 
.25404 
.25432 


.98742 
.96734 
•96727 
•96719 - 
•96712 


.28172 
.26203 
.26235 
.26288 
.26297 


3-82083 
3-81630 
3-81177 
3-80726 
3-80276 


-27004 
•27032 
-27060 
27088 
-27116 


.96285 
.96277 
.96269 
.96261 
-96253 


.28046 
.28077 
.28109 
.28140 
.28172 


3 
3 
3 
3 
3 

3 
3 
3 

i 

3 

3 

3. 

3. 

3. 


56557 
56159 
55761 
55364 
54968_ 

54573 
54179 
53785 
53393 
53001_ 

52609 
52219 
51829 
51441 
51053 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•25460 
•25488 
•25516 
.25545 
.25573 


.96705 
.96697 
.96690 
96682 
.96675 


.26328 
•28359 
.26390 
•26421 
.26452 


3.79827 
3.79378 
3.78931 
3-78485 
3 - 78040 


-27144 
-27172 
-27200 
.27228 
.27256 

.27284 
.27312 
.27340 
.27368 
.27396 
-27424 
-27452 
-27480 
-27508 
.2753ff 


.96246 
.96238 
.96230 
.96222 
.96214 


.28203 
.28234 
.28266 
.28297 
.28329 

.28360 
.28391 
.28423 
.28454 
.28486 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.25601 
.25629 
.25657 
.25685 
•25713 
.25741 
.25769 
.25798 
.25826 
.25854 


.96667 
.96660 
•96653 
.96645 
.96638 


.26483 
.26515 
.26546 
.26577 
.26608 

.26639 
.26670 
.26701 
.26733 
-26764 


3-77595 
3-77152 
3.76709 
3.76263 
3.75828 


.96206 
.96198 
.96190 
.96182 
.96174 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.96630 
.96623 
.96615 
.96608 
.96600 

.96593 


3.75388 
3.74950 
3.74512 
3.74075 
3-73640 


.96166 
.96158 
.96150 
.96142 
.96134 


.28517 
.28549 
.28580 
.28612 
.28643 


3. 
3. 
3. 
3. 
3. 


50666 
50279 
49894 
49509 
49125 


5 

4 
3 

2 

1 


60 


.25882 


.26795 


3-73205 


-27564 


.96126 


-28675 


-3. 


/18741 





/ 


Cos. 


Sin. 


1 Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



75^ 



744 



74^ 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
16° 17° 



f 


Sin, Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 

4 


.27564 
.27592 
.27620 
.27648 
.27676 


.96126 
.96118 
.96110 
.96102 
.96094 


.28675 
.28706 
.28738 
.28769 
.28800 


3-48741 
3.48359 
3.47977 
3.47596 
3.47216 


-29237 
-29265 
-29293 
-29321 
.29348 


•95630 
•95622 
.95613 
.95605 
•95596_ 

.95588 
•95579 
•95571 
•95562 
•95554 


-30573 
-30605 
.30637 
.30669 
.30700 


3.27085 
3^26745 
3 • 26406 
3-26067 
3^25729 

3^25392 
3^25055 
3^24719 
3 •24383 
3 • 24049 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.27704 
.27731 
.27759 
.27787 
.27815 


.96086 
.96078 
•96070 
.96062 
.96054 


•28832 
.28864 
.28895 
.28927 
.28958 


3-46837 
3.46458 
3.46080 
3-45703 
3.45327 


-29376 
- 29404 
-29432 
.29460 
.29487 


•30732 
.30764 
.30796 
•30828 
•30860 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.27843 
.27871 
.27899 
.27927 
.27955 


.96046 
.96037 
.96029 
.96021 
.96013 


.28990 
.29021 
.29053 
.29084 
.29116 


3.44951 
3.44576 
3.44202 
3.43829 
3-43456 


.29515 
-29543 
.29571 
.29599 
-29626 


•95545 
•95536 
.95528 
•95519 
-95511 


•30891 
.30923 
•30955 
•30987 
•31019 


3-23714 
3.23381 
3.23048 
3.22715 
3.22384 


50 

49 
48 
47 
46 


15 
16 
17 
18 

19. 


.27983 
.28011 
.28039 
.28067 
.28095 


.96005 
.95997 
.95989 
.95981 
.95972 


.29147 
.29179 
.29210 
.29242 
.29274 


3.43084 
3.42713 
3.42343 
3.41973 
3.41604 


-29654 
.29682 
.29710 
-29737 
.29765 


•95502 
•95493 
.95485 
.95476 
.95467 


•31051 
•31083 
•31115 
•31147 
-31178 


3.22053 
3 21722, 
3.21392' 
3.21063 
3.20734 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.28123 
.28150 
.28178 
.28206 
.28234 


.95964 
.95956 
.95948 
.95940 
.95931 


.29305 
.29337 
.29368 
.29400 
.29432 


3-41236 
3.40869 
3.40502 
3.40136 
3.39771 


.29793 
.29821 
.29849 
.29876 
.29904 


.95459 
.95450 
.95441 
.95433 
-95424 


•31210 
•31242 
•31274 
•31306 
•31338 


3.20406 
3.20079 
3.19752 
3.19426 
3.19100 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.28262 
.28290 
.28318 
.28346 
.28374 


.95923 
.95915 
.95907 
.95898 
.95890 


.29463 
.29495 
.29526 
.29558 
.29590 


3.39406 
3.39042 
3.38679 
3.38317 
3-37955 


.29932 
•29960 
•29987 
•30015 
•30043 


.95415 
.95407 
.95398 
-95389 
-95380 


•31370 
•31402 
•31434 
•31466 
•31498 


3-18775 
3.18451 
3.18127 
3.17804 
3.17483 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.28402 
.28429 
.28457 
.28485 
.28513 


.95882 
.95874 
.95865 
.95857 
.95849 


.29621 
.29653 
.29685 
.29716 
.29748 


3-37594 
3.37234 
3.36875 
3.36516 
3-36158 

3-35800 
3-35443 
3.35087 
3.34732 
3. 34377 


•30071 
•30098 
-30126 
•30154 
.30182 


.95372 
.95363 
.95354 
.95345 1 
-95337 


.31530 
•31562 
•31594 
.31626 
•31658 


3.17159 
3.16838 
3.16517 
3-16197 
3-15877 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.28541 
.28569 
.28597 
.28625 
.28652 


.95841 
.95832 
.95824 
.95816 
.95807 


.29780 
.29811 
.29843 
•29875 
.29906 


.30209 
.30237 
.30265 
•30292 
.30320 


-95328 
-95319 
.95310 
-95301 
•95293 


•31690 
•31722 
•31754 
•31786 
-31818 


3.15558 
3-15240 
3.14922 
3.14605 
3-14288 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.28680 
.28708 
.28736 
.28764 
.28792 


•95799 
.95791 
.95782 
.95774 
.95766 


.29938 
.29970 
.30001 
.30033 
.30065 


3-34023 
3-33670 
3-33317 
3.32965 
3.32614 


.30348 
.30376 
-30403 
-30431 
.30459 


-95284 
-95275 
-95266 
-95257 
.95248 


•31850 
•31882 
•31914 
•31946 
•31978 


3-13972 
3.13656 
3.13341 
3.13027 
3.12713 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.28820 
.28847 
.28875 
.28903 
.28931 

.28959 
.28987 
.29015 
.29042 
.29070 


.95757 
.95749 
.95740 
•95732 
.95724 


.30097 
.30128 
.30160 
.30192 
.30224 


3.32264 
3.31914 
3-31565 
3.31216 
3-30868 

3.30521 
3.30174 
3.29829 
3.29483 
3.29139 


.30486 
.30514 
.30542 
•30570 
.30597 


•95240 
.95231 
.95222 
.96213 
.95204 


-32010 
•32042 
•32074 
•32106 
•32139 


3.12400 
3.12087 
3.11775 
3.11464 
3-11153 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.95715 
.95707 
.95698 
.95690 
.95681 


•30255 
•30287 
.30319 
.30351 
.30382 


•30625 
.30653 
•30680 
.30708 
.30736 


.95195 
-95186 
.95177 
.95168 
.95159 


•32171 
.32203 
•32235 
.32267 
•32299 

•32331 
•32363 
-32396 
•32428 
•32460 


3-10842 
3-10532 
3-10223 
3-09914 
3-09606 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.29098 
.29126 
.29154 
.29182 
.29209 


.95673 
.95664 
.95656 
.95647 
.95639 


.30414 
.30446 
.30478 
.30509 
.30541 


3.28795 
3.28452 
3-28109 
3-27767 
3.27426 

3-27085 


.30763 
.30791 
.30819 
.30846 
.30874 


.95150 
•95142 
.95133 
.95124 
.95115 


3.09298 
3.08991 
3.08685 
3-08379 
3.08073 


5 

4 
3 

2 

1 


fio„ 


.29237 


.95630 


.30573 


.30902 


-95106 


•32492 


3.07^68 





1 ' 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


f 



73' 



745» 



73** 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
18° 19° 


/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


t 




1 
2 
3 
4 


.30902 
.30929 
.30957 
.30985 
.31012 

.31040 
.31068 
.31095 
.31123 
.31151 


.95106 
.95097 
.95088 
.95079 
.95070 


•32492 
.32524 
•32556 
.32588 
.32621 


3.07768 
3-07464 
3-07160 
3.06857 
3.06554 


•32557 
•32584 
•32612 
•32639 
•32667 
.32694 
.32722 
.32749 
.32777 
•32804 


.94552 
.94542 
■94533 
.94523 
.94514 


.34433 
•34465 
-34498 
.34530 
.34563 


2.90421 
2-90147 
2.89873 
2-89600 
2.89327 


60 

59 

58 

57 

_56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 

45 
44 
43 
42 
41 


5 
6 
7 
8 
9 


.95061 
.95052 
.95043 
.95033 
.95024 


.32653 
.32685 
.32717 
.32749 
•32782 


3-06252 
3.05950 
3.05649 
3-05349 
3-05049 


.94504 
.94495 
.94485 
.94476 
.94466 


•34596 
.34628 
•34661 
•34693 
•34726 


2 •89055 
2.88783 
2^88511 
2.88240 
2.87970 


10 

11 
12 
13 
14 


.31178 
.31206 
•31233 
.31261 
.31289 

•31316 
•31344 
•31372 
•31399 
•31427_ 


.95015 
.95006 
.94997 
.94988 
.94979 


•32814 
.32846 
.32878 
.32911 
.32943 


3-04749 
3-04450 
3-04152 
3-03854 
3-03556 


•32832 
•32859 
•32887 
.32914 
.32942 


.94457 
. 94447 
.94438 
.94428 
.94418 


•34758 
•34791 
.34824 
.34856 
.34889 


2.87700 
2.87430 
2^87161 
2 •86892 
2 •86624 

2^86356 
2^86089 
2.85822 
2.85555 
2.85289 


15 
16 
17 
18 
19 


.94970 
.94961 
.94952 
.94943 
.94933 


.32975 
.33007 
•33040 
•33072 
•33104 


3-03260 
3-02963 
3-02667 
3-02372 
3-02077 


.32969 
.32997 
.33024 
•33051 
.33079 


.94409 
.94399 
.94390 
.94380 
.94370 


.34922 
.34954 
.34987 
.35020 
.35052 


20 

21 
22 
23 
24 


•31454 
•31482 
.31510 
.31537 
.31565 


.94924 
.94915 
.94906 
.94897 
.94888 


•33136 
•33169 
.33201 
.33233 
•33266 


3.01783 
3 • 01489 
301196 
3.00903 
3^00611 


.33106 
•33134 
•33161 
•33189 
•33216 


.94?81 
.94351 
.94342 
.94332 
.94322 


.35085 
.35118 
.35150 
.35183 
.35216 


2.85023 
2.84758 
2.84494 
2^84229 
2.83965 
2.83702 
2.83439 
2.83176 
2.82914 
2 •82653 


40 

39 
38 
37 
36 

35 
34 
33 
32 
31 

30 

29 
28 
27 
36 


25 
26 
27 
28 
29 


.31593 
.31620 
.31648 
.31675 
.31703 


.94878 
.94869 
.94880 
.94851 
.94842 


•33298 
•33330 
.33363 
.33395 
.33427 


3 • 00319 
3-00028 
2^99738 
2-99447 
2-99158 


•33244 
•33271 
•33298 
•33326 
•33353 


.94313 
.94303 
•94293 
.94284 
.94274 


.35248 
.35281 
•35314 
•35346 
.35379 


30 

31 
32 
33 
34 


•31730 
.31758 
.31786 
.31813 
.31841 


.94832 
.94823 
.94814 
.94805 
.94795 


.33460 
.33492 
.33524 
.33557 
.33589 


2-98868 
2-98580 
2-98292 
2-98004 
2-97717 


•33381 
•33408 
.33436 
.33463 
.33490 


.94264 
.94254 
.94245 
.94235 
.94225 


.35412 
•35445 
•35477 
•35510 
•35543 


2.82391 
2.82130 
2.81870 
2^81610 
2.81350 


35 
36 
37 
38 
39 


•31868 
.31896 
.31923 
•31951 
•31979 


.94786 
.94777 
.94768 
.94758 
.94749 


.33621 
.33654 
.33686 
•33718 
•33751 


2-97430 
2-97144 
2-96858 
2-96573 
2.96288 


.33518 
.33545 
.33573 
.33600 
•33627 


.94215 
.94206 
.94196 
.94186 
.94176 


•35576 
•35608 
•35641 
.35674 
.35707 


2.81091 
2.80833 
2.80574 
2.80316 
2.80059 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•32006 
•32034 
•32061 
•32089 
.32116 


.94740 
•94730 
.94721 
.94712 
.94702 


•33783 
•33816 
•33848 
.33881 
•33913 


2-96004 
2-95721 
2-95437 
2.95155 
2-94872 


•33655 
.33682 
.33710 
.33737 
•33764 


.94167 
.94157 
.94147 
.94137 
.94127 


.35740 
.35772 
•35805 
•35838 
.35871 


2.79802 
2.79545 
2.79289 
2.79033 
2.78778 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•32144 
.32171 
.32199 
.32227 
•32254 


.94693 
.94684 
•94674 
•94665 
•94656 


•33945 
•33978 
•34010 
•34043 
•34075 


2-94591 
2-94309 
2.94028 
2.93748 
2-93468 


.33792 
.33819 
.33846 
.33874 
.33901 


.94118 
.94108 
.94098 
.94088 
.94078 


•35904 
•35937 
•35969 
•36002 
.36035 


2.78523 
2.78269 
2.78014 
2.77761 
2.77507 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•32282 
•32309 
•32337 
•32364 
•32392 


•94646 
•94637 
.94627 
.94618 
•94609 


•34108 
•34140 
•34173 
•34205 
34238 


2-93189 
2.92910 
2.92632 
2.92354 
2.92076 


•33929 
.33956 
.33983 
.34011 
.34038 


.94068 
.94058 
. 94049 
.94039 
.94029 

.94019 
.94009 
.93999 
•93989 
.93979 


.36068 
.36101 
.36134 
.36167 
-36199 


2.77254 
2.77002 
2.76750 
2.76498 
2.76247 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.32419 
•32447 
.32474 
•32502 
•32529 


.94599 
•94590 
.94580 
.94571 
•94561 


34270 
•34303 
•34335 
•34368 
•34400 


2.91799 
2.91523 
2.91246 
2.90971 
2-90696 


.34065 
.34093 
.34120 
.34147 
.34175 


.36232 
.36265 
.36298 
.36331 
.36364 


2.75996 
2.75746 
2^75496 
2^75246 
2^74997 


5 
4 
3 
2 
1 


60 


•32557 


•94552 


•34433 


2^90421 


•34202 


•93969 


•36397 


2^74748 


■^ 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. Tan. ) 



71^ 



746 



70' 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
30° 31° 



/ 


Sin. 


Cos. 


Tan. 1 Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


t 




1 

: 2 
3 

4 


.34202 
.34229 
.34257 
.34284 
.34311 


.93969 
.93959 
.93949 
.93939 
.93929 


.36397 
.36430 
.36463 
.36496 
.36529 


2.74748 
2.74499 
2.74251 
2.74004 
2-73756 


.35837 
.35864 
.35891 
.35918 
.35945 


.93358 
.93348 
.93337 
.93327 
.93316 


.38386 
.38420 
.38453 
.38487 
.38520 


2.60509 
2.60283 
2.60057 
2.59831 
2.59606 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.34339 
.34366 
.34393 
.34421 
.34448 


.93919 
.93909 
.93899 
.93889 
•93879 


.36562 
.36595 
.36628 
.36661 
.36694 


2.73509 
2.73263 
2.73017 
2.72771 
2.72526 


.35973 
•36000 
.36027 
.36054 
•36081 


.93306 
.93295 
•93285 
•93274 
•93264 


.38553 
.38587 
.38620 
.38654 
.38687 


2.59381 
2.59156 
2.58932 
2.58708 
2-58484 


55 
54 
53 
52 
51 


10 

7.1 
12 
13 

14 


.34475 
•34503 
•34530 
•34557 
.34584 


.93869 
•93859 
•93849 
.93839 
.93829 


.36727 
.36760 
.36793 
.36826 
•36859 


2.72281 
2.72036 
2.71792 
2.71548 
2.71305 


•36108 
.36135 
.36162 
.36190 
-36217 


•93253 
•93243 
•93232 
•93222 
•93211 


.38721 
.38754 
.38787 
.38821 
•38854 


2-58261 
2.58038 
2.57815 
2.57593 
2.57371 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


•34612 
34639 
•34666 
•34694 
•34721 


.93819 
.93809 
.93799 
.93789 
•93779 


.36892 
.36925 
.36958 
.36991 
.37024 


2.71062 
2.70819 
2.70577 
2.70335 
2-70094 


.36244 
.36271 
.36298 
.36325 
•36352 


.93201 
.93190 
.93180 
.93169 
•93159 


•38888 
•38921 
•38955 
•38988 

•39022 


2.57150 
2.56928 
2-56707 
2-56487 
2.56266 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


•34748 
•34775 
.34803 
•34830 
•34857 


.93769 
.93759 
.93748 
.93738 
.93728 


.37057 
.37090 
.37123 
.37157 
.37190 


2.69853 
2.69612 
2-69371 
2.69131 
2.68892 


•36379 
.36406 
.36434 
.36461 
•36488 


.93148 
.93137 
.93127 
.93116 
.93106 


-39055 
-39089 
.39122 
.39156 
-39190 


2-56046 
2.55827 
2.55608 
2.55389 
2.55170 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•34884 
•34912 
.34939 
.34966 
.34993 


.93718 
.93708 
.93698 
.93688 
.93677 


.37223 
.37256 
.37289 
.37322 
.37355 


2.68653 
2.68414 
2.68175 
2.67937 
2.67700 


•36515 
•36542 
.36569 
.36596 
.36623 


.93095 
-93084 
.93074 
.93063 
.93052 


-39223 
39257 
.39290 
.39324 
.39357 


2.54952 
2.54734 
2.54516 
2.54299 
2.54082 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.35021 
.35048 
.35075 
•35102 
•35130 


.93667 
.93657 
•93647 
.93637 
•93626 


.37388 
.37422 
•37455 
•37488 
•37521 


2.67462 
2.67225 
2.66989 
2.66752 
2^66516 


.36650 
.36677 
•36704 
.36731 
•36758 


.93042 
.93031 
•93020 
•93010 
.92999 


39391 
.39425 
•39458 
•39492 
•39526 


2.53865 
2.53648 
2-53432 
2-53217 
2-53001 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•35157 
•35184 
•35211 
•35239 
•35266 


•93616 
.93606 
.93596 
.93585 
•93575 


•37554 
•37588 
•37621 
.37654 
.37687 


2.66281 
2.66046 
2.65811 
2.65576 
2.65342 


•36785 
•36812 
•36839 
•36867 
.36894 


.92988 
.92978 
•92967 
•92956 
•92945 


•39559 
•39593 
•39626 
•39660 
■39694 


2.52786 
2.52571 
2.52357 
2.52142 
2-51929 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•35293 
•35320 
•35347 
•35375 
.35402 


.93565 
.93555 
.93544 
.93534 
.93524 


.37720 
.37754 
•37787 
•37820 
•37853 


2.65109 
2.64875 
2.64642 
2.64410 
2.64177 


.36921 
•36948 
•36975 
•37002 
•37029 


•92935 
.92924 
•92913 
•92902 
•92892 


•39727 
•39761 
•39795 
•39829 
•39862 


2.51715 
2.51502 
2.51289 
2.51076 
2-50864 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•35429 
•35456 
•35484 
•35511 
•35538 


.93514 
.93503 
.93493 
.93483 
.93472 


.37887 
.37920 
•37953 
•37986 
•38020 


2.63945 
2.63714 
2 •63483 
2^63252 
2^63021 


•37056 
•37083 
•37110 
•37137 
•37164 


•92881 
•92870 
•92859 
.92849 
.92838 


-39896 
-39930 
-39963 
-39997 
.40031 


2-50652 
2 - 50440 
2-50229 
2-50018 
2-49807 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•35565 
.35592 
•35619 
•35647 
•35674 


.93462 
.93452 
.93441 
.93431 
•93420 


.38053 
.38086 
.38120 
•38153 
•38186 


2.62791 
2.62561 
2.62332 
2^62103 
2-61874 


.37191 
.37218 
.37245 
.37272 
•37299 


.92827 
.92816 
.92805 
.92794 
•92784 


-40065 
-40098 
-40132 
-40166 
-40200 


2.49597 
2.49386 
2.49177 
2.48967 
2.48758 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•35701 
•35728 
•35755 
•35782 
•35810 


.93410 
.93400 
.93389 
.93379 
.93368 


.38220 
.38253 
.38286 
.38320 
•38353 


2.61646 
2.61418 
2.61190 
2 •60963 
2 •60736 


•37326 
•37353 
•37380 
•37407 
•37434 


.92773 
•92762 
•92751 
• 92740 
•92729 


-40234 
-40267 
-40301 
.40335 
-40369 


2.48549 
2.48340 
2.48132 
2-47924 
2-47716 


5 

4 
3 
2 
1 


60 


•35837 


.93358 


•38386 


2.60509 


•37461 


.92718 


.40403 


2-47509 





t 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


( 



69' 



747 



68' 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 



33' 



33' 



/ 


Sin. 


Cos.. 


Tan. 


Cot. 


Sin. 1 Cos. 


Tan. 


Cot. 


* 




1 

2 
3 
4 


.37461 
.37488 
.37515 
.37542 
.37569 


.92718 
.92707 
.92697 
.92686 
.92675 


•40403 
.40436 
.40470 
•40504 
.40538 


2.47509 
2.47302 
2.47095 
2.46888 
2.46682 


•39073 
.39100 
-39127 
-39153 
-39180 


-92050 
-92039 
.92028 
.92016 
.92005 


-42447 
.42482 
-42516 
-42551 
-42585 


2^35585 
2-35395 
2-35205 
2-35015 
2-34825 


60 

59 
58 
57 
56 


5 • 

6 

7 

8 

9 


.37595 
.37622 
.37649 
.37676 
.37703 


.92664 
.92653 
.92642 
.92631 
92620 


.40572 
.40606 
.40640 
•40674 
.40707 


2.46476 
2.46270 
2.46065 
2.45860 
2.45655 


-39207 
-39234 
-39280 
-39287 
.39314 


.91994 
-91982 
-91971 
-91959 
-91948 


-42619 
-42654 
-42688 
•42722 
•42757 


2-34636 
2-34447 
2.34258 
2-34069 
2-33881 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.37730 
.37757 
.37784 
.37811 
.37838 


.92609 
.92598 
.92587 
.92576 
.92565 


•40741 
•40775 
.40809 
.40843 
.40877 


2.45451 
2.45246 
2.45043 
2.44839 
2.44636 

2.44433 
2-44230 
2.44027 
2.43825 
2.43623 


-39341 
-39367 
-39394 
.39421 
-39448 


.91936 
-91925 
.91914 
.91902 
.91891 


•42791 
•42826 
.42860 
.42894 
.42929 


2-33693 
2-33505 
2-33817 
2-33130 
2-32943 


50 

49 
48 
47 
46 


15 
16 
17 
18 

19 


.37865 
.37892 
.37919 
.37946 
.37973 


.92554 
.92543 
.92532 
.92521 
.92510 


.40911 
.40945 
.40979 
.41013 
.41047 


-39474 
.39501 
•39528 
•39555 
.39581 

•39608 
■39635 
•39661 
-39688 
-39715 


.91879 
-91868 
-91856 
•91845 
.91833 

.91822 
.91810 
-91799 
-91787 
-91775 


•42963 
.42998 
.43032 
.43067 
.43101 


2-32756 
2-32570 
2-32383 
2-32197 
2-32012 


45 
44 
43 
42 
41 


30 

21 
22 
23 

24 


.37999 
.38026 
.38053 
.38080 
.38107 

.38134 
.38161 
.38188 
.38215 
.38241 


.92499 
.92488 
.92477 
.92466 
.92455 


•41081 
•41115 
•41149 
•41183 
•41217 


2.43422 
2.43220 
2.43019 
2.42819 
2.42618 


.43136 
.43170 
.43205 
.43239 
-43274 


2-31826 
2-31641 
2-31456 
2-31271 
2.31086 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.92444 
.92432 
.92421 
.92410 
.92399 


•41251 
.41285 
.41319 
.41353 
.41387 


2.42418 
2.42218 
2.42019 
2.41819 
2.41620 


•39741 
.39768 
.39795 
.39822 
•39848 


-91764 
-91752 
.91741 
.91729 
•91718 

•91706 
.91694 
.91683 
.91671 
.91660 


.43308 
.43343 
.43378 
-43412 
-43447 


2-30902 
2-30718 
2.30534 
2-30351 
2-30167 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.38268 
.38295 
.38322 
.33349 
.38376 


.92388 
.92377 
•92366 
.92355 
.92343 


.41421 
.41455 
.41490 
.41524 
•41558 


2 41421 
2-41223 
2-41025 
2.40827 
2.40629 


•39875 
•39902 
.39928 
•39955 
.39982 


-43481 
-43516 
-43550 
-43585 
.43620 


2.29984 
2.29801 
2-29619 
2-29437 
2-29254 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.38403 
.38430 
.38456 
.38483 
.38510 


.92332 
.92321 
.92310 
.92299 
.92287 


.41592 
.41626 
.41660 
.41694 
.41728 


2-40432 
2.40235 
2.40038 
2.39841 
2.39645 


.40008 
•40035 
■40062 
•40088 
■40115 

.40141 
•40168 
•40195 
•40221 
.40248 


.91648 
.91636 
.91625 
.91613 
.91601 


•43654 
.43689 
.43724 
.43758 
■43793 


2.29073 
2-28891 
2-28710 
2.28528 
2.28348 


25 
24 
23 
22 
21 


40 

41 
42 
43 

44 


.38537 
.38564 
.38591 
.38617 
.38644 

.38671 
.38698 
.3??725 
.38752 
.38778 


.92276 
.92265 
.92254 
.92243 
.92231 


.41763 
.41797 
.41831 
•41865 
.41899 


2.39449 
2.39253 
2.39058 
2.38863 
2.38668 


.9*1590 
.91578 
.91566 
•91555 
.91543 


.43828 
.43862 
.43897 
.43932 
•43966 


2.28167 
2.27987 
2.27806 
2-27626 
2-27447 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.92220 
.92209 
.92198 
.92186 
.92175 


•41933 
•41968 
.42002 
.42036 
.42070 


2.38473 
2.38279 
2.38084 
2.37891 
2.37697 


.40275 
40301 
.40328 
.40355 
•40381 


.91531 
.91519 
.91508 
.91496 
.91484 


.44001 
.44036 
.44071 
.44105 
.44140 


2.27267 
2.27088 
2.26909 
2.26730 
2.26552 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.38805 
.38832 
.38859 
.38886 
.38912 


.92164 
.92152 
.92141 
•92130 
.92119 


.42105 
•42139 
.42173 
.42207 
.42242 


2.37504 
2.87311 
2.37118 
2-36925 
2.36733 

2-36541 
2.36349 
2-36158 
2-35967 
2.35776 


.40408 

•40434 

•40461 

.40488 

^0_51^ 

-40541 

.40567 

.40594 

•40621 

■40347 


.91472 
.91461 
•91449 
•91437 
•91425 


-44175 
-44210 
- 44244 
-44279 
.44314 


2.26374 
2-26196 
2-26018 
2.25840 
2.25663 


10 

9 
8 

7 
6 


55 
56 
57 
58 
59 


.38939 
.38966 
.38993 
.39020 
.39046 


.92107 
.92096 
.92085 
.92073 
.92062 

.92050 


.42276 
.42310 
.42345 
.42379 

.42413 


.91414 
.91402 
.91390 
.91378 
.91366 

-91355 


-44349 
.44384 
.44418 
.44453 
.44488 


2.25486 
2.25309 
2.25132 
2.24956 
2.24780 


5 
4 
3 
2 
1 


60 


.39073 


•42447 


2.35585 


•40674 


•44523 


2-24604 


-_0 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



^t 



748 



66° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 



2^' 



35^ 



t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 

.42262 
.42288 
-42315 
-42341 
.42367 


Cos. 


Tan. Cot. 


/ 




1 

2 
3 
4 


.10674 
.40700 
.40727 
.40753 
.40780 


.91355 
.91343 
.91331 
.91319 
.9x307 


.44523 
.44558 
•44593 
.44627 
.44662 


2.24604 
2.24428 
2.24252 
2.24077 
2.239U2 


.90631 
.90618 
-90606 
.90594 
.90o82 


-46631 
.46666 
.46702 
-46737 
.46772 


2.14451 
2.14288 
2.14125 
2.13963 
2-13801 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.40806 
.40833 
-40860 
.40886 
.40913 


.91295 
.91283 
.91272 
.91260 
.91248 

.91236 
.91224 
.91212 
.91200 
•91188 


.44697 
.44732 
.44767 
.44802 
•44837 


2.23727 
2.23553 
2.23378 
2.23204 
2-23030 


.42394 
.42420 

.42446 
- 424^73 
•42499 


.90569 
.00557 
-90545 
.90532 
-90520 


-46808 
.46843 
.46879 
.46914 
.4695Q 


2.13839 
2.13477 
2.13316 
2.13154 
2.12993 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.40939 
.40966 
.40992 
.41019 
•41045 


.44872 
.44907 
.44942 
.44977 
.45012 


2.22857 
2.22683 
2.22510 
2.22337 
2.22164 


.42525 
-42552 
-42578 
-42604 
-42631 


-90507 
-90495 
.90483 
-90470 
.90458 

•90446 
-90433 
-90421 
-90408 
-90396 

-90383 
-90371 
-90358 
-90346 
•90334 


.46985 
.47021 
.47056 
.47092 
.47128 


2.12832 
2.12671 
2.12511 
2.12350 
2 12190 


50 

49 
48 
47 
46 


15 
16 
17 
18 
1? , 


.41072 
.41098 
.41125 
.41151 
.41178 


.91178 
•91164 
•91152 
.91140 
.91128 


.45047 
.45082 
.45117 
.45152 
•45187 


2.21992 
2.21819 
2.21647 
2.21475 
2.21304 


.42657 
.42683 
.42709 
.42736 
.42762 


.47163 
.47199 
.47234 
.47270 
-47305 


2.12030 
2.11871 
2.11711 
2.11552 
2.11392 


45 
44 
43 
42 


30 

21 
22 
23 
24 


.41204 
.41231 
.41257 
.41284 
•41310 


.91116 
.91104 
.91092 
.91080 
91068 


.45222 
.45257 
.45292 
•45327 
.45362 


2-21132 
2.20961 
2.20790 
2.20619 
2.20449 


.42788 

.42815 
.42841 
.42867 
-42894 


-47341 
-47377 
.47412 
.47448 
.47483 


2.11233 
2.11075 
2.10916 
2.10758 
2.10600 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.41337 
.41363 
.41390 
.41416 
•41443 


•91056 
•91044 
•91032 
•91020 
.91008 


.45397 
•45432 
.45467 
.45502 
•45538 


2.20278 
2-20108 
2.19938 
2.19769 
2.19599 


-42920 
-42946 
-42972 
-42999 
.43025 


.90321 
.90309 
•90296 
-90284 
-90271 


•47519 
.47555 
.47590 
-47626 
.47662 


2^10442 
2.10284 
2.10126 
2.09969 
2.09811 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.41469 
.41496 
.41522 
.41549 
.41575 


.90998 
.90984 
.90972 
.90960 
.90948 


•45573 
.45608 
.45643 
.45678 
•45713 


2.19430 
2.19261 
2.19092 
2.18923 
2.18755 


•43051 
•43077 
•43104 
-43130 
.43156 


-90259 
-90246 
.90233 
.90221 
,90208 


.47698 
. 7733 
.47769 
.47805 
•47840 


2.09654 
2-09498 
2-09341 
2-09184 
2.09028 


30 

29 
28 
27 
26 


35 
36 
37 
38 
38 


.41602 
.41628 
.41655 
•41681 
•41707 


.90936 
•90924 
•90911 
•90899 
•90887 


.4571:8 
.45784 
.45819 
.45854 
•45889 


2^18587 

2-18419 
2.18251 
2-18084 
2.17916 


•43182 
•43209 
•43235 
-43261 
•43287 


-90196 
-90183 
-90171 
-90158 
-90146 


•47876 
•47812 
.47948 
.47984 
• 4801.9 


2-08872 
2-08716 
2.08560 
2-08405 
2.08250 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•41734 
•41760 
•41787 
•41813 
•41840 


•90875 
•90863 
•90851 
•90839 
•90826 


.45924 
.45960 
.45995 
.46030 
.46065 


2-17749 
2-17582 
2-17416 
2.17249 
2.17083 


•43313 
•43340 
43366 
•43392 
.43418 


-90133 
-90120 
-90108 
-90095 
-90082 


•48055 
.4809J 
•48127 
•48163 

■ ABirn 


2-08094 
2-07939 
2.07785 
2.07630 
2.07476 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•41866 
.41892 
.41919 
.41945 
.41972 


•90814 
.90802 
.90790 
.90778 
.90766 


.46101 
.46136 
•46171 
.46206 
.46242 


2-16917 
2-16751 
2.16585 
2.16420 
2.16255 


• 43445 
43471 
•43497 
-43523 
•43549 


-90070 
-90057 
-90045 
-90032 
.90019 


.48234 
.48270 
.48306 
.48342 
•48378 


2.07321 
2-07167 
2-07014 
2-06860 
2. 06706 


15 
14 
13 
. 12 
11 


50 

51 
52 
53 
54 


.41998 
.42024 
•42051 
.42077 
•42104 


.90753 
.90741 
.90729 
.90717 
90704 


.46277 
.46312 
•46348 
•46383 
.46418 


2.16090 
2.15925 
2.15760 
2.15596 
2-15432 


-43575 
•43602 
-43628 
.43854 
■43680 


-90007 
-89994 
-89981 
-89968 
•89956 


.48414 
-48450 
.48486 
.48521 
•48557 


2-06553 
2-06400 
2.06247 
2.06094 
2.05842 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.42130 
.42156 
.42183 
.42209 
•42235 


.90692 
•90680 
.90668 
•90655 
•90643 


.46454 
.46489 
•46525 
.46560 
•46595 


2-15268 
2-15104 
2.14940 
2 14777 
2-14614 


.43706 
-43733 
-43759 
-43785 
.43811 


•89943 
.89930 
.89918 
•89905 
.89892 


.48593 
.48629 
.48665 
.48701 
-48737 


2.05790 
2.05637 
2.05485 
2.05333 
2.05182 


5 

4 
3 
2 
1 


60 


.42262 


•90631 


•46631 


2.14451 


-43837 


.89879 


.48773 


2.05030 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



65^ 



749 



64' 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
36° 37° 



t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


.43837 
.43863 
.43889 
.43916 
.43942 


.89879 
.89867 
.89854 
.89841 
.89828 


.48773 
.48809 
.48845 
.48881 
.48917^ 

.48953 
.48989 
.49026 
.49062 
.49098_ 

.49134 
.49170 
.49206 
.49242 
.49278 

.49315 
.49351 
.49387 

.49423 
.49459 


2.05030 
2.04879 
2.04728 
2.04577 
2.04426 
2.04276 
2.04125 
2.03975 
2.03825 
2.03675 


.45399 
.45425 
.45451 
.45477 
.45503 


.89101 
.89087 
.89074 
.89061 
.89048 


.50953 
.50989 
.51026 
.51063 
.51099 


1.96261 
1.96120 
1.95979 
1.95838 
1.95698 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.43968 
.43994 
.44020 
.44046 
.44072 


.89816 
.89803 
.89790 
.89777 
.89764 


.45529 
.45554 
.45580 
.45606 
.45632 


.89035 
.89021 
.89008 
.88995 
.88981 


.51136 
.51173 
.51209 
.51246 
.51283_ 

.51319 
.51356 
.51393 
.51430 
.51467 


1.95557 
1.95417 
1.95277 
1.95137 
1.94997 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.44098 
.44124 
.44151 
.44177 
.44203 


.89752 
.89739 
.89726 
.89713 
.89700 

.89687 
.89674 
.89662 
.89649 
.89636 


2.03526 
2.03376 
2.03227 
2.03078 
2.02929 


.45658 
.45684 
.45710 
.45736 
.45762 


.88968 
.88955 
.88942 
.88928 
.88915 

.88902 
.88888 
.88875 
.88862 
•88848 


1.94858 
1.94718 
1.94579 
1 . 94440 
1.94301 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.44229 
.44255 
.44281 
.44307 
.44333 


2.02780 
2.02631 
2.02483 
2.02335 
2.02187 


.45787 
.45813 
.45839 
.45865 
.45891 


.51503 
.51540 
.51577 
.51614 
.51651 


1.94162 
1.94023 
1.93885 
1.93746 
1.93608 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.44359 
.44385 
.44411 
.44437 
.44464 


.89623 
.89610 
.89597 
.89584 
.89571 


.49495 
.49532 
.49568 
.49604 
.49640 


2.02039 
2.01891 
2.01743 
2.01596 
2.01449 


.45917 
.45942 
.45968 
•45994 
.46020 


.88835 
.88822 
.88808 
.88795 
.88782 

.88768 
.88755 
.88741 
.88728 
.88715 


.51688 
.51724 
.51761 
.51798 
.51835 


1.93470 
1.93332 
1.93195 
1.93057 
1.92920 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.44490 
.44516 
.44542 
.44568 
.44594 


.89558 
.89545 
.89532 
.89519 
.89506 

. 89493 
.89480 
.89467 
.89454 
.89441 


.49677 
.49713 
.49749 
.49786 
.49822 


2.01302 
2.01155 
2.01008 
2.00862 
2.00715 


.46046 
.46072 
.46097 
.46123 
•46149 


.51872 
.51909 
.51946 
.51983 
.52020 


1.92782 
1.92645 
1.92508 
1.92371 
1.92235 


35 
34 
33 
32 
31 


30 

81 
32 
33 
34 


.44620 
.44646 
.44672 
.44698 
.44724 


.49858 
.49894 
.49931 
.49967 
. 50004_ 

.50040 
.50076 
.50113 
.50149 
.50185 


2.00569 
2-00423 
2.00277 
2.00131 
1.99986 


.46175 
.46201 
.46226 
.46252 
.46278 


.88701 
.88688 
.88674 
.88661 
.88647 


.52057 
.52094 
.52131 
.52168 
.52205 

.52242 
.52279 
.52316 
.52353 
.52390 


1.92098 
1.91962 
1.91826 
1.91690 
1.91554 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.44750 
.44776 
.44802 
.44828 
.44854 


.89428 
.89415 
.89402 
.89389 
.89376 


1.99841 
1.99695 
1.99550 
1.99406 
1.99261 


.46304 
.46330 
.46355 
.46381 
•46407 


.88634 
.88620 
.88607 
.88593 
.88580 


1.91418 
1.91282 
1.91147 
1.91012 
1.90876 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.44880 
.44906 
.44932 
.44958 
•44984 

.45010 
.45036 
.45062 
.45088 
.45114 


.89363 
.89350 
.89337 
.89324 
.89311 


.50222 
.50258 
.50295 
.50331 
.50368 


1.99116 
1.98972 
1.98828 
1.98684 
1.98540 


.46433 
.46458 
.46484 
.46510 
.46536 


.88566 
.88553 
.88539 
.88526 
.88512 


.52427 
.52464 
.52501 
.52538 
.52575 

. 52613 
.52650 
.52687 
.52724 
.52761 


1.90741 
1-90607 
1.90472 
1.90837 
1.90203 


30 

19 
18 
17 
16 


45, 1 

46 

47 

48 

49 


.89298 
.89285 
.89272 
.89259 
.89245 


.50404 
.50441 
.50477 
.50514 
.50550 


1.98396 
1.98253 
1.98110 
1.97966 
1.97823 


.46561 
.46587 
•46613 
•46639 
.46664 

.46690 
.46716 
.46742 
.46767 
.46793 


.88499 
.88485 
.88472 
.88458 
.88445 


1.90069 
1.89935 
1.89801 
1.89667 
1.89533 


15 

14 
13 
12 
11 


50 

51 
52 
53 
54 


.45140 
.45166 
.45192 
.45218 
.45243 


.89232 
.89219 
.89206 
.89193 
.89180 

.89167 
.89153 
.89140 
.89127 
.89114 


.50587 
.50623 
50660 
.50696 
.50733 

.50769 
•50806 
•50843 
•50879 
50916 


1.97681 
1.97538 
1.97395 
1.97253 
_1_^711L 
1.96969 
1.96827 
1.96685 
1.96544 
1.96402 


.88431 
« 88417 
.88404 
.88390 
.88377 


.52798 
.52836 
.52873 
.52910 
.52947 


1.89400 
1.89266 
1.89133 
1.89000 
1.88867 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.45269 
.45295 
.45321 
.45347 
.45373 


.46819 
.46844 
.46870 
.46896 
•46921 


.88363 
.88349 
.88336 
.88322 
.88308 


.52985 
.53022 
.53059 
•53096 
.53134 


1.88734 
1.88602 
1.88469 
1.88337 
1.88205 


5 

4 
3 

2 

1 


60 


.45399 


.89101 


•50953 


1.96261 


•46947 


.88295 


.53171 


1.88073 





/ 


Cos. Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



63^ 



750 



63' 



O'ABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
28° 39° 


1 
2 
3 
4 
5 
6 
7 
8 
9 

10 

11 
12 
13 

14 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 


.46947 
.46973 
.46999 
.47024 
.47050 


.88295 
.88281 
.88267 
.88254 
.88240 


.53171 
.53208 
53246 
.53283 
.53320 

.53358 
.53395 
.53432 
53470 
.53507 


1. 88073 
1.87941 
1.87809 
1.87677 
1.87546 


•48481 
•48506 
.48532 
.48557 
.48583 


87462 
.87448 
.87434 

87420 
.87406 


.55431 
•55469 
•55507 
•55545 
•55583 


1 80405 
1.80281 
1.80158 
1.80034 
1.79911 


60 

59 
58 
57 
56 


.47076 
.47101 
.47127 
.47153 
.47178 


88226 

88213 

.88199 

.88185 

88172 


1.87415 
1.87283 
1.87152 
1.87021 
1.86891 


.48608 
.48634 
.48659 
48684 
•48710 


.87391 
•87377 
.87363 
.87349 
.87335 


•55621 
.55659 
.55697 
.55736 
.55774 




79788 

79665 

.79542 

•79419 

•79296 


55 
54 
53 
52 
51 


.47204 
.47229 
.47255 
.47281 
.47306 


.88158 
.88144 
.88130 
.88117 
.88103 


.53545 
.53582 
.53620 
•53657 
.53694 


1.86760 
1.86630 
1.86499 
1.86369 
1.86239 


.48735 
.48761 
=48786 
.48bll 
•48837 


•87321 
.87306 
.87292 
.87278 
•87264 


.55812 
.55850 
.55888 
.55926 
.55964 




•79174 
7905] 
•78929 
.78807 
.78685 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19, 


.47332 
•47358 
.47383 
.47409 
.47434 


.88089 
.88075 
.88062 
.88048 
.88034 


.53732 
.53769 
.53807 
.53844 
.53882 

.53920 
.53957 
.53995 
.54032 
.54070 


1.86109 
1.85979 
1.85850 
1.85720 
1.85591 


.48862 
.48888 
.48913 
.48938 
•48964 


.87250 
.87235 
•87221 
.87207 
.87193 


.56003 
.56041 
.56079 
.56117 
.56156 




•78563 
.78441 
.78319 
.78198 
•78077 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


.47460 
.47486 
.47511 
.47537 
.47562 


.88020 
•88008 
.87993 
.87979 
.87965 


-1.85462 
i. 85333 
1.85204 
1.85075 
1.84946 


•48989 
.49014 
.49040 
.49065 
•49090 


.87178 
.87164 
.87150 
.87136 
.87121 


.56194 
.56232 
.56270 
.56309 
.56347 




•77955 
•77834 
•77713 
.77592 
•77471 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.47588 

•47614 
•47639 
.47665 
.47690 


.87951 
.87937 
.87923 
.87909 
.87896 


.54107 
.54145 
•54183 
-54220 
•54258 


1.84818 
1.84689 
1.84561 
1.84433 
1.84305 


•49116 
.49141 
.49166 
.49192 
•49217 


.87107 
87093 
.87079 
.87064 
.87050 


.56385 
•56424 
.56462 
.56501 
•56539 




.77351 
•77230 
77110 
.76990 
•76869 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.47716 
.47741 
.47767 
.47793 
.47818 


.87882 
.87868 
.87854 
.87840 
.87826 


.54296 
.54333 
.54371 
. 54409 
• 54446 


1.84177 
1.84049 
1.83922 
1.83794 
1.83667 


•49242 
•49268 
•49293 
•49318 
.49344 


.87036 
.87021 
•87007 
.86993 
.86978 


.56577 
.56616 
.56654 
.56693 
.58731 




.76749 
.76629 
.76510 
.76390 
•76271 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.47844 
.47869 
.47895 
.47920 
.47946 


.87812 
.87798 
•87784 
.87770 
.87756 


. 54484 
.54522 
.54560 
.54597 
.54635 


1.83540 
1.83413 
1.83286 
1.83159 
1.83033 


.49369 
.49394 
•49419 
.49445 
•49470 


.86964 
.86949 
.86935 
.86921 
•86906 


.56769 
•56808 
•56846 
.56885 
.56923 




.76151 

.76032 

.75913 

75794 

75675 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.47971 
.47997 
•48022 
•48048 
•48073 


.87743 
.87729 
.87715 
.87701 
.87687 


.54673 
.54711 
. 54748 
.54786 
•54824 


1.82906 
1.82780 
1.82654 
1.82528 
1.82402 


•49495 
•49521 
•49546 
•49571 
.49596 


.86892 
.86878 
.86863 
.86849 
.86834 


.56962 
•57000 
•57039 
•57078 
.57116 




75556 
75437 
75319 
75200 
75082 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•48099 
•48124 
.48150 
.48175 
.48201 


.87673 
.87659 
.87645 
.87631 
.87617 


.54862 
.54900 
.54938 
.54975 
.55013 


1.82276 
1.82150 
1.82025 
1.81899 
1.81774 


.49622 
.49647 
.49672 
.49697 
•49723 


•86820 
.86805 
.86791 
.86777 
.86762 


.57155 
.57193 
•57?32 
•57271 
.57309 




74964 
74846 
74728 
74610 
74492 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.48226 
.48252 
48277 
.48303 
.48328 


.87603 
.87589 
.87575 
.87561 
•87546 


.55051 
.55089 
.55127 
.55165 
•55203 


1.81649 
1.81524 
1.81399 
1.81274 
1.81150 


.49748 
.49773 
.49798 
.49824 
.49849 


.86748 
.86733 
.86719 
.86704 
•86690 


•57348 
•57386 
.57425 
•57464 
.57503 


I 


74375 
.4257 
74140 
74022 
73905 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.48354 
.48379 
.48405 
.48430 
.48456 


•87532 
.87518 
.87504 
•87490 
•87476 


.55241 
.55279 
.55317 
.55355 
.55393 


1.81025 
1.80901 
1.80777 
1.80653 
1.80529 


.49874 
.49899 
.49924 
.49950 
•49975 


.86675 
.86661 
.86646 
.86632 
•86617 


•57541 
.57580 
.57619 
.57657 
•57696 




73788 
73671 
73555 
73438 
73321 


5 

4 
3 

2 

1 


60 


•48481 

Cos. 


•87462„, 

Sin. 


•55431 


1.80405 


•50000 


•86603 


•57735 


1. 


73205 





f 


Cot. 


Tan. 


Cos, 


Sin. 


Cot. 


Tan. 1 


7 






61 


L^ 


75 


1 


6 


0° 









TABLE IX.- 


-NATURAL SINES. COSINES, TANGENTS, AND COTANGENTS | 
30° 31° 1 


1 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


1 




1 
2 
3 
4 


.50000 
.50025 
.50050 
.50076 
.50101 


.86603 
.86588 
.86573 
.86559 
.86544_ 

.86530 
.86515 
.86501 
.86486 
.86471 

.86457 
.86442 
.86427 
.86413 
.86398 


.57735 
.57774 
.57813 
.57851 
.57890 


1.73205 
1.73089 
1.72973 
1.72857 
1.72741 


.51504 
.51529 
.51554 
.51579 
.51604 


.85717 
.85702 
.85687 
.85672 
.85657 


•60086 
•60126 
•60165 
•60205 
•60245 


1.66428 
1.66318 
1.66209 
1.66099 
1.65990 


60 

59 
58 
57 
56f 

55 
54 
53 
52 
51 

50 

49 
48 
47 
43 


5 
6 
7 
8 
9 


.50126 
.50151 
.50176 
.50201 
.50227 


.57929 
.57968 
.58007 
.58046 
.58085 


1.72625 
1.72509 
1.72393 
1.72278 
1.72163 


.51628 
.51653 
.51678 
.51703 
.51728 


•85642 
.85627 
.35612 
.85597 
.85582 


•60284 
•60324 
•60364 
.60403 
.60443 


.1^65881 
1.65772 
1.65663 
1-65554 
1.65445 


10 

11 

12 
13 

14. 


.50252 
.50277 
.50302 
.50327 
.50352 


.58124 
.58162 
.58201 
.58240 
.58279 


1.72047 
1.71932 
1.71817 
1.71702 
1.71588 


.51753 
.51778 
.51803 
.51828 
.51852 


•85567 
.85551 
.85536 
.85521 
.85506 


.60483 
.60522 
.60562 
.60602 
•60642 


1.65337 
1.65228 
1.65120 
1.65011 
1.64903 


15 
16 
17 
18 

19 


.50377 
.50403 
.50428 
.50453 
.50478 


.86384 
.86369 
.86354 
.86340 
.86325 


.58318 
.58357 
.58396 
.58435 
.58474 


1.71473 
1.71358 
1.71244 
1.71129 
1.71015 


.51877 
.51902 
.51927 
.51952 
•51977 


.85491 
.85476 
.85461 
.85446 
.85431 


.60681 
.60721 
.60761 
.60801 
.60841 


1.64795 
1.64687 
1.64579 
1.64471 
1-64363 


45 
44 
43 
42 
-41 
40 
39 
38 
37 
36 


20 

21 
22 
23 
24 


.50503 
.50528 
•50553 
.50578 
.50603 


.86310 
.86295 
.86281 
.86266 
.86251 


.58513 
.58552 
.58591 
.58631 
.58670 


1.70901 
1.70787 
1.70673 
1-70560 
1.70446 


.52002 
.52026 
•52051 
•52076 
•52101 


.85416 
.85401 
.85385 
.85370 
.85355 

.85340 
.85325 
.85310 
•85294 
.85279 


.60881 
.60921 
.60960 
.61000 
•61040 


1.64256 
1.64148 
1-64041 
1.63934 
1.63826 


25 
26 
27 
28 
29 


.50628 
.50654 
.50679 
. 50704 
.50729 


.86237 
.86222 
o 86207 
.86192 
.86178 


58709 
.58748 
.58787 
.58826 
.58865 


1.70332 
1.70219 
1.70106 
1.69992 
1.69879 


•52126 
•52151 
•52175 
.52200 ! 
.52225 i 


.61080 
.61120 
.61160 
.61200 
.61240 


1.63719 
1.63612 
1.63505 
1.63398 
1.63292 


35 
34 
33 
32 
31 

30 

29 
28' 
27 
26 


30 

31 
32 
33 
"34 


.50754 

.50779 

•50804 

50829 

50854 


.86163 
.86148 
.86133 
.86119 
.86104 


.58905 
.58944 
.58983 
.59022 
.59061 


1-69766 
1.69653 
1.69541 
1.69428 
1.69316 


.52250 
.52275 
-52299 
•52324 
•52349 


.85264 
.85249 
.85234 
.85218 
.85203 


.61280 
.61320 
.61360 
.61400 
.61440 


1.63185 
1.63079 
1.62972 
1.62866 
1.62760 


35 
36 
37 
38 
39„ 


.50879 
•50904 
.50929 
.50954 
.50979 


.86089 
.86074 
.86059 
.86045 
.86030 


.59101 
.59140 
.59179 
59218 
.59258 


1.69203 
1.69091 
1.68979 
1.68866 
1.68754 


•52374 
•52399 
. 52423 
. 52448 
•52473 


.85188 
.85173 
.85157 
.85142 
.85127 


.61480 
.61520 
.61581 
.61601 
.61641 


1.62654 
1.62548 
1.62442 
1.62336 
1.62230 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.51004 

.51029 

.51054 

51079 

51104 


.86015 
.86000 
85985 
•85970 
.85956 


.59297 
.59336 
.59376 
.59415 
.59454 


1.68643 
1.68531 
1.68419 
1.68308 
1.68196 


•52498 
•52522 
•52547 
.52572 
•52597 


.85112 
.85096 
.85081 
.85066 
.85051 


.61681 
•61721 
.61761 
.61801 
.61842 


1.62125 
3.62019 
1.61914 
1.61803 
1.61703 


20 

19 
18 
171 
16! 


45 
46 
47 
48 
49 


.51129 
.51154 
.51179 
.51204 
.51229 


.85941 
.85926 
.85911 
.85896 
.85881 


.59494 
.59533 
.59573 
.59612 
.59651 


1.68085 
1.67974 
1.67863 
1.67752 
1.67641 


.52621 
.52646 
.52671 
.52696 
.52720 


.85035 
.85020 
.85005 
•84989 
•84974 

.84959 
. 84943 
.84928 
.84913 
84897 


.61882 
.61922 
•61962 
. 62003 
.62043 


1.61598 
1.61493 
1.61388 
1.61283 
1.61179 


15 
14 
13 1 
12 
11 


50 

51 
52 
53 
54 


.51254 
.51279 
.51304 
.51329 
.51354 


.85866 
.85851 
.85836 
.85821 
.85806 


.59691 
•59730 
.59770 
.59809 
•59849 


1.67530 
1.67419 
1.67309 
1. 67198 
1.67088 

1.66978 
1.66867 
1.66757 
1.66647 
1.66538 

1.66428 


.52745 
.52770 
.52794 
.52S19 
.52844 


.62083 
•62124 
.62164 
.62204 
.62245 


1.61074 
1.60970 
1- 60865 
1.60761 
1.60657 


10 

9 
8 
7 
6 

5 
4 
3 
2 
1 



55 
56 
57 
58 
59 


.51379 
51404 
.51429 
.51454 
.51479 


.85792 
.85777 
.85762 
.85747 
85732 


.59888 
.59928 
•59967 
.60007 
■ 60046 

•60086 


.52869 
.52893 
.52918 
.52943 
.52967 
. 52992 


.84882 
•84868 
.84851 
.84836 
.84820 


.62285 
.62325 
.62366 
.62406 
:J_2446_ 

•62487 


1.60553 
1.60449 
1.60345 
1.60241 
1.60137 
1 . 60033 


60. 


.51504 


•85717 


.84805 


' / 


Cos. 1 Sin. 1 Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 



59^ 



752 



68' 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS. 



33' 



33' 



• 


Sin. 


Cos 


Tan. 


Cot 


Sin. 


Cos. 


Tan. 


Cot. 


• 




1 

2 
3 
4 


.52992 
.53017 
.53041 
.53066 
.53091 

.53115 
.53140 
.53164 
.53189 
.53214 


.84805 
.84789 
.84774 
.84759 
.84743 


•62487 
.62527 
.62568 
.62608 
.62649 


1.60033 
1.59930 
1.59826 
1.59723 
1.59620 


•54464 
•54488 
•54513 
•54537 
.54561 


•83867 
.83851 
.83835 
.83819 
.83804 


•64941 
•64982 
•65024 
•65065 
•65106 


1.53986 
1.53888 
1.53791 
1.53693 
1.53595 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.84728 
.84712 
.84697 
.84681 
.84666 


.62689 
.62730 
.62770 
.62811 
.62852 


1.59517 
1.59414 
1.59311 
1.59208 
1.59105 

1.59002 
1.58900 
1.58797 
1.58695 
1-58593 


■^4585 

5^610 

• 54b635 

•54659 

54683 


.83788 
.83772 
.83756 
.83740 
.83724 


.65148 
.65189 
•65231 
•65272 
•65314 


1.53497 
1.53400 
1.53302 
1.53205 
1.53107 
1.53010 
1.52913 
1.52816 
1.52719 
1.52622 


55 
54 
53 
5L 
51 


10 

11 
12 
13 
14 


.53238 
.53263 
.53288 
.53312 
.53337 


.84650 
.84635 
.84619 
.84604 
.84588 


.62892 
.62933 
.62973 
•63014 
.63055 
.63095 
.63136 
.'63177 
.63217 
•63258 


•54708 
•54732 
•54756 
•54781 
•54805 


.83708 
•83692 
•83676 
•83660 
.83645 


•65355 
•65397 
•65438 
•65480 
-65521 


50 

49 
48 
47 
46 


15 
16 
17 
18 

19 


.53361 
.53386 
.53411 
.53435 
.53460 


.84573 
.84557 
.84542 
.84526 
.84511 


1.58490 
1.58388 
1.58286 
1.58184 
1.58083 


•54829 
•54854 
•54878 
-54902 
•54927 


•83629 
•83613 
•83597 
•83581 
.83565 


.65563 
.65604 
.65646 
.65688 
-65729 


1.52525 
1.52429 
1.52332 
1.52235 
1^52139 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.53484 
.53509 
.53534 
.53558 
.53583 


.84495 
.84480 
.84464 
. 84448 
.84433 

.84417 
.84402 
.84386 
.84370 
.84355 


•63299 
.63340 
•63380 
.63421 
. 63462 


1.57981 
1.57879 
1.57778 
1.57676 
1.57575 


54951 
•54975 
•54999 
•55024 
•55048 


.83549 
.83533 
.83517 
•83501 
•83485 


.65771 
•65813 
.65854 
•65896 
•65938 


1.52043 
1.51946 
1.51850 
1.51754 
1.51658 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.53607 
.53632 
.53656 
-53681 
.53705 


•63503 
.63544 
.63584 
•63625 
.63666 


1.57474 
1.57372 
1.57271 
1.57170 
1.57069 


•55072 
•55097 
•55121 
•55145 
•55169 


.83469 
•83453 
•83437 
•83421 
•83405 


.65980 
.66021 
.66063 
•66105 
•66147 


1.51562 
1.51466 
1.51370 
1.51275 
1.51179 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.53730 
.53754 
.53779 
.53804 
.53828 


.84339 
•84324 
•84308 
.84292 
.84277 


•63707 
.63748 
.63789 
.63830 
.63871 


1.56969 
1.56868 
1.56767 
1.56667 
1.56566 


••55194 
•55218 
•55242 
•55266 
•55291 


•83389 
•83373 
•83356 
•83340 
•83324 


•66189 
•66230 
.66272 
.66314 
•66356 


1.51084 
1.50988 
1.50893 
1.50797 
1.50702 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.53853 
.53877 
.53902 
.53926 
.53951 


.84261 
.84245 
.84230 
.84214 
.84198 


.63912 
.63953 
.63994 
.64035 
•64076 


1.56466 
1.56366 
1.56265 
1.56165 
1.56065 


•55315 
•55339 
•55363 
•55388 

•55412 


•83308 
•83292 
.83276 
•83260 
•83244 


.66398 
.66440 
.66482 
.66524 
•66566 


1^50607 
1.50512 
1.50417 
1.50322 
1.50228 


25 
24 
23 
22 
21 


40 

41 
42 
43 

44 


.53975 
.54000 
.54024 
.54049 
.54073 


•84182 
.84167 
.84151 
•84135 
.84120 


•64117 
.64158 
.64199 
. 64240 
.64281 

.64322 
.64363 
. 64404 
. 64446 
.64487 


1.55966 
1.55866 
1.55766 
1.55666 
1.55567 


•55436 
•55460 
•55484 
•55509 
•55533 


•83228 
•83212 
^83195 
•83179 
•83163 


•66608 
.66650 
.66692 
.66734 
.66776 


1.50133 
1.50038 
1.49944 
1.49849 
1.49755 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.54097 
.54122 
.54146 
.54171 
.54195 


.84104 
•84088 
•84072 
.84057 
.84041 


1.55467 
1.55368 
1.55269 
1.55170 
1.55071 


•55557 
.55581 
.55605 
.55630 
55654 


•83147 
•83131 
•83115 
•83098 
•83082 


.66818 
.66860 
.66902 
.66944 
:66986 


1.49661 
1.49566 
1.49472 
1.49378 
1.49284 

1.49190 
1.49097 
1.49003 
1.48909 
1.48816 


15 
14 
13 
12 
11 


50 

51 
59, 
53 
54 


•54220 
. 54244 
.54269 
.54293 
.54317 

.54342 
•54366 
.54391 
.54415 
. 54440 


.84025 
.84009 
.83994 
•83978 
•83962 


.64528 
.64569 
•64610 
.64652 
•64693 

•64734 
.64775 
•64817 
.64858 
64899 

•64941 


1.54972 
1.54873 
1.54774 
1.54675 
1.54576 


•55678 
.55702 
.55726 
.55750 
.55775 

.55799 
.55823 
•55847 
•55871 
•55895 


•83066 
•83050 

83034 
•83017 

83001 


.67028 
.67071 
.67113 
.67155 
.67197 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•83946 
•83930 
•83915 
•83899 
•83883 

83867 


1.54478 
1-54379 
1.54281 
1.54183 
1.54085 


•82985 
82969 
•82953 
•82936 
•82920 


.67239 
.67282 
.67324 
.67366 
•67409 


1.48722 
1.48629 
1.48536 
1.48442 
1-48349 


5 

4 
3 

2 

1 


60 


. 54464 


1.53986 


55919 


.82904 


•67451 


1-48256 


O 




Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


' 



5r 



753 



66' 



TABLE IX.— N^ rURAL S NES, COSINES, TANGENTS, AND COTANGENTS. 







34° 






35° 








Sin. 
.55919 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


f 





•82904 


.67451 


1.48256 


-57358 


•81915 


.70021 


1.42815 


6(1 


X 


.55943 


.82887 


•67493 


1.48163 


-57381 


-81899 


•70064 


1.42726 


59 


2 


.55968 


.82871 


.67536 


1.48070 


-57405 


•81882 


.70107 


1.42638 


58 


a 


.55992 


.82855 


.67578 


1.47977 


-57429 


•81865 


.70151 


1.42550 


5' 


j4_ 


.56016 


82839 


•67620 


1.47885 


-57453 


.81848 
.81832 


.70194 
.70238 


1.42462 


5f 
5! 


5 


.56040 


.82822 


.67663 


1.47792 


-57477 


1.42374 


6 


.56064 


.82806 


.67705 


1.47699 


.57501 


.81815 


.70281 


1.42286 


54 


7 


.56088 


.82790 


.67748 


1.47607 


.57524 


.81798 


.70325 


1.42198 


5S 


8 


.56112 


.82773 


.67790 


1.47514 


.57548 


.81782 


.70368 


1.42110 


52^ 


9 


.56136 
.56160 


.82757 


•67832 


1.47422 


-57572 


•81765 


.70412 


1.42022 


51 


10 


.82741 


.67875 


1.47330 


.57596 


.81748 


.70455 


1.4.1934 


50 


11 


.56184 


.82724 


.67917 


1.47238 


.57619 


.81731 


•70499 


1.41847 


49 


12 


.56208 


.82708 


.67960 


1.47146 


.57643 


•81714 


•70542 


1.41759 


48 


13 


.56232 


.82692 


.68002 


1.47053 


.57667 


•81698 


•70586 


1.41672 


47 


14 


.56256 
.56280 


.82675 


■68045 


1.46962 


-57691 


■81681 


.70629 


1.41584 


46 
45 


15 


.82659 


.68088 


1.46870 


.57715 


-81664 


.70673 


1.41497 


16 


.56305 


.82643 


.68130 


1.46778 


.57738 


•81647 


.70717 


1.41409 


44 


17 


.56329 


.82626 


.68173 


1.46686 


-57762 


.81631 


.70760 


1.41322 


43 


18 


.56353 


.82610 


.68215 


1.46595 


•57786 


.81614 


.70804 


1.41235 


42 


19 


.56377 


•82593 


•68258 


1.46503 


-57810 


.81597 


.70848 


1.41148 


40 


30 


.56401 


.82577 


.68301 


1.46411 


-57833 


.81580 


.70891 


1.41061 


21 


.56425 


.82561 


.68343 


1.46320 


-57857 


.81563 


.70935 


1.40974 


39 


22 


. 56449 


.82544 


.68386 


1.46229 


-57881 


.81546 


.70979 


1.40887 


38 


23 


•56473 


.82528 


.68429 


1.46137 


•57904 


.81530 


.71023 


1.40800 


37 


24 


■56497 


•82511 


.68471 


1.46046 


■57928 


•81513 


.71066 


1-40714 


36 

35 


25 


.56521 


.82495 


.68514 


1.45955 


-57952 


.81496 


.71110 


1-40627 


26 


.56545 


.82478 


.68557 


1.45864 


•57976 


■81479 


•71154 


1-40540 


34 


27 


.56569 


.82462 


.68600 


1.45773 


•57999 


.81462 


•71198 


1.40454 


33 


28 


■56593 


.82446 


.68642 


1.45682 


-58023 


.81445 


.71242 


1-40367 


32 


29 


■56617 


•82429 


■68685 


1.45592 


•58047 


.81428 


•71285 


1-40281 


31 
30 


30 


.56641 


.82413 


.68728 


1.45501 


.58070 


.81412 


.71329 


1.40195 


31 


.56665 


.82396 


.68771 


1.45410 


.58094 


.81395 


.71373 


1.40109 


29 


32 


.56689 


.82380 


.68814 


1.45320 


.58118 


.81378 


.71417 


1.40022 


28 


33 


.56713 


.82363 


.68857 


1.45229 


.58141 


.81361 


.71461 


1.39936 


27 


34 


■56736 


.82347 


•68900 


1-45139 


-58165 


.81344 


.71505 


1-39850 


26! 
25 


35 


.56760 


.82330 


•68942 


1.45049 


-58189 


.81327 


.71549 


1-39764 


36 


.56784 


.82314 


68985 


1.44958 


-58212 


.81310 


.71593 


1^ 39679 


24 


37 


.56803 


.82297 


•69028 


1.44868 


-58236 


.81293 


.71637 


1.395-3 


23 


38 


.56832 


.82281 


•69071 


1-44778 


-58260 


.81276 


.71681 


1-39507 


22 


39 


■ 568.V,S. 
.56880 


■82264 
•82248 


■69114 


1-44688 


-58283 


•81259 


•71725 


1-39421 


20 


40 


•69157 


1.44598 


-58307 


•81242 


.71769 


.1.39336 


41 


.56904 


.82231 


•69200 


1.44508 


-58330 


-81225 


.71813 


1.39250 


19 ' 


42 


.56928 


.82214 


•69243 


1.44418 


-58354 


-81208 


.71857 


1.39165 


18 1 


43 


.56952 


.82198 


•69286 


1.44329 


-58378 


-81191 


.71901 


1.39079 


17 


44 


.56976 


■82181 


■69329 


1-44239 


■58401 


-81174 


•71946 


1-38994 


16 
15 1 


45 


.57000 


•82165 


■69372 


1.44149 


■58425 


-81157 


-71990 


1.38909 


46 


.57024 


•82148 


•69416 


1.44060 


. 58449 


•81140 


. 72034 


1.38824 


14! 


47 


.57047 


.82132 


•69459 


1.43970 


■58472 


.81123 


. 72078 


1.38738 


13 


48 


.57071 


.82115 


•69502 


1.43881 


58498 


81106 


.72122 


1.38653 


12 


49 


.57095 


•82098 


■69545 


1-43792 
1-43703 


■58519 


■81089 


.72167 


1-38568 


11 
10 


50 


.57119 


.82082 


■69588 


.58543 


•81072 


.72211 


1.38484 


51 


.57143 


.82065 


•69631 


1.43614 


■58567 


•81055 


•72255 


1.38399 


9 


52 


.57167 


.82048 


•69675 


1-43525 


■58590 


•81038 


•72299 


1.38314 


8 


53 


.57191 


.82032 


.69718 


1-43436 


■58614 


•81021 


• 72344 


1.38229 


7 


54 


.57215 


• 82015 


•69761 


1-43347 


58637 


•81004 


•72388 

. 72432 


1.38145 


6 
5 


55 


.57238 


•81999 


•69804 


1.43258 


■58661 


-80987 


1.38060 


56 


.57262 


81982 


•69847 


1.43169 


■58684 


.80970 


•72477 


1.37976 


4 


57 


.57286 


■81965 


•69891 


1.43080 


-58708 


-80953 


•72521 


1.37891 


3 


58 


.57310 


■81949 


•69934 


1-42992 


-58731 


.80936 


•72565 


1.37807 


2 


59 


.57334 
.57358 


81932 


•69977 


1-42903 
1-42815 


■58755 


-80919 


-72610 


1-37722 
1-37638 


1 



/ I 


60 


•81915 


.70021 


-58779 


-80902 


-72654 


' 


Cos. 


Sin. 


Cot. 


Tan, 


Cos. 


Sin. 


Cot. 


Tan. 






5 


5** 


71 


54 


5^ 


1° 







TABLE IX.— NATURAL SINES, COSINES, TANGENTS. AND COTANGENTS. 
36'' 370 





Sin. 


Cos. 


Tan. 


Cot. 1 Sin. ' Cos. 


Tan. 


Cot. 


r 




1 

2 
3 
4 


58779 
•58802 
-58826 
.58849 
.58873 


.80902 
.80885 
.80867 
.80850 
.80833 
.80816 
.80799 
.80782 
.80765 
.80748 


.72654 
.72899 
.72743 
.72788 
.72832 


1.37638 
1.37554 
1-37470 
1-37386 
1^37302 


•60182 
60205 
■60228 
•60251 
• 60274 


•79864 
.79846 
.79829 
.79811 
.79793 


-75355 
-75401 
•75447 
•75492 
•75538 


1.32704 
1^32624 
1.32544 
1.32464 
1^ 32384 


"60 

59 
58 
57 
56 


5 

6 
7 
8 
9 


58896 
.58920 
.58943 
.58967 

58990 


.72877 
.72921 
.72966 
.73010 
.73055 

.73100 
.73144 
73189 
.73234 
•73278 


1.37218 
1.37134 
1.37050 
1.36967 
1.36883 


. 60298 
.60321 
.60344 
.60367 
.60390 


.79776 
•79758 
.79741 
.79723 
•79706 


.75584 
.75629 
.75675 
.75721 
.75767 


1.32304 
1.32224 
1.32144 
1.32064 
1.31984 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.59014 
.59037 
.59061 
59084 
.59108 


.80730 
.80713 
.80696 
.80679 
.80662 


1.36800 
1.36716 
1.36633 
1.36549 
1.36466 


. 60414 
.60437 
•60460 
.60483 
60506 


.79688 
.79671 
.79653 
•79635 
.79618 

•79600 
•79583 
.79565 
.79547 
.79530 


.75812 
.75858 
.75904 
.75950 
.75996 


1.31904 
1.31825 
1.31745 
1.31666 
131586 


50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
88 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 


15 
16 
17 
18 


.59131 
•59154 
•59178 
•59201 
•59225 


.80644 
.80627 
.80610 
.80593 
.80576 


•73323 
•73368 
•73413 
.73457 
•73502 


1.36383 
1.36300 
1.36217 
1.36134 
1.3605] 


-60529 
-60553 
-60576 
-60599 
•60622 


.76042 
.76088 
•76134 
.76180 
76226 


1-31507 
1^31427 
1.31348 
1.31269 
1-31190 


20 

21 
22 
23 
24 


•59248 
•59272 
.59295 
.59318 
.59342 


.80558 
.80541 
.80524 
.80507 
.80489 


•73547 
•73592 
•73637 
•73681 
.73726 


1.35968 
1.35885 
1.35802 
1.35719 
1.35637 


•60645 
-60668 
.60691 
.60714 
•60738 


79512 
.79494 
.79477 
.79459 
-79441 


•76272 
•76318 
.76364 
•76410 
.76456 


1-31110 
1.31031 
1.30952 
1.30873 
1. 30795 


25 
26 
27 
28 
29 


.59365 
.59389 
•59412 
•59436 
.59459 


.80472 
80455 
.80438 
.80420 
. 80403 


•73771 
•73816 
•73861 
.73906 
•73951 


1.35554 
1.35472 
1.35389 
1.35307 
1.35224 


•60761 
.60784 
•60807 
•60830 
•60853 


. 79424 
-79406 
.79388 
-79371 
-79353 


.76502 
•76548 
•76594 
•76640 
.76686 


1.30716 
1.30637 
1-30558 
1-30480 
1-30401 


30 

31 
32 
33 
34 

35 
36 
37 
38 
39_ 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 

62. 


•59482 
.59506 
•59529 
•59552 
59576 


.80386 
•80368 
.80351 
•80334 
•80316 


•73996 
.74041 
.74086 
.74131 
•74176 


1.35142 
1.35060 
1.34978 
1.34896 
1-34814 


•60876 
•60899 
•60922 
.60945 
.60968 


•79335 
•79318 
•79300 
•79282 
-79264 


.76733 
.76779 
.76825 
•76871 
•76918 


1.30323 
1.30244 
1^ 30166 
1. 30087 
1-30009 


.59599 
.59622 
•59646 
.59669 
.59693 


•80299 
•80282 
•80264 
.80247 
•80230 


• 74221 
•74267 
.74312 
.74357 

• 74402 


1.34732 
1.34650 
1.34568 
1.34487 
1-34405 


.60991 
.61015 
.61038 
61061 
•61084 


79247 
.79229 
.79211 
.79193 
-79176 


•76964 
•77010 
.77057 
.77103 
•77149 


1.29931 
1.29853 
1^29775 
1.29696 
1^ 29618 


.59716 
.59739 
•59763 
•59786 
.59809 


.80212 
.80195 
.80178 
.80160 
.80143 


. 74447 
. 74492 
.74538 
.74583 
.74628 


1.34323 
1.34242 
1.34160 
1.34079 
1-33998 


•61107 
.61130 
-61153 
-61176 
•61199 


•79158 
•79140 
•79122 
•79105 
-79087 


.77196 
.77242 
.77289 
.77335 
•77382 


1.29541 
1.29463 
1.29385 
1.29307 
1-29229 


"So 

19 
18 
17 
16 

15 

it 

12 
11 


59832 
.59856 
.59879 
.59902 
.59926 


.80125 
•80108 
.80091 
.80073 
.80056 


.74674 
.74719 
.74764 
.74810 
.74855 


1.33916 
1.33835 
1.33754 
1.33673 
1.33592 


•61222 
-61245 
-61268 
-61291 
-61314 


.79069 
•79051 
•79033 
•79016 
•78998 


•77428 
.77475 
.77521 
.77568 
•77615 


1^29152 
1.29074 
1.28997 
1.28919 
1.28842 


.59949 
.59972 
.59995 
.60019 
•60042 


.80038 
.80021 
.80003 
.79986 
.79968 


.74900 
.74946 
.74991 
.75037 
.75082 


1.33511 
1.33430 
1.33349 
1.33268 
1.33187 


•61337 
.61360 
.61383 
Ul406 
.61429 


•78980 
•78962 
.78944 
.78926 
•78908 


.77661 
•77708 
.77754 
•77801 
•77848 


1.28764 
1.28687 
1.28610 
1.28533 
1.28456 


10 

9 
8 
7 


•60065 
•60089 
•60112 
•60135 
•60158 


.f9951 
.79934 
.79916 
.79899 
.79881 


.75128 
.75173 
.75219 
.75264 
.75310 


1.33107 

1.33026 
1.32946 
1.32865 
1.32785 


.61451 
.61474 
.61497 
.61520 
.61543 


.78891 
•78873 
•78855 
•78837 
78819 


.77895 
.77941 
.77988 
.78035 
.78082 


1.28379 
1.28302 
1.28225 
1.28148 
1.28071 


5 
4 
3 
2 
1 


•60182 


.79864 


.75355 


1.32704 


.61566 


•78801 
Sin. 


•78129 


1.27994 





i- — 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Cot. 


Tan. 


-4: 


1 




5 


5« 


75, 


5 


5'^ 


i" 







TABLE IX. 



-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
38^ 39° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


.61566 
.61589 
.61612 
•61635 
.61658 


.78801 
.78783 
.78765 
.78747 
.78729 


•78129 
•78175 
•78222 
•78269 
•78316 


1 
1 

1 

-, 
i 


27994 
27917 
27841 
27764 
27688 


•62932 
•62955 
.62977 
•63000 
•63022 


.77715 
.77696 
.77678 
•77660 
•77641 


•80978 
•81027 
•81075 
•81123 
•81171 


1.23490 
1^23416 
1.23343 
1.23270 
1.23196 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.61681 
•61704 
•61726 
•61749 
•61772 

•61795 
•61818 
•61841 
• 61864 
•61887 


.78711 
.78694 
.78676 
•78658 
•78640 


.78363 
.78410 
•78457 
•78504 
•78551 




27611 
27535 
27458 
27382 
27306 


•63045 
•63068 
•63090 
•63113 
•63135 

•63158 
•63180 
•63203 
•63225 
•63248 


•77623 
•77605 
77586 
•77568 
.77550 


.81220 
.81268 
•81316 
•81364 
•81413 


1.23123 
1.23050 
1^22977 
1^22904 
1.22831 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


•78622 
.78604 
.78586 
.78568 
•78550 


•78598 
.78645 
.78692 
.78739 
•78786 


1 


27230 
27153 
27077 
27001 
26925 


.77531 
•77513 
•77494 
.77476 
.77458 


•81461 
.81510 
.81558 
.81606 
•81655 


1-22758 
1.22685 
1.22612 
1.22539 
1.22467 


50 

49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 

30 

19 
18 
17 
16 

15' 

14 

13 

12 

11 

10 

9 

8 

7i 

6 

5 
4 
3 

2 
1 




15 
16 
17 
18 
19 


•61909 
•61932 
•61955 
•61978 
•62001 


•78532 
•78514 
.78496 
.78478 
•78460 


.78834 
•78881 
•78928 
.78975 
•79022 




26849 
26774 
26698 
26622 
26546 


•63271 
•63293 
•63316 
•63338 
•63361 


.77439 
.77421 
. 77402 
.77384 
.77366 


•81703 
.81752 
.81800 
.81849 
•81898 


1.22394 
1.22321 
1.22249 
1.22176 
1^22104 


20 

21 

22 
23 
24 


•62024 
•62046 
•62069 
•62092 
•62115 


. 78442 
. 78424 
.78405 
.78387 
.78369 


.79070 
.79117 
.79164 
.79212 
•79259 




26471 
26395 
26319 
26244 
26169 


•63383 
•63406 
•63428 
•63451 
•63473 


.77347 
.77329 
•77310 
•77292 
.77273 


•81946 
.81995 
.82044 
.82092 
.82141 


1.22031 
1.21959 
1.21886 
1.21814 
1.21742 


25 
26 
27 
28 
29 


•62138 
•62160 
•62183 
•62206 
•62229 


.78351 
.78333 
.78315 
.78297 
•78279 


.79306 
.79354 
. 79401 
• 79449 
•79496 




26093 
26018 
25943 
25867 
25792 


•63496 
•63518 
•63540 
•63563 
•63585 


.77255 
.77236 
.77218 
.77199 
•77181 


.82190 
.82238 
.82287 
•82336 
•82385 


1.21670 
1.21598 
1.21526 
1.21454 
1.21382 


30 

31 
S2 
33 
34 


•62251 
.62274 
.62297 
.62320 
•62342 


.78261 
. 78243 
.78225 
•78206 
•78188 


•79544 
•79591 
•79639 
•79686 
•79734 




25717 
25642 
25567 
25492 
25417 


.63608 
.63630 
.63653 
•63675 
.63698 


.77162 
.77144 
.77125 
.77107 
.77088 


.82434 
.82483 
.82531 
.82580 
•82629 


1.21310 
1.21238 
1.21166 
1.21094 
1.21023 


35 
36 
37 
38 
39 


.62365 
.62388 
.62411 
.62433 
•62456 


•78170 
•78152 
.78134 
.78116 
•78098 


•79781 
•79829 
•79877 
•79924 
.79972 

•80020 
.80067 
.80115 
.80163 
.80211 


1 


25343 
25268 
25193 
25118 
25044 

24969 
.24895 
.24820 
•24746 
•24672 


-63720 
•63742 
•63765 
•63787 
•63810 


.77070 
.77051 
.77033 
.77014 
.76996 


•82678 
.82727 
.82776 
.82825 
.82874 


1.20951 
1.20879 
1.20808 
1.20736 
1-20665 


40 

41 
42 
43 
44 


.62479 
.62502 
•62524 
.62547 
•62570 


.78079 
.78061 
•78043 
•78025 
.78007 


•63832 
•63854 
•63877 
•63899 
•63922 


.76977 
.76959 
• 76940 
•76921 
•76903 


.82923 
.82972 
.83022 
.83071 
•83120 


1.20593 
1.20522 
1.20451 
1.20379 
1.20308 


45 
46 
47 
48 
49 


•62592 
.62615 
.62638 
•62660 
•62683 


.77988 
.77970 
•77952 
•77934 
.77916 


•80258 
•80306 
.80354 
.80402 
.80450 




•24597 

•24523 

.24449 

24375 

24301 


•63944 
•63966 
•63989 
•64011 
•64033 


•76884 
•76866 
•76847 
•76828 
•76810 


•83169 
•83218 
•83268 
•83317 
•83366 


1.20237 
1.20166 
1.20095 
1.20024 
1-19953 


50 

51 
52 
53 
54 


•62706 
•62728 
•62751 
.62774 
•62796 


.77897 
.77879 
•77861 
•77843 
•77824 


•80498 
•80546 
•80594 
•80642 
•80690 




24227 
24153 
24079 
24005 
23931 


•64056 
•64078 
•64100 
•64123 
•64145 


.76791 
.76772 
.76754 
.76735 
•76717 


•83415 
•83465 
•83514 
•83564 
•83613 


1.19882 
1.19811 
1.19740 
1.19669 
1.19599 


55 
56 
57 
58 
59 


•62819 
.62842 
.62864 
.62887 
•62909 


.77806 
.77788 
.77769 
•77751 
•77733 


•80738 
•80786 
•80834 
•80882 
.80930 




23858 
23784 
23710 
23637 
23563 


•64167 
•64190 
•64212 
•64234 
•64256 


.76698 
.76679 
.76661 
.76642 
•76623 


-83662 
•83712 
•83761 
•83811 
83860 


1.19528 
1.19457 
1^19387 
1.19316 
1-19246 


60 


•62932 


•77715 


80978 




23490 


•64279 


•76604 


•83910 


1.19175 


/ 


Cos. 


Sin. 


Cot. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 


51'' 756 5 


[)" 




!• 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
40° 41° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


r 





•64279 


•76604 


•83910 


1^19175 


•65606 


.75471 


•86929 


1.15037 


60 


1 


•64301 


.76586 


•83960 


1^19105 


•65628 


.75452 


•86980 


1.14969 


59 


2 


•64323 


•76567 


•84009 


1^19035 


•65650 


.75433 


•87031 


1.14902 


5ft 


3 


•64346 


•76548 


•84059 


1^18964 


•65672 


.75414 


•87082 


1.14834 


57 


4 


•64368 
•64390 


•76530 


•84108 


1^ 18894 


•65694 


.75395 


•87133 


1-14767 


56 


5 


•76511 


•84158 


1^ 18824 


•65716 


.75375 


•87184 


1-14699 


55 


6 


•64412 


•76492 


•84208 


1^18754 


•65738 


.75356 


•87236 


1.14632 


54 


7 


•64435 


•76473 


•84258 


1^ 18684 


•65759 


.75337 


•87287 


1.14565 


53 


8 


•64457 


.76455 


•84307 


1.18614 


•65781 


.75318 


•87338 


1.14498 


52 


9 


. 64479 


.76436 


.84357 


1.18544 


•65803 
•65825 


•75299 


■87389 


1-14430 


51 


10 


•64501 


.76417 


•84407 


1.18474 


•75280 


•87441 


1^ 14363 


50 


11 


• 64524 


.76398 


•84457 


1.18404 


•65847 


•75261 


•87492 


1.14296 


49 


12 


. 64546 


•76380 


•84507 


1.18334 


•65869 


•75241 


•87543 


1.14229 


48 


13 


•64568 


.76361 


•84556 


1.18264 


•65891 


•75222 


•87595 


1.14162 


47 


14 


•64590 


•76342 


•84606 


1.18194 


•65913 


•75203 


•87646 


1.14095 


46 


15 


•64612 


.76323 


•84656 


1.18125 


•65935 


.75184 


•87698 


1^14028 


45 


16 


•64635 


.76304 


•84706 


1.18055 


•65956 


.75165 


•87749 


1^13961 


44 


17 


•64657 


.76286 


•84756 


1^17986 


•65978 


•75146 


•87801 


1^ 13894 


43 


18 


•64679 


.76267 


.84806 


1^17916 


•66000 


•75126 


•87852 


1.13828 


42 


19 . 


•64701 


•76248 


•84856 


1.17846 


.66022 


■75107 
•75088 


•87904 


1^13761 


,41 


30 


•64723 


.76229 


•84906 


1-17777 


• 66044 


•87955 


1.13694 


40 


21 


•64746 


.76210 


•J4956 


1.17708 


•66066 


•75069 


•88007 


1.13627 


39 


22 


•64768 


.76192 


•85006 


1-17638 


•66088 


•75050 


•88059 


1.13561 


38 


23 


•64790 


.76173 


•85057 


1-17569 


•66109 


•75030 


•88110 


1.13494 


37 


24 


•64812 


•76154 


•85107 


1^17500 


.66131 


•75011 


■88162 


1^ 13428 


36 


25 


• 64834 


•76135 


•85157 


1^17430 


.66053 


•74992 


•88204 


1^ 13361 


35 


26 


•64856 


•76116 


•85207 


1^17361 


•66175 


.74973 


•88265 


1^13295 


34 


27 


•64878 


•76097 


•85257 


1^17292 


•66197 


•74953 


•88317 


1-13228 


33 


28 


•64901 


•76078 


•85308 


1^17223 


•66218 


• 74934 


•88369 


1^13162 


32 


29 


•64923 


•76059 


.85358 


1^17154 


• 66240 


•74915 


•88421 


1.13096 


31 


30 


•64945 


•76041 


.85408 


1^17085 


•66262 


•74896 


•88473 


1.13029 


30 


31 


•64967 


.76022 


.85458 


1^ 17016 


.66284 


•74876 


•88524 


1.12963 


29 


32 


•64989 


.76003 


.85509 


1^16947 


.66306 


•74857 


•88576 


1.12897 


28 


33 


•65011 


.75984 


.85559 


1^16878 


•66327 


•74838 


•88628 


1.12831 


27 


34 


.65033 
•65055 


.75965 


•85609 


1.16809 


•66349 


•74818 


■88680 


1.12765 


26 


35 


•75946 


•85660 


1.16741 


•66371 


•74799 


•88732 


1.12699 


25 


36 


•65077 


•75927 


•85710 


1^16672 


•66393 


• 74780 


•88784 


1^12633 


24 


37 


•65100 


•75908 


•85761 


1^16603 


•66414 


• 74760 


•88836 


1^12567 


23 


38 


•65122 


.75889 


•85811 


1^16535 


•66436 


• 74741 


•88888 


1-12501 


22 


39 


.65144 


.75870 


•85862 


1-16466 


•66458 


.7472? 


•88940 


1-12435 


21 


40 


•65166 


.75851 


•85912 


1- 16398 


. 66480 


■ 74703 


•88992 


1.12369 


30 


41 


•65188 


.75832 


•85963 


1^16329 


•66501 


■ 74683 


.89045 


1.12303 


19 


42 


.65210 


•75813 


•86014 


1.16261 


•66523 


■ 74664 


.89097 


1.12238 


18 


43 


.65232 


.75794 


•86064 


1^16192 


•66545 


• 74644 


.89149 


1.12172 


17 


44 


•65254 


•75775 


•86115 


1^16124 


•66566 


■74625 


•89201 


1.12106 


16 


45 


•65276 


•75756 


•86166 


M6056 


66588 


• 74606 


•89253 


1.12041 


15 


46 


•65298 


•75738 


86216 


1^ 15987 


•66610 


•74586 


•89306 


1.11975 


14 


47 


•65320 


•75719 


•86267 


1^15919 


•66632 


•74567 


•89358 


1.11909 


13 


48 


.65342 


•75700 


•86318 


1^15851 


■66653 


• 74548 


•89410 


1.11844 


12 


49 


.65364 


■75680 
.75661 


•86368 


1^ 15783 


•66675 


• 74528 


.89463 


1.11778 


11 


50 


.65386 


•86419 


1.15715 


.66697 


• 74509 


•89515 


1.11713 


10 


51 


•65408 


•75642 


•86470 


1.15647 


.66718 


• 74489 


•89567 


1.11648 


9 


52 


•65430 


•75623 


•86521 


1.15579 


. 66740 


• 74470 


•89620 


1.11582 


8 


53 


.65452 


•75604 


•86572 


1.15511 


•66762 


•74451 


•89672 


1.11517 


7 


54 


.65474 


•75585 


•86623 


1.15443 


•66783 


•74431 


•89725 


1-11452 


6 


55 


•65496 


•75566 


•86674 


1.15375 


•66805 


.74412 


•89777 


1-11387 


5 


56 


•65518 


•75547 


•86725 


1.15308 


•66827 


•74392 


.89830 


1.11321 


4 


57 


•65540 


.75528 


•86776 


1.15240 


66848 


•74373 


.89883 


1.11256 


3 


58 


.65562 


.75509 


•86827 


1.15172 


.66870 


•74353 


.89935 


1.11191 


2 


59 


•65584 


.75490 


.86878 


1.15104 


.66891 


• 74334 


•89988 


1-11126 


1 


60^ 


.65606 


.75471 


•86929 


1^15037 
Tan. 


•66913 


•74314 


•90040 


1^11061 







Cos. 


Sin. 


Cot. 


Cos. 


Sin. 


Cot. 


Tan. 


f 



49^ 



757 



48^ 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
dS'' 43° 



/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


f 




1 

2 
3 

4 


.66913 
66935 
.66956 
.66978 
.66999 


.74314 
.74295 
.74276 
.74256 
.74237 


.90040 
.90093 
.90146 
.90199 
.90251 


1.11061 
1 10996 
1.10931 
1.10867 
1.10802 


-68200 
•68221 
.68242 
.68264 
•68285 


•73135 
.73116 
.73096 
.73076 
.73056 


.93252 
.93308 
.93360 
-.93415 
.93469 


1.07237 
1.07174 
1-07112 
1.07049 
1-06987 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.67021 
67043 
.67064 
•67086 
.67107 


.74217 
•74198 
74178 
•74159 
•74139 


.90304 
.90357 
.90410 
.90463 
.90516 


1.10737 
1.10672 
1.10607 
1.10543 
1.10478 


68306 
.68327 
.68349 
.68370 
.68391 


.73036 
.73016 
.72996 
.72976 
.72957 


.93524 
.93578 
.93633 
.93688 
.93742 

.93797 
.93852 
.93906 
.93961 
.94016 


1-06925 
1.06862 
1.06800 
1.06738 
1-06676 


55 
54 
53 
52 
51 


10 

11 
12 
13 

U 


.67129 
.67151 
.67172 
.67194 
.67215 

.67237 
.67258 
.67280 
.67301 
.67-^23 


.74120 
.74100 
. 74080 
.74061 
. 74041 


.90569 
.90621 
.90674 
90727 
.90781 


1.10414 
1.10349 
1.10285 
1.10220 
1-10156 


.68412 
. 68434 
.68455 
.68476 
.68497 


.72937 
.72917 
.72897 
.72877 
.72857 


1-06613 
1.06551 
1.06489 
1.06427 
1 06365 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


. 74022 
- 74002 
.73983 
.73963 
. 73944 


.90834 
.90887 
.90940 
.90993 
.91046 


1.10091 
1.10027 
1.09963 
1.09899 
1.09834 


.68518 
.68539 
.68561 
.68582 
. 68603 

.68624 
.68645 
.68666 
.68688 
. 68709 


.72837 
.72817 
.72797 
.72777 
.72757 


.94071 
.94125 
.94180 
.94235 
.94290 


1.06303 
1.06241 
1.06179 
1.06117 
1-06056 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


.67344 
.67366 
.67387 
.67409 
.67430 


.73924 
.73904 
. 73885 
.73865 
.73846 


.91099 
.91153 
.91206 
.91259 
.91313 


1.09770 
1.09706 
1.09642 
1.09578 
1.09514 


.72737 
.72717 
•72697 
•72677 
.72657 


.94345 
.94400 
•94455 
.94510 
.94565 


1-05994 
1.05932 
1-05870 
1-05809 
1-05747 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


67452 
.67473 
•67495 
.67516 
.67538 


.73826 
.73806 
•73787 
•73767 
•73747 


.91366 
.91419 
.91473 
.91526 
•91580 


1.09450 
1.09386 
1.09322 
1-09258 
1.09195 


.68730 
.68751 
.68772 
•68793 
.68814 


.72637 
.72617 
.72597 
.72577 
.72557 


.94620 
.94676 
.94731 
.94786 
.94841 


1^05685 
1-05624 
1-05562 
1.05501 
1-05439 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•67559 
•67580 
•67602 
•67623 
•67645 


.73728 
•73708 
•73688 
•73669 
.7^,849 


.91633 
.91687 
.91740 
•91794 
•91847 


1.09131 
1.09067 
1.09003 
1-08940 
1-08876 


.68835 
.68857 
•68378 
•68899 
•68920 


.72537 
.72517 
.72497 
.72477 
.72457 


•94896 
•94952 
•95007 
v95062 
•95118 


1-05378 
1-05317 
1-05255 
1-05194 
1. 05133 


30 

29 
28 
2? 
26 


35 
36 
37 
38 


•67666 
• 67688 
•67709 
•67730 
67752 


.73629 
.73610 
•73590 
73570 
•73551 


•91901 
•91955 
•92008 
•92062 
•92116_ 


1-08813 
1-08749 
1-08686 
1.08622 
1-08559 


• 68941 
•68962 

• 68983 

• 69004 
•69025 


.72437 
.72417 
.72397 
•72377 
.72357 


•95173 
•95229 
.95284 
.95340 
.95395 


1-05072 
1.05010 
1.04949 
1-04888 
1-04827 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.67773 
•67795 
•67816 
•67837 
•67859 


•73531 
•73511 
.73491 
.73472 
.73452 


•92170 
.92224 
.92277 
•92331 
.92385 


1^ 08496 
1.08432 
1.08369 
1.08306 
1.08243 


• 69046 
•69067 
-69088 
-69109 
.69130 

.69151 
.69172 
.69193 
.69214 
.69235 


.72337 
.72317 
.72297 
.72277 
.72257 


.95451 
.95506 
.95562 
.95618 
.95673 


1-04766 
1-04705 
1-04644 
1-04583 
1-04522 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•67880 
•67901 
•67923 
•67944 
•67965 


•73432 
•73413 
.73393 
•73373 
•73353 


•92439 
•92493 
.92547 
.92601 
•92655 

.92709 
.92763 
.92817 
.92872 
.92926 

.92980 
.93034 
.93088 
•93143 
•93197 


1.08179 
1-08116 
1^ 08053 
1.07990 
1.07927 


.72236 
•72216 
.72196 
.72176 
.72156 


.95729 
.95785 
.95841 
.95897 
.95952 


1-04461 
1-04401 
1-04340 
1-04279 
1-04218 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•67987 
•68008 
•68029 
•68051 
•68072 


•73333 
•73314 
.73294 
-73274 
•73254 


1.07864 
1.07801 
1.07738 
1.07676 
1.07613 


.69256 
-69277 
-69298 
.69319 
.69340 


.72136 
.72116 
.72095 
.72075 
.72055 


.96008 
.96064 
.96120 
.96176 
•96232 


1.04158 
1.04097 
1.04036 
1.03976 
1.03915 


10 
9 
8 
7 

6 


55 
56 
57 
58 
59 


•68093 
.68115 
.68136 
•68157 
.68179 
•68200 


•73234 
•73215 
•73195 
.73175 
.73155 


1.07550 
1.07487 
1.07425 
1.07362 
1^07299 


.69361 
69382 

• 69403 
69424 
69445 


.72035 
•72015 
.71995 
.71974 
•71954 


.96288 
96344 

-96400 
96457 

•96513 


1. 03855 
1.03794 
1.03734 
1-03674 
1.03613 


5 

4 
3 

2 

1 


60 


.73135 


•93252 


1.07?37 


69466 


71934 


•96569 


1 03553 


^ 


-7- 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 





4.T 



758 



46« 



TABLE IX.— NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS 
44° 44° 



t 


Sin. 


Cos. 

•71934 
•71914 
.71894 
•71873 
71853 


Tan. 


Cot. 


60 

59 
58 
57 
56 


/ 


Sin. i Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


69466 
•69487 
•69508 
•69529 
•69549 

.69570 
•69591 
•69612 
•69633 
•69654 


•96569 
•96625 
-96681 
.96738 
.96794 


1.03553 
1.03493 
1.03433 
1.03372 
1.03312 


30 

31 
32 
33 
34 


.70091 
.70112 
.70132 
.70153 
.70174 


.71325 
.71305 
.71284 
.71264 
•71243 


•98270 
.98327 
.98384 
.98441 
•98499 

.98556 
.98613 
.98671 
•98728 
•98786 


1.01761 
1^01702 
1^01642 
1^01583 
1.01524 


30 

29 
28 
27 
26 


5 
6 
7 
8 
9 


.71833 
.71813 
.71792 
.71772 
.71752 


•96850 
•96907 
.96963 
.97020 
.97076 


1.03252 
1^03192 
1.03132 
1.03072 
1.03012 


55 
54 
53 
52 
51 


35 
36 
37 
38 
39 

40 

41 
42 
43 
44 


.70195 
•70215 
•70236 
•70257 
•70277 


•71223 
•71203 
.71182 
•71162 
•71141 


1.01465 
1.01406 
1.01347 
1.01288 
1.01229 


25 
24 
23 
22 
21 


10 

11 
12 
13 
14 


•69675 
•69696 
.69717 
.69737 
.69758 

•69779 
.69800 
.69821 
.69842 
•69862 


.71732 
.71711 
.71691 
•71671 
•71650 


.97133 
.97189 
.97246 
.97302 
•97359 


1 02952 
1^02892 
1-02832 
1.02772 
1.02713 


50 

49 
48 
47 
46 


•70298 
•70319 
•70339 
•70360 
•70381 


•71121 
.71100 
•71080 
.71059 
.71039 


98843 
.98901 
.98958 
.99016 
.99073 


1-01170 
1.01112 
1.01053 
1^00994 
1^00935 


20 

19 
18 
17 
16 


15 
16 
17 
18 
19 


.71630 
.71610 
.71590 
.71569 
.71549 


.97416 
.97472 
.97529 
.97586 
.97643 


1.02653 
1.02593 
1.02533 
1.02474 
1.02414 


45 
44 
43 
42 


45 
46 
47 
48 
49 


• 70401 

• 70422 
70443 

• 70463 

• 70484 


•71019 
.70998 
•70978 
•70957 
70937 


.99131 
.99189 
.99247 
.99304 
•99362 


1.00876 
1.00818 
1^00759 
1.00701 
1.00642 


15 
14 
13 
12 
11 


20 

21 
22 
23 
24 


.69883 
.69904 
•69925 
•69946 
•69966 


.71529 
.71508 
.71488 
•71468 
.71447 


•97700 
.97756 
•97813 
•97870 
•97927 


1.02355 
1.02295 
1.02236 
1.02176 
1.02117 


40 

39 
38 
37 
36 


50 

51 
52 
53 
54 


.70505 
.70525 
•70546 
•70567 
•70587 


•70916 
•70896 
•70875 
•70855 
• 70834 


.99420 
.99478 
•99536 
.99594 
•99652 


1^00583 
1^00525 
1-00467 
1-00408 
1-00350 


10 
9 
8 
7 
6 


25 
26 
27 
28 
29 


•69987 
. 70008 
•70029 
. 70049 
. 70070 


.71427 
.71407 
.71386 
.71366 
•71345 

• 71325 


.97984 
98041 
.98098 
.98155 
.98213 

•98270 


1.02057 
1.01998 
1.01939 
1.01879 
1.01820 


35 
34 
33 
32 
31 


55 
56 
57 
58 
59 


•70608 
•70628 
•70649 
•70670 
• 70690 


•70813 
•70793 
•70772 
•70752 
•70731 


.99710 
.99768 
.99826 
.99884 
.99942 


1.00291 
1.00233 
1^00175 
1^00116 
1-00058 


5 

4 
3 

2 

1 


^0. 


.70091 
Cos. 


1^01761 


30 


60 


•70711 


•70711 
Sin. 


1 . 00000 


1^00000 





/ 


Sin. 


Cot. 


Tan. 


/ 


/ 


Cos. 


Cot. 


Tan. 


"^ 



45^ 



759 



45' 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



0^ 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


9 




1 

2 
3 
4 


.00000 
.00000 
.00000 
.00000 
.00000 


.00000 
.00000 
.00000 
.00000 
.00000 


.00015 
.00016 
.00016 
.00017 
.00017 




00015 
00016 
00016 
00017 
00017 


.00061 
.00062 
.00063 . 
.00064 
.00065 


.00061 
.00062 
.00063 
.00064 
.00065 

.00066 
.00067 
.00068 
.00069 
.00070 


.00137 
.00139 
.00140 
.00142 
•00143 




00137 
00139 
00140 
00142 
00143 




1 

2 
3 
4 


5 
6 
7 
8 
9 


.00000 
.00000 
.00000 
.00000 
.00000 


.00000 
.00000 
.00000 
.00000 
.00000 


.00018 
.00018 
.00019 
.00020 
.00020 




00018 
00018 
00019 
00020 
00020 


.00066 
.00067 
.00068- 
.00069 
.00070 


.00145 
.00146 
•00148 
•00150 
.00151 




00145 
.00147 
.00148 
.00150 
.00151 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.00000 
.00001 
.00001 
.00001 
.00001 


.00000 
.00001 
.00001 
.00001 
.00001 


.00021 
.00021 
.00022 
.00023 
.00023 




00021 

00021 

.00022 

.00023 

.00023 


.00071 
•00073 
.00074 
•00075 
•00076 


.00072 
.00073 
.00074 
.00075 
•00076 


.00153 
.00154 
.00156 
.00158 
.00159 




.00153 
.00155 
.00156 
.00158 
.00159 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.00001 
.00001 
.00001 
.00001 
.00002 


.00001 
.00001 
.00001 
.00001 
.00002 


.00024 
.00024 
.00025 
.00026 
.00026 




.00024 
.00024 
.00025 
.00026 
.00026 


•00077 
•00078 
.00079 
.00081 
•00082 


•00077 
•00078 
•00079 
•00081 
.00082 


.00161 
.00162 
•00164 
.00166 
.00168 




.00161 
.00163 
.00164 
.00166 
.00168 


15 
16 
17 
18 

19. 


io 

21 
22 
23 
24 


.00002 
.00002 

00002 
.00002 

00002 


.00002 
.00002 
.00002 
.00002 
-00002 

.00003 
.00003 
.00003 
.00003 
.00004 


.00027 
.00028 
.00028 
.00029 
.00030 




00027 
00028 
00028 
00029 
00030 


.00083 
00084 
00085 
00087 

.00088 


•00083 
•00084 
•00085 
.00087 
.00088 


.00169 
•00171 
•00173 
•00174 
.00176 




•00169 
•00171 
.00173 
•00175 
•00176 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.00003 
00003 
. 00003 
.00003 
.00004 


.00031 
.00031 
.00032 
.00033 
00034 




.00031 

00031 

00032 

.00033 

.00034 


.00089 
00090 
.00091 
.00093 
.00094 


•00089 
.00090 
.00091 
•00093 
.00094 


• 0017*i 
•00179 
•00181 
.00183 
.00185 




.00178 
.00180 
•00182 
•00183 
.00185 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.00004 
•00004 
• 00004 
.00005 
•00005 


.00004 
.00004 
.00004 
.00005 
.00005 


00034 
.00035 
.00036 
.00037 
.00037 




00034 
00035 
00036 
00037 
00037 


00095 
.00096 
.00093 
.00099 
.00100 


•00095 
.00097 
00098 
.00099 
•00100 


00187 

00188 

•00190 

•00192 

•00194 




.00187 
.00189 
.00190 
.00192 
.00194 


30 

31 
32 
33 
34 


35 
36 
37 
38 
,39 


00005 

•00005 

•00006 

00006 

00006 


.00005 
.00005 
.00006 
00006 
.00006 


.00038 
.00039 
.00040 
.00041 
•00041 




00038 
00039 
00040 
00041 
00041 


.00102 
.00103 
.00104 
.00106 
•00107 


.00102 
.00103 
•00104 
.00106 
•00107 


.00196 

:ooi97 

.00199 
.00201 
.00203 




•00196 
•00198 
•00200 
.00201 
•00203 


35 
36 
37 
38 

39 


40 

41 
42 
43 
44 


00007 
00007 
.00007 
00008 
00008 


•00007 
.00007 
•00007 
.00008 
■00008 

00009 

.00009 

.00009 

..00010 

.00010 


.00042 
.00043 
00044 
.00045 
.00046 




00042 
00043 
00044 
00045 
00046 


•00108 
.00110 
.00111 
.00112 
•00114 


•00108 
.00110 
•00111 
•00113 
00114 

.00115 
.00117 
.00118 
.00120 
.00121 

.00122 
.00124 
.00125 
.00127 
•00128 


.00205 
•00207 
.00208 
.00210 
00212 


- 


.00205 
•00207 
•00209 
•00211 
•00213 

00215 
^0216 
00218 
00220 
00222 


40 

41 
42 
43 
44 


45 
46 
47 
47 
49 


00009 
00009 
00009 
00010 
00010 


.00047 

00048 

00048 

.00049 

.00050 


- 


00047 
00048 
00048 
00049 
00050 

00051 
00052 
00053 
00054 
00055 


.00115 
.00U7 
.00118 
.00119 
•00121 


■00214 
00216 
.00218 
.00220 
•00222 


'3 


50 

51 
52 
53 
54 


00011 
■00011 
00011 
00012 
00012 


.00011 
.00011 
.00011 
.00012 
.00012 


.00051 
.00052 
.00053 
.00054 
.00055 


.00122 
.00124 
.00125 
.00127 
•00128 


00224 
.00226 
.00228 

00230 
•00232 


- 


00224 
00226 
00228 
00230 
_00232_ 
00234 
00236 
C0238 
00240 
00242 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


00013 
•00013 
00014 
00014 
00015 


.00013 
.00013 
.00014 
.00014 
.00015 


■00056 
00057 
.00058 
■00059 
■00060 




00056 
00057 
00058 
0U059 
00060 


00130 
•00131 
.00133 
.00134 
.00136 


•00130 
.00131 
00,133 
.00134 
•00136 


.00234 
.00236 
00238 
.00240 
.00242 


55 
56 
57 
58 

59 


60 


•00015 


.00015 


.00061 


.00061 


00137 


•00137 


•00244 




00244 


60 



760 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



6° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.00244 
.00246 
.00248 
.00250 
•00252 


.00244 
.00246 
.00248 
.00250 
.00252 
.00254 
.00257 
.00259 
.00261 
00263 

.00265 
.00267 
.00269 
.00271 
.00274 

.00276 
.00278 
•00280 
.00282 
.00284 


.00381 
00383 
•00386 
.00388 
.00391 


.00382 
•00385 
•00387 
.00390 
•00392 


.00548 
•00551 
.00554 
•00557 
.00560 


.00551 
.00554 
.00557 
•00560 
•00563 


00745 
•00749 

00752 
•00756 
-00760 


.00751 
•00755 
.00758 
.00762 
00765 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.00254 
.00256 
.00258 
.00260 
•00262 


.00393 
.00396 
.00398 
.00401 
.00404 

.00406 
.00409 
.00412 
.00114 
.00417 


•00395 
•00397 
•00400 
•00403 
.00405 


.00563 
.00566 
.00569 
.00572 
•00576 


•00566 
•00569 
.00573 
•00576 
•00579 


•00763 
•00767 
•00770 
.00774 
•00778 


.00769 
•00773 
•00776 
•00780 
-00784 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


00264 
.00266 
.00269 
•00271 

00273 


.00408 
.00411 
•00413 
.00416 
.00419 


00579 

•00582 

00585 

00588 

.00591 


.00582 
.00585 
.00588 
•00592 
•00595 


•00781 
.00785 
•00789 
00792 
•00796 


.00787 
.00791 
•00795 
•00799 
-00802 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


00275 
.00277 
.00279 
•00281 
•00284 


.00420 
.00422 
.00425 
.00428 
•00430 


•00421 
.00424 
•00427 
•00429 
•00432 


00594 
.00598 
•00601 
•00604 
•00607 


•00598 
•00601 
•00604 
.00608 
00611 


-00800 
.00803 
.00807 
.00811 
-00814 


.00806 
.00810 
.00813 
.00817 
-00821 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.00286 
00288 
•00290 
•00293 
•00295 


.00287 
.00289 
.00291 
.00293 
•00296 


•00433 
.00436 
.00438 

• 00441 

• 00444 


•00435 
•00438 
.00440 
• 00443 
•00446 


.00610 
•00614 
.00617 
.00620 
•00623 


00614 
•00617 
.00621 
•00624 
•00627 


.00818 
.00822 
.00825 
00829 
-00833 


.00825 
•00828 
•00832 
•00836 
•00840 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•00297 
•00299 
-00301 
•00304 
•00306 


.00298 
.00300 
.00302 
.00305 
.00307 


•00447 
.00449 
.00452 
.00455 
.00458 


•00449 
•00451 
.00454 
•00457 
•00460 


•00626 
•00630 
•00633 
•00636 
• 00640 


.00630 
.00634 
•00637 
•00640 
•00644 


00837 
.00840 
.00844 
.00848 
.00852 


.00844 
.00848 
.00851 
.00855 
.00859 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•00308 
.00311 
.00313 
00315 
.00317 


.00309 
.00312 
.00314 
.00316 
.00318 


.00460 
.00463 
.00466 
.00469 
.00472 


.004b3 
.00465 
.00468 
.00471 
•00474 


•0C643 
•00646 
.00649 
•00653 
•00656 


.00647 
•00650 
•00654 
•00657 
.00660 


.00856 
00859 
-00863 
-00867 
•00871 


.00863 
.00867 
.00871 
.00875 
•00878 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•00320 
•00322 
•00324 
•00327 
•00329 


.00321 
.00323 
.00326 
.00328 
.00330 


.00474 
.00477 
00480 
•00483 
•00486 


•00477 
•00480 
•00482 
•00485 
.00488 


•00659 
•00663 
•0C666 
•00669 
•00673 


•00664 
•00667 
•00671 
•00674 
.00677 


•00875 

•00878 

•00882 

00886 

00890 


.00882 
00886 
.00890 
.00894 
.00898 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•00332 
•00334 
■00336 
.00339 
•00341 


.00333 
.00335 
.00337 
•00340 
.00342 


.00489 
.00492 
•00494 
•00497 
•00500 


.00491 
•00494 
.00497 
.00500 
•00503 


•00676 
•00680 
•00683 
•00686 
•00690 


.00681 
.00684 
.00688 
.00691 
.00695 


•00894 
•00898 
•00902 
•00906 
•00909 


.00902 
.00906 
.00910 
.00914 
•00918 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•00343 

.00346 

•00348 

00351 

00353 

•00356 
•00358 
00361 
•0036d 
•00365 


.00345 
.00347 
00350 
.00352 
.00354 


•00503 
•00506 
00509 
.00512 
.00515 


•00506 
•C0509 
.00512 
•00515 
•00518 


•00693 
00607 
•00700 
•00703 
•00707 


.00698 
.00701 
.00705 
.00708 
.00712 


-00913 
00917 
00921 
00925 

•00929 


.00922 
.00926 
.00930 
.00934 
00938 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.00357 
.00359 
.00362 
.00364 
.00367 


•00518 
.00521 
.00524 
.00527 
.00530 


.00521 
.00524 
.00527 
•00530 
•00533 


.00710 
.00714 
•00717 
.00721 
.00724 


.00715 
.00719 
.00722 
.00726 
.00730 


.00933 
00937 
. 00941 
.00945 
•00949 


•00942 
.00946 
.00950 
.00954 
•00958 


50 

51 
52 
53 
54 


55 
5b 
57 
58 
59 


00368 
.00370 
.00373 
•00375 
•00378 


.00369 
.00372 
•00374 
00377 
.00379 


•00533 
.00536 
•00539 
•00542 
00545 


.00536 
.00539 
.00542 
.00545 
00548 


.00728 
.00731 
.00735 
•00738 
.00742 


.00733 
.00737 
.00740 
.00744 
.00747 


•00953 
•00957 
.00961 
00965 
•00969 


.00962 
.00966 
.00970 
.00975 
•00979 


55 
56 
57 
58 
59 


60 


•00881 


.00382 


00548 


.00551 


.00745 


.00751 


.00973 


•00983 


60 



761 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



8= 



10^ 



ir 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


■/ 




1 

2 
3 
4 


.00973 
.00977 
.00981 
.00985 
.00989 


.00983 
.00987 
•00991 
.00995 
.00999 


.01231 
•01236 
.01240 
.01245 
.01249 




01247 
01251 
01256 
01261 
01265 


.01519 
.01524 
•01529 
.01534 
.01540 


.01543 
.01548 
.01553 
.01558 
-01564 


•01837 
.01843 
.01848 
.01854 
•01860 


.01872 
•01877 
•01883 
•01889 
•01895 




1 
2 
3 
4 


5 
6 
7 
8 
9 


00994 
•00998 
.01002 
.01006 
.01010 


•01004 
.01008 
.01012 
.01016 
.01020 


.01254 
.01259 
•01263 
■01268 
•01272 




01270 
01275 
01279 
01284 
01289 


.01545 
•01550 
.01555 
.01560 
.01565 


.01569 
.01574 
.01579 
.01585 
.01590 


.01865 

01871 

.01876 

.01882 

01888 


.01901 
.01906 
.01912 
•01918 
.01924 


5 

6 
7 
8 
9 


10 

11 
12 
13 
14 


.01014 
.01018 
.01022 
.01027 
.01031 


•01024 
.01029 
•01033 
.01037 
.01041 


.01277 
.01282 
•01286 
•01291 
•01296 




01294 
01298 
01303 
01308 
01313 


•01570 
•01575 
•01580 
•01586 
•01591 


.01595 
.01601 
.01606 
.01611 
.01616 


.01893 
.01899 
.01904 
.01910 
•01916 


.01930 
.01936 
.01941 
.01947 
•01953 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.01035 
.01039 
■01043 
•01047 
•01052 


.01046 
.01050 
.01054 
.01059 
.01063 


.01300 
.01305 
.01310 
.01314 
01319 




01318 
01322 
01327 
01332 
01337 


•01596 
■01601 
.01606 
.01612 
.01617 


.01622 
.01627 
.01633 
.01638 
.01643 


.01921 
.01927 
.01933 
.01939 
•01944 


.01959 
.01965 
.01971 
.01977 
.01983 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


•01056 

01060 

01064 

.01069 

•01073 


.01067 
.01071 
.01076 
.011180 
.0rD84 

.01089 
.01093 
.01097 
•01102 
.01106 


•01324 
•01329 
•01333 
•01338 
.01343 

•01348 
.01352 
.01357 
•01362 
.01367 




01342 
01346 
01351 
01356 
01361 


.01622 
.01627 
.01632 
.01638 
.01643 


.01649 
.01654 
•01659 
•01665 
.01670 

•01676 
.01681 
•01687 
.01692 
.01698 


•01950 
.01956 
.01961 
•01967 
•01973 


.01989 
.01995 
.02001 
.02007 
•02013 

.02019 
.02025 
.02031 
.02037 
.02043 

.02049 
.02055 
.02061 
.02067 
.02073 


20 

21 

22 
23 
24 


25 
26 
27 
28 
29 


.01077 
.01081 
.01086 
•01090 
.01094 




01366 
01371 
01376 
01381 
01386 


.01648 
.01653 
.01659 
•01664 
.01669 


01979 
•01984 
•01990 
.01996 
.02002 


25 
26 
27 
28 

29 


30 

31 
32 
33 
34 


.01098 
.01103 
01107 
.01111 
.01116 


•01111 
.01115 
.01119 
.01124 
.01128 


.01371 
.01376 
•01381 
.01386 
.01391 




01391 
01395 
01400 
01405 
01410 


.01675 
.01680 
•01685 
.01690 
•01696 


.01703 
.01709 
.01714 
.01720 
•01725 


.02008 
.02013 
.02019 
•02025 
•02031 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


■01120 
.01124 
.01129 
.01133 
.01137 


.01133 
.01137 
.01142 
.01146 
•01151 

.01155 
.01160 
.01164 
.01169 
01173 


.01396 
-01400 
.01405 
.01410 
•014^5 

•01420 
.01425 
.01430 
-01435 
.01439 




01415 
01420 
01425 
01430 
01435 


.01701 
.01706 
.01712 
•01717 
.01723 


•01731 
.01736 
.01742 
.01747 
•01753 


•02037 
•02042 
.02048 
.02054 
■02060 


.02079 
.02085 
.02091 
.02097 
•02103 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


01142 
•01146 
.01151 
.01155 
•01159 




01440 
01445 
01450 
01455 
01461 


•01728 
01733 
.01739 
.01744 
.01750 


•01758 
•01764 
•01769 
.01775 
•01781 


■02066 
.02072 
.02078 
.02084 
.02090 


.02110 
.02116 
.02122 
.02128 
.02134 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


.01164 
.01168 
■01173 
.01177 
•01182 


.01178 
.01182 
.01187 
.01191 
.01196 


.01444 
.01449 
.01454 
•01459 
•01464 




.01466 
01471 
.01476 
.01481 
.01486 


•01755 
.01760 
■01766 
•01771 
•01777 


01786 
•01792 
•01793 
•01803 
.01809 

•01815 
.01820 
.0.1826 
•01832 
01837 


.02095 
.02101 
.02107 
.02113 
•02119 


.02140 
.02146 
.02153 
.02159 
.02165 

.02171 
.02178 
.02184 
.02190 
.02196 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.01186 
.01191 
.01195 
.01200 
.01204 


.01200 
.01205 
.01209 
.01214 
.01219 


•01469 
.01474 
.01479 
•01484 
.01489 




01491 

.01496 

.01501 

01506 

01512 


.01782 
.01788 
.01793 
.0179^w 
01804 


.02125 
.02131 
.02137 
.02143 
.02149 


50 

51 
52 
53 
54 


55 
56 
57 
58 

59 


.01209 
.01213 
.01218 
•01222 
■ 01227 


.01223 
•01228 
.01233 
.01237 
.01242 


.01494 
.01499 
.01504 
.01509 
■01514 




01517 
01522 
.01527 
01532 
01537 


.01810 
.01815 
.01821 
.01826 
.01832 


•0184a 
•01849 
•U1854 
•01860 
.01866 


.02155 
.02161 
.02167 
.02173 
.02179 


.02203 
•02209 
•02215 
.C2221 
.02228 


55 
56 
57 
58 
59 


60 


01231 


.01247 


.01519 




01543 


.01837 


.01872 


•02185 


.02234 


60 



762 



TABLE X— NATURAL VERSED SINES AND EXTERNAL SECANTS, 



13° 



13^ 



14^ 



15^ 



f 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.02185 
.02191 
•02197 
.02203 
.02210 


.02234 
.02240 
.02247 
.02253 
.02259 


.02563 
.02570 
.02576 
.02583 
.02589 


.02630 
.02637 
.02644 
.02651 
.02658 


.02970 
.02977 
.02985 
.02992 
.02999 


.03061 
.03069 
.03076 
•03084 
•03091 


.03407 
•03415 
•03422 
.03430 
•03438 


.03528 
.03536 
.03544 
.03552 
■03560 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.02216 
.02222 
.02228 
.02234 
.02240 


.02266 
.02272 
.02279 
.02235 
.02291 

.02298 
■02304 
.02311 
.02317 
.02323 


•02596 
.02602 
.02609 
.02616 
.02622 


.02665 
.02672 
.02679 
.02686 
.02693 

.02700 
.02707 
•02714 
.02721 
.02728 


.03006 
.03013 
.03020 
.03027 
•03034 


.03099 
.03106 
.03114 
.03121 
•03129 

.03137 
.03144 
.03152 
•03159 
-03167 


•03445 
•03453 
03460 
•03468 
.03476 


.03568 
.03576 
.03584 
.03592 
.03601 


5 

6 
7 
8 
9 


10 

11 
12 
13 

14 


.02246 
.02252 
.02258 
.02265 
•02271 


.02629 
•02635 
.02642 
.02649 
.02655 


.03041 
•03048 
.03055 
•03063 
.03070 


.03483 
.03491 
•03498 
•03506 
•03514 


.03609 
.03617 
.03625 
.03633 
.03642 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


02277 
.02283 
■02289 
.02295 
•02302 


.02330 
.02336 
.02343 
.02349 
.02356 


.02662 
.02669 
•02675 
•02682 
.02689 


.02735 
.02742 
.02749 
•02756 
.02763 


.03077 
.03084 
.03091 
•03098 
•03106 


•03175 
•03182 
•03190 
•03198 
•03205 


•03521 
•03529 
.03537 
-03544 
•03552 


.03650 
.03658 
•03666 
.03674 
•03683 


15 
16 
17 
18 

.19 


30 

21 
22 
23 
24 


.02308 
.02314 
.02320 
.02327 
.02333 


.02362 
.02369 
.02375 
.02382 
.02388 


•02696 
•02702 
•02709 
.02716 
.02722 


.02770 
.02777 
.02784 
.02791 
•02799 


•03113 
•03120 
•03127 
•03134 
.03142 


-03213 
•03221 
•03228 
.03236 
•03244 


-03560 
-03567 
03575 
.03583 
-035G0 


.03691 
.03699 
.03708 
.03716 
.03724 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.02339 
.02345 
•02352 
•02358 
•02364 


.02395 
.02402 
.02408 
•02415 
.02421 


•02729 
•02736 
.02743 
.02749 
.02756 


•02806 
.02813 
.02820 
.02827 
•02834 


.03149 
•03156 
.03163 
■03171 
.03178 


.03251 
.03259 
•03267 
•03275 
-03282 


-03598 
-03606 
-03614 
-03621 
-03629 


.03732 
.03741 
.03749 
.03758 
•03766 


2§ 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•02370 
02377 
.02383 
.02389 
•02396 


.02428 
.02435 
.02441 
.02448 
.02454 


.02763 
.02770 
.02777 
.02783 
.02790 


.02842 
.02849 
.02856 
.02863 
.02870 


-03185 
•03193 
•03200 
•03207 
•03214 


•03290 
•03298 
.03306 
.03313 
•03321 


-03637 
.03645 
.03653 
.03660 
-03668 


.03774 
.03783 
-03791 
.03799 
03808 


30 

31 
32 
33 
34 


36 
37 
38 
39 


•02402 

• 02408 
.02415 

• 02421 
•02427 


.02461 
.02468 
.02474 
.02481 
.02488 


.02797 
.02804 
•02811 
•02818 
•02824 


.02878 
.02885 
.02892 
.02899 
.02907 


-03222 
.03229 
.03236 
.03244 
-03251 


.03329 
•03337 
.03345 
•03353 
03360 


-03676 
-03684 
-03692 
-03699 
•03707 


-03816 
-03825 
.03833 
.03842 
-03850 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•02434 
•02440 
.02447 
•02453 
•02459 


.02494 
.02501 
.02508 
.02515 
.02521 


•02831 
•02838 
•02845 
02852 
•02859 


.02914 
.02921 
.02928 
.02936 
.02943 

.02950 
.02958 
.02965 
.02972 
.02980 

.02987 
.02994 
•03002 
.03009 
.03017 


•03258 
•03266 
•03273 
•03281 
.03288 


•03368 
•03376 
•03384 
.03392 
.03400 


03715 
■03723 
-03731 
-03739 
-03747 


.03858 
.03867 
.03875 
.03884 
•03892 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•02466 
•02472 
•02479 
• 02485 
•02492 


.02528 
.02535 
.02542 
.02548 
.02555 


.02866 
.02873 
.02880 
.02887 
.02894 


•03295 
•03303 
•03310 
•03318 
03325 


•03408 
•03416 
.03424 
•03432 
•03439 


-03754 
-03762 
03770 
.03778 
.03786 


.03901 
.03909 
.03918 
.03927 
.03935 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•02498 
•02504 
•02511 
•02517 
02524 

•02530 
•02537 
.02543 
•02550 
•02556 


.02562 
.02569 
.02576 
.02582 
.02589 


•02900 
•02907 
•02914 
.02921 
•02928 


03333 
.03340 
.03347 
-03355 
.03362 


•03447 
•03455 
.03463 
•03471 
•03479 


.03794 
.03802 
.03810 
.03818 
-03826 


.03944 
.03952 
•03961 
•03969 
•03978 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.02596 
.02603 
.02610 
.02617 
•02624 


•02935 
•02942 
02949 
•02956 
•02963 


.03024 
.03032 
.03039 
03046 
.03054 


.03370 
.03377 
03385 
.03392 
.03400 


.03487 
•03495 
•03503 
.03512 
•03520 


-03834 
•03842 
.03850 
.03858 
-03866 


-03987 
.03995 
.04004 
•04013 
•04021 


55 
56 
57 
58 
59 


60 


-02563 


•02630 


.02970 


.03061 


03407 


.03528 


.03874 


.04030 


60 



763 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



16= 



17^ 



18^ 



19= 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec 


/ 




1 

2 
3 
4 


•03874 
•03882 
•03890 
•03898 
• 03906 


.04030 
.04039 
.04047 
•04056 
.04065 


.04370 
•04378 
•04387 
•04395 
•04404 


•04569 
•04578 
•04588 
•04597 
.04606 


.04894 
.04903 
.04912 
.04921 
.04930 


.05146 
•05156 
•05166 
•05176 
.05186 


.05448 
.05458 
.05467 
•05477 
.05486 


.05762 
05773 
.05783 
•05794 
.05805 




1 
2 
3 
4 


5 
6 
7 
8 
9 


• 03914 
.03922 
.03930 
.03938 
.03946 


.04073 
.04082 
.04091 
.04100 
.04108 


.04412 
.04421 
.04429 
.04438 
.04446 


.04616 
•04625 
04635 
.04644 
•04653 


•04939 
.04948 
•04957 
•04967 
.04976 


.05196 
.05206 
.05216 
.05226 
.05236 


•05496 
.05505 
.05515 
.05524 
.05534 


.05815 
.05826 
•05836 
•05847 
.05858 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.03954 
.03963 
.03971 
•03979 
•03987 


.04117 
04126 
.04135 
.04144 
.04152 


.04455 
.04464 
.04472 
.04481 
.04489 


.04663 
•04672 
•04682 
.04691 
.04700 


•04985 
•04994 
•05003 
•05012 
•05021 


•05246 
•05256 
•05266 
•05276 
.05286 


•05543 
•05553 
•05562 
•05572 
•05582 


.05869 
.05879 
.05890 
.05901 
.05911 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.03995 
.04003 
.04011 
• 04019 
•04028 


•04161 
•04170 
•04179 
•04188 
.04197 


.04498 
.04507 
.04515 
.04524 
•04533 


•04710 
.04719 
•04729 
•04738 
.04748 


•05030 
.05039 
.05048 
.05057 
.05067 


•05297 
•05307 
•05317 
•05327 
.05337 


.05591 
•05601 
05610 
.05620 
•05630 


•05922 
•05933 
•05944 
•05955 
.05965 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


•04036 
•04044 
•04052 
04060 
•04069 


•04206 
•04214 
•04223 
•04232 
.04241 


.04541 
.04550 
.04559 
.04567 
•04576 


.04757 
•04767 
•04776 
•04786 
•04795 


.05076 
.05085 
.05094 
.05103 
•05112 


.05347 
.05357 
•05367 
•05378 
.05383 


•05639 
.05649 
.05658 
.05668 
.05678 


.05976 
•05987 
•05998 
•06009 
.06020 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.04077 
.04085 
.04093 
04102 
•04110 


.04250 
•04259 
•04268 
•04277 
•04286 


.04585 
.04593 
•04602 
•04611 
.04620 


•04805 
•04815 
•04824 
04834 
•04843 


.05122 
.05131 
•05140 
.05149 
•05158 


•05398 
•05408 
.05418 
•05429 
.05439 


.05687 
.05697 
05707 
.05716 
.05726 


.06030 
.06041 
•06052 
•06063 
•06074 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.04118 
04126 

•04135 
04143 

.04151 


•04295 
•04304 
•04313 
•04322 
.04331 


•04628 
.04637 
•04646 
04655 
.04663 


•04853 
•04863 
.04872 
•04882 
•04891 


.05168 
.05177 
.05186 
.05195 
•05205 


.05449 
•05460 
•05470 
.05480 
•05490 


.05736 
.05746 
.05755 
•05765 
.05775 


•06085 
•06096 
•06107 
•06118 
.06129 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.04159 
.04168 
.04176 
.04184 
•04193 


.04340 
.04349 
•04358 
.04367 
.04376 


•04672 
.04681 
•04690 
.04699 
•04707 


•04901 
•04911 
04920 
.04930 
.04940 


•05214 
.05223 
.05232 
.05242 
•05251 


•05501 
.05511 
.05521 
.05532 
.05542 


.05785 
.05794 
.05804 
.05814 
.05824 


•06140 
•06151 
•06162 
•06173 
.06184 


35 
36 
37 
38 
-39 


40 

41 
42 
43 
44 


•04201 
•04209 
.04218 
.04226 
•04234 


•04385 
•04394 
•04403 
•04413 
.04422 


-04716 
.04725 
.04734 
.04743 
•04752 


•04950 
.04959 
.04969 
-04979 
•04989 


.05260 
.05270 
.05279 
-.05288 
.05298 


•05552 
.05563 
.05573 
.05584 
.05594 


•05833 
•05843 
.05853 
•05863 
.05873 


.06195 
•06206 
•06217 
•06228 
.06239 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•04243 
.04251 
04260 
•04268 
•04276 


•04431 
•04440 
•04449 
•04458 
•04468 


.04760 
•04769 
.04778 
.04787 
.04796 


•04998 
•05008 
•05018 
•05028 
.05038 


•05307 
•05316 
•05326 
•05335 
•05344 


.05604 
•05615 
.05625 
•05636 
.05646 


.05882 
.05892 
.05902 
.05912 
.05922 


.06250 
.06261 
-06272 
•06283 
•06295 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•04285 
.04293 
.04302 
•04310 
•04319 


•04477 
•04486 
•04495 
•04504 
.04514 


.04805 
.04814 
•04323 
.04832 
•04841 


•05047 
•05057 
.05067 
•05077 
.05087 


.05354 
.05363 
.05373 
.05382 
•05391 


.05657 
.05667 
.05678 
.05688 
.05699 


.05932 
.05942 
.05951 
.05961 
05971 


•06306 
•06317 
•06328 
.06339 
.06350 


50 

51 
52 
53 
54 


55 
56 
57 
58 

59 


.04327 
.04336 
.04344 
•04353 


.04523 
•04532 
•04541 
.04551 
•04560 


•04850 
• 04.858 
•04867 
.04876 
04885 


•05097 
•05107 
•05116 
.05126 
•05136 


.05401 
•05410 
•05420 
.05429 
•05439 


.05709 
.05720 
•05730 
.05741 
.05751 


•05981 
.05991 
.06001 
.06011 
-06021 


•06362 
.06373 
.06384 
.06395 
.06407 


55 
56 
57 
58 
59 


60 


04370 


•04569 


•04894 


.05146 


•05448 


.05782 


•06031 


•06418 


60 



764 



TAB1.E X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
30° 21° 33° 33° 



/ 


Vers. 


Exc sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.G6031 
•06041 
•06051 
•06061 
•06071 


•06418 
.06429 
.06440 
.06452 
06463 


•06642 
•06652 
.06663 
.Q6673 
.06684 


.07115 
.07126 
•07138 
•07150 
•07162 


•07282 
.07293 
•07303 
.07314 
.07325 


•07853 
.07866 
•07879 
.07892 
-07904 


.07950 
.07961 
.07972 
.07984 
•07995 


•08636 
•08649 
•08663 
•08676 
•08690 




1 
2 
3 

4 


5 
6 
7 
8 
9 


•06081 
.06091 
•06101 
•06111 
06121 


.06474 
.06486 
06497 
.06508 
.06520 

.06531 
.06542 
.06554 
.06565 
.06577 


.06694 
.06705 
•06715 
.06726 
.06736 


.07174 
.07186 
.07199 
.07211 
•07223 


•07336 
•07347 
•07358 
•07369 
•07380 


.07917 

07930 

07943 

.07955 

•07968 


•08006 
.08018 
.08029 
.08041 
•08052 


.08703 
•08717 
•08730 
•08744 
•08757 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•06131 
.06141 
•06151 
•06161 
•03171 


. 06747 
.06757 
•06768 
•06778 
•06789 


•07235 
.07247 
•07259 
•07271 
.07283 


•07391 
•07402 
•07413 
.07424 
.07435 


•07981 
•07994 
•08006 
.08019 
.08032 


•08064 
.08075 
.08086 
.08098 
.08109 


•08771 
•08784 
.08798 
.08811 
.08825 

«08839 
.08852 
.08866 
.08880 
.08893 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


•06181 
•06191 
• 06201 
•06211 
•06221 


•06588 
.06600 
.06611 
.06622 
.06634 


•06799 
•06810 
•06820 
•06831 
•06841 


•07295 
•07307 
•07320 
•07332 
.07344 


.07446 
•07457 
•07468 
•07479 
.07490 


•08045 
.08058 
.08071 
•08084 
.08097 


•08121 
.08132 
.08144 
•08155 
08167 


15 
13 
17 
18 
1? 


30 

21 
22 
23 
24 


.06231 
.06241 
.06252 
.06262 
•06272 


.06645 
.06657 
.06868 
.06680 
•06891 


• 0t)852 
•06863 
.06873 
.06884 
.06894 


.07356 
.07368 
.07380 
.07393 
.07405 


.07501 
•07512 
.07523 
.07534 
.07545 


•08109 
.08122 
.08135 
.08148 
.08161 


.08178 
•03190 
•08201 
•08213 
.08225 

.08236 

08248 

.08259 

•08271 

08282 


.08907 
.08921 
08934 
.08948 
.08962 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•06282 
.06292 
.06302 
.06312 
•06323 


.06703 
.06715 
.06726 
.06738 
.06749 


•06905 
.06918 
•06926 
•06937 
•06948 


.07417 
.07429 
.07442 
•07454 
.07486 


•07556 
•07568 
•07579 
.07590 
.07601 


.08174 
.08087 
.08200 
•08213 
.08226 


.08975 
.08989 
•09003 
•09017 
.09030 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•08333 
.06343 
.06353 
.06363 
•06374 


•06761 
.06773 
.06784 
.06796 
.06807 


•06958 
•06969 
•06980 
•06990 
.07001 


.07479 
.07491 
.07503 
.07516 
.07528 


•07612 
07623 
.07634 
.07645 
•07657 


.08239 
.08252 
.08265 
.08278 
•08291 


.08294 
•08306 
•08317 
.08329 
08340 


.09044 
.09058 
.09072 
.09086 
.09099 


30 

31 
32 
33 

34 


35 
36 
37 
38 
39 


•06384 
•06394 
.06404 
•06415 
.06425 


.06819 
•06831 
•06843 
•06854 
•06866 


.07012 
.07022 
•07033 
.07044 
•07055 


.07540 
.07553 
.07565 
.07578 
•07590 


.07668 
•07679 
.07690 
■07701 
•07713 


•08305 
.08318 
.08331 
.08344 
.08357 


•08352 
•08364 
.08375 
.08387 
•08399 


.09113 
09127 
-09141 
.09155 
•09169 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•06435 
•06445 
•06456 
.06466 
.06476 


.06878 
.06889 
•06901 
•06913 
•06925 


•07065 
.07076 
•07087 
•07098 
.07108 


.07o02 
•07615 
•07627 
•07040 
•07652 


•07724 
•07735 
.07746 
.07757 
.07769 


.08370 
.08383 
08397 
.08410 
.08423 


.08410 
•08422 
08434 
•08445 
.08457 


09183 
.09197 
.09211 
.09224 
.09238 


40 

41 

42 
48 
44 


45 
46 
47 
48 
49 


06486 
.06497 
.06507 
■06517 
• 06528 


.06936 
-06948 
.06960 
.06972 
06984 


.07119 
•07130 
•07141 
.07151 
.07162 


.07665 
.07677 
.07690 
.07702 
•07715 


07780 
.07791 
•07802 

07814 
.07825 


.08436 
•08449 
.08463 
-08476 
•08489 


08469 
.08481 
•08492 
•08504 
.08516 


.09252 
.09266 
.09280 
.09294 
09308 


45 
46 
47 
48 

49 


50 

51 
52 
53 
54 


•06538 
•06548 
•06559 
06569 
•06580 


•06995 
.07007 
•07019 
•07031 
.07043 


•07173 
•07184 
•07195 
•07206 
.07216 


.07727 
.07740 
.07752 
.07765 
.07778 


•07836 

07848 

07859 

•07870 

•07881 


.08503 
.08516 
.08529 
.08542 
.08556 


.08528 
•08539 
•08551 
.08563 
•08575 


. .09323 

.09337 

.09351 

.09365 

09379 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.06590 

.06600 

06611 

06621 

•06632 


.07055 
.07067 
•07079 
•07091 
.07103 


.07227 
•07238 
•07249 
•07260 
•07271 


.07790 
•07803 
.07816 
.07828 
.07841 


•07893 
•07904 
.07915 
.07927 
07938 


.08569 
•08582 
•08596 
.08069 
•08623 


.08586 

08598 

•08610 

.08622 

08634 


•09393 
.09407 
-09421 
•09435 
09449 


55 
56 
57 
58 

59 


60 


06642 


•07115 


•07282 


.07853 


.07950 


.08636 


•08645 


•09464 


60 



765 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
34° 25° 26° 21° ' 


/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


1 
2 
3 
4 




1 
2 
3 
4 


08645 
.08657 
.08669 
.08681 
.08693 


.09464 
.09478 
.09492 
.09506 
•09520 

.09535 
.09549 
.09563 
.09577 
.09592 


.09369 
.09382 
.09394 
.09406 
.09418 




10338 
10353 
10368 
10383 
10398 


10121 
.10133 
.10146 
.10159 
-10172 


•11260 
•11276 
•11292 
•11308 
•11323 


.10899 
10913 
•10926 
.10939 
•10952 


.12233 
.1224b 
.12266 
.12283 
•12299 


5 
6 
7 
8 
9 


.08705 
.08717 
08728 
.08740 
.08752 


.09431 
.09443 
.09455 
09468 
.09480 




10413 
10428 
10443 
10458 
10473 


.10184 
•10197 
•10210 
.10223 
•10236 


•11339 
11355 
.11371 
.11387 
•11403 


.10965 
.10979 
•10992 
.11005 
.11019 


.12316 
.12333 
.12349 
•12366 
•12383 


5 
6 
7 
8 

9 


10 

11 
12 
13 
14 


.08764 
•08776 
.08788 
.08800 
.08812 


.09603 
.09620 
.09635 
.09649 
.09663 


.09493 
.09505 
.09517 
.09530 
.09542 




10488 
10503 
10518 
10533 
10549 


• 10248 
•10261 
•10274 
•10287 
•10300 


•11419 
.11435 
.11451 
.11467 
•11483 


•11032 
•11045 
•11058 
11072 
•11085 


•12400 
.12416 
.12433 
.12450 
•12467 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.08824 
.08836 
.08848 
.08860 
•08872 


.09678 
.09o92 
.09707 
.09721 
^09735 


09554 
•09567 
•09579 
•09592 
•09604 




10564 
10579 
10594 
10609 
10625 


•10313 
•10326 
•10338 
•10351 
-10364 


.11499 
11515 
.11531 
.11547 
.11563 


•11098 
•11112 
•11125 
•11138 
•11152 


•12484 
•12501 
.12518 
•125o4 
.12551 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.08884 
.08896 
•08908 
.08920 
.08932 


.09750 
.09764 
.09779 
.09793 
.09808 


.09617 
.09629 
•09642 
•09654 
•09666 




10640 
10655 
10670 
10686 
10701 


•10377 
•10390 
. 10403 
.10416 
10429 


•11579 
•11595 
■11611 
•11627 
•11643 


•11165 
•11178 
•11192 
•11205 
•11218 


•12568 
•12585 
•12602 
•12619 
12636 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.08944 
.08956 
.08968 
•08980 
.08992 


.09822 
.09837 
.09851 
.09866 
.09880 


•09679 
.09691 
.09704 
.09716 
•09729 




10716 
10731 
10747 
10762 
10777 


.10442 
•10455 
•10468 
•10481 
.10494 


•11659 
•11675 
•11691 
•11708 
•11724 


•11232 
•11245 
•11259 
•11272 
.11285 


.12653 
.12670 
.12687 
.12704 
.12721 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•09004 
•09016 
•09028 
•09040 
•09052 


.09895 
•09909 
.09924 
•09939 
•09953 


•09741 
•09754 
•09767 
•09779 
•09792 




1079J 
10808 
10824 
10839 
10854 


•10507 
•10520 
•10533 
•10546 
.10559 


.11740 
.11756 
.11772 
.11789 
-11805 


•11299 
■11312 
•11326 
•11339 
•11353 


•12738 

•12755 

•12772 

•12789. 

.12807 


30 

31 
32 
33 
34, 


35 
36 
37 
38 
39 


.09064 
.09076 
.09089 
•09101 
.09113 


•09968 
•09982 
•09997 
.10012 
•10026 


•09804 

09817 

09829 

•09842 

.09854 




10870 
10885 
10901 
10916 
10932 


.10572 
•10585 
•10598 
•10611 
.10624 


•11821 
•11838 
•11854 
•11870 
•11886 


•11366 
•11380 
•11393 
•11407 
.11420 


•12824 
.12841 
.12858 
.1-875 
•12892 


35 ' 
36 
37 
38 

39 


40 

41 
42 
43 
44 


•09125 
•09137 
•09149 
•09161 
.09174 


.10041 
.10055 
.10071 
.10085 
.10100 

.10115 
.10130 
.10144 
.10159 
•10174 


•09867 
•09880 
.09892 
•09905 
.09918 




10947 
10963 
10978 
10994 
11009 


.10637 
.10650 
•10663 
•10676 
•10689 


.11903 
•11919 
•11936 
•11952 
•11968 


.11434 
•11447 
•11461 
•11474 
.11488 


12910 
.12927 
.12944 
•12961 
.12979 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•09186 
.09198 
.09210 
•09222 
•09234 


•09930 
•09943 
.09955 
.09968 
•09981 




11C25 
11041 
.11056 
11072 
11087 


•10702 
•10715 
•10728 
.10741 
•10755 


•11985 
•12001 
•12018 
•12034 
.12051 


.11501 
•11515 
•11528 
■11542 
•11555 


•12996 
•13013 
•13031 
•13048 
13065 


45 
46 
47 
48 
49 


60 

51 
52 
53 
54 


•09247 
•09259 
09271 
.09283 
.09296 


.10189 
.10204 
.10218 
.10233 
.10248 

.10263 
.10278 
.10293 
.10308 
,.10323 


•09993 
•10006 
•10019 
•10032 
. 10044 




11103 
11119 
11134 
11150 
11166 


•10768 
•10781 

10794 
•10807 

10820 


•12067 
. 12084 
.12100 
•12117 
.12133 


•11569 
•11583 
•11598 
11610 
.11623 


•13083 
•13100 
•13117 
•13135 
13152 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•09308 
•09320 
•09332 
•09345 
.09357 


•10057 
•10070 
•10082 
•10095 
•10108 




11181 
11197 
11213 
11229 
11244 


•10833 
•10847 
•10860 
•10873 
•10886 


•12150 
•12166 
•12183 
.12199 
•12216 


.11637 
•11651 
•11664 
•11678 
•11692 


.13170 
•13187 
•13205 
•13222 
.13240 


55 
56 
57 
58 
59 


60 


•09369 


.10338 


•10121 




11260 


•10899 


.12233 


•11705 


•13257 


60 



766 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
28° 29° 30° 31° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 




' 1 

2 
3 
4 


.11705 
.11719 
.11733 
.11746 
.11760 




.13257 
.13275 
.13292 
.13310 
.13327 


.12538 
.12552 
.12566 
.12580 
.12595 




14335 
14354 
14372 
14391 
.14409 


.13397 
.13412 
.13427 
.13441 
•13456 




15470 
15489 
15509 
15528 
.15548 


• 14283 
.14298 
.14313 
.14328 

• 14343 


.16663 
.16684 
.16704 
.16725 
.16745 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.11774 
.11787 
•11801 
.11815 
.11828 




.13345 
.13362 
.13380 
.13398 

.13415 


.12609 
.12623 
.12637 
.12651 
•12665 




14428 

.14446 

.14465 

14483 

14502 


.13470 
.13485 
.13499 
.13514 
•13529 




.15567 
.15587 
.15606 
.15626 
•15645 


.14358 
.14373 
. 14388 
. 14403 
•14418 


.16766 
.16786 
.16806 
.16827 
.16848 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.11842 
.11856 
.11870 
.11883 
.11897 




.13433 

.13451 

.13468 

13486 

13504 


.12679 
.12694 
.12708 
.12722 
•12736 




.14521 
.14539 
.14558 
.14576 
14595 


.13543 
.13558 
•13573 
•13587 
•13602 




•15665 
.15684 
.15704 
.15724 
.15743 


• 14433 

• 14449 
. 14464 
. 14479 
. 14494 


.16868 
.16889 
.16909 
.16930 
.16950 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.11911 
.11925 
.11938 
.11952 
.11966 




.13521 

.13539 

.13557 

13575 

13593 


.12750 
.12765 
.12779 
.12793 
•12807 




.14614 
.14632 
.14651 
.14670 
14689 


•13616 
•13631 
•13646 
•13660 
.13675 




.15763 
.15782 
.15802 
.15822 
.15841. 


•14509 
.14524 
.14539 
.14554 
.]'^5e9 


.16971 
.16992 
.17012 
.17033 
•17054 


15 
16 
17 
18 
-19 


20 

21 
22 
23 
24 


.11980 
.11994 
.12007 
.12021 
.12035 




13610 
13628 
13646 
13664 
13682 


.12822 
.12836 
.12850 
.12864 
.12879 




.14707 
.14726 
.14745 
.14764 
.14782 


•13690 
•13705 
•13719 
•13734 
•13749 




.15861 
.15881 
.15901 
.15920 
.15940 


•14584 
.14599 
.14615 
.14630 
•14645 


.17075 
.17095 
.17116 
.17137 
•17158 


20 

21. 
22 
23 

24 


25 
26 
27 
28 
29 


.12049 
•12063 
.12077 
•12091 
12104 




13700 
13718 
13735 
13753 
13771 


.12893 
.12907 
.12921 
.12936 
.12950 




.14801 
.14820 
.14839 
.14858 
14877 


.13763 
.13778 
.13793 
.13808 
.13822 




.15960 
.15980 
.16000 
.16019 
16039 


•14660 
.14675 
.14680 
.14706 
.i4721 


.17178 
.17199 
.17220 
•17241 
.17262 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.12118 
•12132 
.12146 
-12160 
12174 




13789 
13807 
13825 
13843 
13861 


.12964 
.12979 
.12993 
.13007 
.13022 

.13036 
.13051 
.13065 
.13079 
•13094 




14896 
14914 
14933 
14952 
14971 


•13837 
.13852 
.13867 
.13881 
•13896 




16059 
16079 
16099 
16119 
16139 


.14736 
.14751 
.14766 
•14782 
.14797 


•17283 
.17304 
.17325 
.17346 
•17367 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.12188 
.12202 
.12216 
•12230 
•12244 




13879 
13897 
13916 
13934 
13952 




14990 
15009 
15028 
15047 
15066 


•13911 
•13926 
•13941 
•13955 
.18970 




16159 
16179 
16199 
16219 
16239 


.14812 
.14827 
.14843 
.14858 
.14873 


.17388 
.17409 
.17430 
.17451 
.17472 


35 
36 
37 
38 
-39 


40 

41 
42 
43 

44 


.12257 
.12271 
.12285 
.12299 
.12313 




13970 
13988 
14006 
14024 
14042 


•13108 
.13122 
.13137 
.13151 
.13166 




15085 
1510J 
15124 
1514.3 
15162 


13985 
• 14000 
-14015 
. 14030 
•14044 




16259 
16279 
16299 
16319 
16339 


.14888 
.14904 
.14819 
.14934 
•14949 


.17493 
.17514 
.17535 
.17556 
.17577 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.12327 
.]234i 
.12355 
.12369 
12383 




14061 
14079 
14097 
14115 
14134 


.13180 
13195 
.13209 
.13223 
.13238 




15181 
15200 
15219 
15239 
15258 


14059 

14074 

•14089 

•14104 

14119 




16359 
16380 
16490 
16420 
16440 


.14965 
.14980 
.14995 
.15011 
•15026 


.17598 
.17620 
.17641 
.17662 
.1768-3- 


45 
46 
47 
48 
49 


50 

51 
1 52 
, 53 

54 


.12397 
.12411 
.12425 
. 12439 
•12454 




14152 
14170 
14188 
14207 
14225 


.13252 
.13267 
.13281 
.13296 
.13310 




15277 
15296 
15315 
15335 
15354 


•14134 
•14149 
.14164 
•14179 
•14194 




16460 
16481 
16501 
16521 
16541 


15041 
.15057 
.15072 
.15087 
•15103 


.17704 
.17726 
.17747 
.17768 
•17790 


50 

51 
52 
53 
54 


!55 
56 
57 
58 
59 


.12468 
.12482 
.12496 

.12510 
.12524 




14243 
14262 
14280 
14299 
14317 


.13325 
.13339 
.13354 
.13368 
.13383 




15373 
15393 
15412 
15431 
15451 


.14208 
.14223 
.14238 
.14253 
•14268 




16562 
16582 
16602 
16623 
16643 


15118 
.15134 
•15149 
.15164 
.15180 


•17811 
.17832 
.17854 
.17875 
.17896 


55 
56 
57 
58 
59 


60»" 


.12538 




14335 


.13397 


. 


15470 


•14283 




16663 


•15195 


.17918 


60 














7i 


)7 













TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS/ 





33^ 


> 


33' 


> 


34 


3 


35 


3 




/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.15195 
.15211 
.15226 
.15241 
.15257 




.17918 
.17939 
.17961 
.17982 
.18004 


.16133 
.16149 
.16165 
.16181 
.16196 




.19236 
.19259 
.19281 
.19304 
.19327 


.17096 
•17113 
.17129 
.17145 
.17161 




.20622 
.20645 
.20669 
•20693 
.20717 


.18085 
.18101 
•18118 
•18135 
•18152 




.22077 
.22102 
.22127 
.22152 
.22177 


O 

1 
2 
3 

4 


5 
6 
7 
8 
9 


.15272 
.15288 
.15303 
.15319 
.15334 




.18025 
.18047 
.18068 
.18090 
•18111 


.16212 
.16228 
.16244 
.16260 
.16276 




.19349 
.19372 
.19394 
.19417 
.19440 


.17178 
.17194 
.17210 
.17227 
.17243 




.20740 
.20764 
.20788 
.20812 
•20836 


•18168 
•18185 
•18202 
•18218 
•18235 




.22202 

.22227 

.22252' 

•22277 

.22302 


5 

6 
7 
8 

9 


iO 

11 
12 
13 
14 


.15350 
.15365 
.15381 
.15396 
.15412 




•18133 
•18155 
.18176 
.18198 
.18220 


.16292 
.16308 
.16324 
•16340 
•16355 




.19463 

.19485 

.19508 

19531 

19554 


.17259 
.17276 
.17292 
.17308 
.17325 




•20859 
.20883 
.20907 
.20931 
.20955 


.18252 
.18269 
.18286 
.18302 
•18319 




.22327 
.22352 
.22377 
.22402 
.22428 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.15427 
•15443 
.15458 
.15474 
.15489 




.18241 
18263 
18285 

.18307 
18328 


•16371 
•16387 
•16403 
•16419 
•16435 




19576 
19599 
19622 
19645 
19668 


.17341 
.17357 
.17374 
.17390 
•17407 




.20979 
.21003 
.21027 
.21051 
.21075 


•18336 
•18353 
•18369 
.18386 
• 18403 




.22453 
.22478 
.22503 
.22528 
•22554 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


.15505 
.15520 
.15536 
.15552 
.15567 




18350 
18372 
18394 
18416 
18437 


•16451 
•16467 
• 16483 
•16499 
-16515 




19691 
19713 
19736 
19759 
19782 


•17423 
•17439 
•17456 
•17472 
•17489 




.21099 
.21123 
.21147 
.21171 
•21195 


•18420 
•18437 
•18454 
•18470 
•18487 




.22579 
.22604 
.22629 
.22655 
.22680 


30 

21 

22 
23 
24 


25 
26 
27 
28 
29 


.15583 
-15598 
•15614 
•15630 
.15645 




18459 
18481 
18503 
18525 
18547 


•16531 
•16547 
•16563 
•16579 
■16595 




19805 
19828 
19851 
19874 
19897 


•17505 
•17522 
•17538 
•17554 
•17571 




•21220 
.21244 
.21268 
.21292 
.21316 


•18504 
•18521 
•18538 
•18555 
.18572 

•18588 
•18605 
•18622 
•18639 
•18656 




.22706 
.22731 
.22756 
.22782 
.22807 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•15661 
•15676 
.15692 
.15708 
•15723 




18569 

18591, 

18613 

18635 

18657 


•16611 
•16627 
•16644 
•16660 
•16676 




19920 
19944 
19967 
19990 
20013 


•17587 
•17604 
•17620 
•17637 
•17653 




.21341 
.21365 
.21389 
.21414 
.21438 




.22833 

.22858 

.22884 

22909 

22935 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•15739 
•15755 
•15770 
•15786 
.15802 




18679 
.18701 
.18723 
.18745 

18767 


•16692 
•16708 
•16724 
•16740 
•16756 




20036 
20059 
20083 
20106 
20129 


•17670 
•17686 
•17703 
•17719 
-17736 




21462 
21487 
21511 
21535 
21560 


•18673 
•18690 
•18707 
•18724 
•18741 




22960 
22986 
23012 
23037 
23063 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•15818 
•15833 
•15849 
•15865 
•15880 




.18790 
18812 
18834 
18856 
18878 


•16772 
•16788 
•16805 
•16821 
.16837 




20152 
20176 
20199 
20222 
20246 


•17752 
•17769 
•17786 
•17802 
•17819 




21584 
21609 
21633 
21658 
21682 


•18758 
•18775 
•18792 
•1880^ 
•18826 




23089 
23114 
23140 
23166 
23192 


40 

41^ 

42 
43 
44 


45 
46 
47 
48 
49 


•15896 
•15912 
•15928 
•15943 
.15959 




18901 
18923 
18945 
18967 
18990 


•16853 
•16869 
.16885 
.16902 
•16918 




20269 
20292 
20316 
20339 
20363 


•17835 
•17852 
•17868 
•17885 
•17902 

•17918 
•17935 
•17952 
•17968 
•17985 




21707 
21731 
21756 
21781 
21805 


18843 
•18860 
•18877 
•18894 
•18911 




23217 
23243 
23269 
23295 
23321 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•15975 
•15991 
•16006 
•16022 
•18038 




19012 
19034 
19057 
19079 
19102 


•16934 
•16950 
•16966 
•16983 
•16999 




20386 
20410 
20433 
20457 
20480- 




21830 
21855 
21879 
21904 
21929 


•18928 
•18945 
•18962 
•18979 
•18996 




23347 
23373 
23399 
23424 
23450 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


• 16054 
•16070 
•16085 
.16101 
.16117 




19124 
19146 
19169 
19191 
19214 


•17015 
•17031 
•17047 
•17064 
•17080 




20504 
20527 
20551 
20575 
20598 


•18001 
•18018 
•18035 
•18051 
•18068 




21953 
21978 
22003 
22028 
22053 


•19013 
•19030 
•19047 
•19064 
•19081 




23476 
23502 
23529 
23555 
23581 


55 
56 
57 
58 
59 


60 


.16133 




19236 


•17096 




20622 


•18085 




22077 


•19098 . 


23607 


60 



768 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS 
36° 37° 38° 39° 



# 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. • 


Ex. sec. 


Vers. 


Ex. sec. 


f 




1 

2 
3 
4 


.19098 
.19115 
.19133 
.19150 
.19167 




23607 
23633 
23659 
23685 
23711 


•20136 
.20154 
.20171 
.20189 
.20207 


.25214 
.25241 
.25269 
.25296 
.25324 


•21199 
•21217 
•21235 
•21253 
•21271 


•26902 
.26931 
.26960 
.26988 
.27017 


•22285 
•22304 
.22322 
.22340 
.22359 


.28676 
.28706 
.28737 
.28767 
•28797 


O 

1 
2 
3 
4 


5 
6 
7 
8 

9 


.19184 
.19201 
.19218 
.19235 
.19252 




23738 
23764 
23790 
23816 
23843 


.20224 
.20242 
.20259 
.20277 
.20294 


.25351 
.25379 
.25406 
.25434 
.25462 


.21289 
•21307 
.21324 
.21342 
•21360 


.27046 
.27075 
.27104 
.27133 
•27162 


.22377 
.22395 
.22414 
•22432 
•22450 


.28828 
.28858 
.28889 
.28919 
.28950 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.19270 
.19287 
.19304 
.19321 
.19338 




23869 
23895 
23922 
23948 
23975 


.20312 
.20329 
.20347 
.20365 
.20382 


.25489 
.25517 
.25545 
.25572 
.25600 

.25628 
.25656 
.25683 
.25711 
•25739 


•21378 
•21396 
•21414 
•21432 
•21450 

•21468 
•21486 
•21504 
•21522 
•21540 


.27191 
•27221 
•27250 
.27279 
•27308 


•22469 
•22487 
•22506 
•22524 
•22542 


.28980 
.29011 
.29042 
.29072 
.29103 


10 

1^ 

13 
14 


15 
16 
17 
18 
19, 


.19356 
.19373 
.19390 
.19407 
.19424 




24001 
24028 
24054 
24081 
24107 


.20400 
.20417 
.20435 
.20453 
.20470 


•27337 
.27366 
.27396 
•27425 
•27454 


•22561 
•22579 
•22598 
•22616 
•22634 


.29133 
•29164 
.29195 
.29226 
•29256 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.19442 
.19459 
.19476 
.19493 
.19511 




24134 
24160 
24187 
24213 
24240 


.20488 
.20506 
.20523 
-20541 
•20559 


.25767 
.25795 
.25823 
•25851 
•25879 


•21558 
•21576 
•21595 
•21613 
•21631 


•27483 
.27513 
.27542 
•27572 
•27601 


•22653 
•22671 
•22690 
•22708 
•22727 


•29287 
.29318 
.29349 
.29380 
•29411 


20 

21 
22 
23 
24 


25 
26 
27 
28 

29 


.19528 
.19545 
.19562 
.19580 
•19597 




24267 
24293 
24320 
24347 
24373 


.20576 
.20594 
.20612 
•20629 
•20647 


•25907 
.25935 
.25963 
.25991 
.26019 


•21649 
•21667 
•21685 
•21703 
•21721 


•27630 
.27660 
.27689 
.27719 
•27748 


•22745 
.22764 
•22782 
•22801 
•22819 


•29442 
.29473 
.29504 
.29535 
.29566 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.19614 
.19632 
.19649 
.19666 
.19684 




24400 
24427 
24454 
24481 
24508 


•20665 
•20682 
•20700 
•20718 
•20736 


.26047 
.26075 
.26104 
.26132 
•26160 


•21739 
•21757 
•21775 
•21794 
•21812 


•27778 
.27807 
.27837 
.27867 
•27896 


•22838 
•22856 
22875 
.22893 
•22912 


.29597 
.29628 
.29659 
.29690 
•29721 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.19701 
.19718 
.19736 
.19753 
.19770 




24534 
24561 
24588 
24615 
24642 


.20753 
.20771 
•20789 
•20807 
•20824 


•26188 
.26216 
.26245 
.26273 
•26301 


•21830 
•21848 
•21866 
•21884 
•21902 


•27926 
•27956 
.27985 
.28015 
•28045 


•22930 
•22949 
•22967 
•22986 
•23004 


•29752 
.29784 
.29815 
.29846 
.29877 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.19788 
.19805 
.19822 
.19840 
.19857 




34669 
24696 
24723 
24750 
24777 


•20842 
•20860 
•20878 
•20895 
•20913 


.26330 
.26358 
.26387 
.26415 
•26443 


21921 
•21939 
•21957 
•21975 
•21993 


.28075 
.28105 
.28134 
.28164 
.28194 


•23023 
•23041 
•23060 
•23079 
•23097 


.29909 
.29940 
.29971 
.30003 
.30034 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.19875 
.19892 
.19909 
.19927 
.19944 




24804 
24832 
24859 
24886 
24913 


.20931 
•20949 
•20967 
•20985 
•21002 


.26472 
.26500 
.26529 
.26557 
.26586 

.26615 
.26643 
.26672 
.26701 
.26729 


•22012 
•22030 
•22048 
•22066 
•22084 


.28224 
•28254 
.28284 
•28314 
•28344 


•23116 
•23134 
•23153 
•23172 
•23190 


.30066 
.30097 
.30129 
.30160 
•30192 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.19962 
.19979 
.19997 
.20014 
.20032 




.24940 
.24967 
.24995 
.25022 
.25049 


•21020 
•21038 
•21056 
•21074 
•21092 


•'22103 
•22121 
•22139 
•22157 
•22176 


.28374 
.28404 
.28434 
.28464 
.28495 


•23209 
•23228 
-23246 
•23265 
•23283 


•30223 
.30255 
.30287 
.30318 
.30350 


50 

51 
52 
53 
54 


55 
56 
67 
58 

59 


. 20049 
.20066 
.20084 
•20101 
•20119 




.25077 
.25104 
.25131 
.25159 
• 25186 


•21109 
•21127 
.21145 
•21163 
•21181 


.26758 
.26787 
.26815 
.26844 
•26873 


•22194 
•22212 
•22231 
•22249 
•22267 


.28525 
.28555 
.28585 
.28615 
•28646 


•23302 
•23321 
•23339 
•23358 

•23377 


.30382 
.30413 
.30445 
.30477 
•30509 


55 
56 
57 
56 
59 


60 


.20136 




.25214 


•21199 


.26902 


•22285 


.28676 


•23396 


.30541 


Off 



769 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



40° 



4r 



42' 



43* 



1 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


•23396 
.23414 
.23433 
.23452 
•23470 


.30541 
.30573 
.30605 
.30636 
.30668 


.24529 
.24548 
.24567 
.24586 
.24605 


.32501 
.32535 
.32568 
.32602 
.32636 


.25686 
.25705 
.25724 
.25744 
.25763 


.34563 
.34599 
.34634 
.34669 
.34704 


•26865 
•26884 
.26904 
.26924 
•26944 


.36733 
.36770 
.36807 
.36844 
•36831 




1 
2 
3 

4 


5 
6 
7 
8 
9 


.23489 
.23508 
.23527 
.23545 
.23564 


.30700 
.30732 
.30764 
.30796 
.30829 


.24625 
.24644 
.24663 
.24682 
.24701 


.32669 
.32703 
.32737 
.32770 
.32804 


.25783 
.25802 
.25822 
•25841 
•25861 


.34740 
.34775 
.34811 
.34846 
•34882 


.26964 
.26984 
.27004 
.27024 
•27043 


.36919 
.36956 
.36993 
.37030 
.37068 


5 
6 
7 
8 
9 


10 

.11 
12 
13 
14 


.23583 
.23602 
.23620 
.23639 
•23658 


.30861 
.30893 
.30925 
.30957 
.30989 


•24720 
.24739 
.24759 
•24778 
•24797 


.32838 
.32872 
.32905 
.32939 
.32973 


•25880 
•25900 
.25920 
.25939 
•25959 


.34917 
.34953 
.34988 
.35024 
•35060 


o 27063 
•27083 
•27103 
.27123 
.27143 


.37105 
.37143 
.37180 
.37218 
•37255 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


• 23677 
.23696 
.23714 
.23733 
•23752 


.31022 
.31054 
.31086 
.31119 
.31151 


.24816 
•24835 
•24854 
•24874 
•24893 


.33007 
.33041 
.33075 
.33109 
•33143 


.25978 
.25998 
.26017 
.26037 
•26056 


.35095 
.35131 
.35167 
.35203 
•35238 


.27163 
.27183 
•27203 
.27223 
•27243 


.37293 
.37330 
.37368 
.37406 
•37443 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


•23771 
.23790 
.23808 
.23827 
.23846 

.23865 
.23884 
.23903 
•23922 
•23941 


.31183 
.31216 
.31248 
.31281 
.31313 


•24912 
.24931 
.24950 
•24970 
•24989 


.33177 
.33211 
.33245 
.33279 
.33314 


.26076 
.26096 
.26115 
•26135 
•26154 


.35274 
.35310 
.35346 
.35382 
•35418 


.27263 
.27283 
•27303 
•27323 
.27343 


.37481 
.37519 
.37556 
.37594 
•37632 


30 

21 

22 
23 
24 


25 
26 
27 
28 
29 


.31346 
.31378 
.31411 
.31443 
.31476 


•25008 
•25027 
.25047 
.25066 
.25085 


.33348 
.33382 
.33416 
.33451 
.33485 


.26174 
.26194 
.26213 
.26233 
.26253 


.35454 
.35490 
.35526 
.35562 
•35598 


•27363 
•27383 
•27403 
.27423 
.27443 


.37670 
.37708 
.37746 
.37784 
•37822 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.23959 
.23978 
.23997 
.24016 
•24035 


.31509 
.31541 
.31574 
.31607 
.31640 


•25104 
•25124 
•25143 
.25162 
.25182 

.25201 
.25220 
.25240 
•25259 
.25278 


.33519 
.33554 
.33588 
.33622 
.33657 

.33691 
.33726 
.33760 
.33795 
•33830 


.26272 
.26292 
.26312 
.26331 
.26351 


.35634 
.35670 
.35707 
.35743 
•35779 


.27463 
•27483 
•27503 
•27523 
•27543 


.37860 
.37898 
.37936 
.37974 
.38012 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.24054 
.24073 
.24092 
.24111 
•24130 


.31672 
.31705 
.31738 
.31771 
.31804 


.26371 
.26390 
.26410 
.26430 
. 26449 


.35815 
.35852 
.35888 
.35924 
.35961 


•27563 
.27583 
•27603 
•27623 
•27643 


.38051 
.38089 
.38127 
.38165 
.38204 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.24149 
.24168 
.24187 
.24206 
.24225 


.31837 
.31870 
.31903 
.31936 
.31969 


.25297 
.25317 
.25336 
•25356 
.25375 


.33864 
.33899 
.33934 
.33968 
.34003 


.26469 
.26489 
•26509 
•26528 
.26548 


.35997 
.36034 
.36070 
.36107 
.36143 


•27663 
•27683 
•27703 
.27723 
.27743 


.38242 
.38280 
.38319 
.38357 
•38396 


40 

41 
42 
43 
44 


45 
46 
47 
48 

49 


.24244 
.24262 
.24281 
.24300 
•24320 


.32002 
.32035 
.32068 
.32101 
.32134 


•25394 
.25414 
•25433 
•25452 
.25472 


.34038 
.34073 
.34108 
.34142 
•34177 


•26568 
•26588 
•26607 
•26627 
.26647 


.36180 
.36217 
.36253 
.36290 
•36327 


.27764 
.27784 
•27804 
.27824 
.27844 


.38434 
.38473 
.38512 
.38550 
•38589 

.38628 
.38666 
.38705 
.38744 
.38783 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.24339 
.24358 
.24377 
.24396 
• 24415 


.32168 
.32201 
.32234 
.32267 
.32301 

.32334 
.32368 
.32401 
.32434 
.32468 


.25491 
.25511 
.25530 
.25549 
-25569 

.25588 
.25608 
.25627 
.25647 
.25666 


.34212 
.34247 
.34282 
.34317 
.34352 


.26667 
.26686 
.26706 
.26726 
.26746 


.36363 
.36400 
•36437 
.36474 
•36511 


.27864 
•27884 
•27905 
•27925 
•27945 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.24434 
.24453 
.24472 
.24491 
•24510 


.34387 

.34423 
.34458 
.34493 
.34528 


•26766 
•26785 
.26805 
.26825 
•26845 


.36548 
.36585 
.36622 
.36659 
.36696 


.27965 
.27985 
o 28005 
•28026 
.28046 


.38822 
.38860 
.38899 
.38938 
•38977 


55 
56 
57 
58 
59 


60 


.24529 


.32501 


•25686 


.34563 


.26865 


.36733 


•28066 


.39016 


60 



770 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



44^ 



45^ 



46' 



47* 



• 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.28066 
.28086 
.28106 
.28127 
.28147 


.39016 
.39055 
.39095 
.39134 
.39173 


•29289 
•29310 
•29330 
.29351 
•29372 


.41421 
.41463 
.41504 
.41545 
.41586 


.30534 
.30555 
.30576 
.30597 
.30618 


.43956 
.43999 
.44042 
.44086 
.44129 


.31800 
.31821 
.31843 
.31864 
31885 


.46628 
.46674 
.46719 
.46765 
•46811 




1 
2 
3 
4 


5 
6 
7 
8 

9 


.28167 
.28187 
.28208 
.28228 
.28248 

.28268 
.28289 
.28309 
.28329 
.28350 


.39212 
.39251 
.39291 
.39330 
.39369 


.29392 
.29413 
.29433 
.29454 
.29475 


.41627 
.41669 
.41710 
.41752 
.41793 


.30639 
.30660 
.30681 
.30702 
.30723 


.44173 
.44217 
.44260 
.44304 
.44347 


.31907 
.31928 
.31949 
.31971 
.31992 


•46857 
•46903 
.46949 
.46995 
47041 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.39409 
.39448 
.39487 
.39527 
.39566 


.29495 
.29516 
.29537 
.29557 
.29578 


.41835 
.41876 
.41918 
.41959 
.42001 


.30744 
.30765 
.30786 
•30807 
.30828 


.44391 
.44435 
.44479 
.44523 
.44567 


.32013 
•32035 
•32056 
•32077 
.32099 


.47087 
•47134 
.47180 
.47226 
.47272 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


.28370 
.28390 
.28410 
.28431 
.28451 


.39606 
.39646 
.39685 
.39725 
.39764 


.29599 
.29619 
.29640 
.29661 
.29681 


.42042 
.42084 
.42126 
.42168 
•42210 


.30849 
.30870 
.30891 
•30912 
•30933 


.44610 
.44654 
.44698 
.44742 
.44787 


•32120 
•32141 
•32163 
•32184 
•32205 


.47319 
.47365 
.47411 
.47458 

.47504 


15 
18 
17 
18 
19 


20 

21 
22 
23 
24 


.28471 
.28492 
.28512 
.28532 
.28553 


.39804 
.39844 
.39884 
.39924 
.39963 


.29702 
.29723 
.29743 
.29764 
.29785 


.42251 
.42293 
.42335 
.42377 
.42419 


•30954 
.30975 
.30996 
•31017 
.31038 


.44831 
.44875 
.44919 
.44963 
.45007 


.32227 
•32248 
•32270 
•32291 
•32312 


.47551 
.47598 
.47644 
.47691 
.47738 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.28573 
.28593 
.28614 
.28634 
.28655 


.40003 
.40043 
.40083 
.40123 
.40163 


^29805 
.29826 
.29847 
.29868 
•29888 


.42461 
.42503 
.42545 
.42587 
.42630 


•31059 
.31080 
.31101 
.31122 
•31143 


.45052 
.45096 
.45141 
.45185 
.45229 


•32334 
•32355 
•32377 
.32398 
•32420 


.47784 
.47831 
.47878 
.47925 
.47972 


25 
26 
27 
28 
29 


30 

31 
32 
33 

34 


.28675 
.28695 
.28716 
.28736 
.28757 


.40203 
.40243 
.40283 
.40^24. 
.40364 


.29909 
.29930 
.29951 
.29971 
.29992 


.42672 
.42714 
.42756 
.42799 
.42841 


.31165 
•31186 
•31207 
•31228 
•31249 


.45274 
.45319 
.45363 
.45408 
.45452 


•32441 
•32462 
•32484 
•32505 
.32527 


.48019 
.48066 
.48113 
.48160 
.48207 


30 

31 
32 
33 
34 


35 
36 
37 
38 

39 


.28777 
.28797 
.28818 
.28838 
.28859 


.40404 
.40444 
.40485 
.40525 
.405^^5 


.30013 
.30C34 
.30054 
.30075 
.30096 


.42883 
.42926 
.42968 
.43011 
.^■3053 


•31270 
•31291 
•31312 
•31334 
.31355 


.45497 
.45542 
.45587 
.45631 
.45676 


•32548 
•32570 
•32591 
•32613 
32634 


.48254 
.48301 
.48349 
.48396 
48443 


35 
36 
37 
38 
39 


40 

41 
42 
4d 
44 


.28879 
.28900 
.28920 
.28941 
.28961 


.40606 
.40646 
.40687 
.40727 
.40768 


.30117 
.30138 
.30158 
.30179 
.30200 


.43096 
.43139 
.43181 
.43224 
.43267 


•31376 
.31397 
•31418 
.31439 
•31461 


.45721 
.45766 
.45811 
.45856 
.45901 


.32656 
.32677 
.32699 
•32720 
.32742 


.48491 
.48538 
.48586 
.48633 
.48681 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.28981 
.29002 
.29022 
.29043 
.29063 


.40808 
.40849 
.40890 
.40930 
.40971 


.30221 
.30242 
.30263 
.30283 
.30304 


•43310 
.43352 
.43395 
.43438 
.43481 


•31482 
•31503 
•31524 
.31545 
•31567 


.45946 
.45992 
-46037 
-46082 
-46127 


•32763 
•32785 
.32806 
.32828 
• 32849 


•48728 
.48776 
.48824 
•48871 
•48919 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.29084 
.29104 
.29125 
.29145 
.29166 


.41012 
.41053 
.41093 
.41134 
.41175 


.30325 
.30346 
.30367 
.30388 
.30409 


.43524 
.43567 
.43610 
.43653 
.43696 


•31588 
.31609 
.31630 
.31651 
.31673 


.46173 
.46218 
.46263 
.46309 
.46354 


•32871 
•32893 
.32914 
•32GS6 
.32Pf7 


.48967 
.49015 
.49063 
.49111 
.49159 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.29187 
.29207 
.29228 
.29248 
.29269 


.41216 
.41257 
.41298 
.41339 
.4.1RR0 


.30430 
.30451 
.30471 
•30492 
•30513 


.43739 
.43783 
.43826 
.43869 
./19Q12 


.31694 
.31715 
.31736 
.31758 
.31779 


.46400 
.46445 
.46491 
.46537 
.46582 


•32979 
•33GC1 

•33022 
•33044 
. ??nP5 


.49207 
.492E5 
.49303 
.49351 


55 
56 
57 
58 
59 


60 


.29289 


.41421 


•30534 j .43956 1 


.31800 


.46628 


.33087 


.49448 


60 



771 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTa 
48° 49° 50° 61*» 



» 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


•33087 
.33109 
.33130 
.33152 
.33173 


.49448 
.49496 
.49544 
.49593 
.49641 


.34394 
.34416 
.34438 
.34460 
.34482 


.52425 
.52476 
.52527 
.52579 
•52630 


.35721 
.35744 
.35766 
.35788 
•35810 


•55572 
.55626 
.55680 
.55734 
.55789 


.37068 
.37091 
.37113 
.37136 
•37158 


.58902 
.58959 
.59016 
.59073 
.59130 




1 
2 
3 

4 


5 
6 
7 
8 
9 


.33195 
.33217 
.33238 
.33260 
.33282 


.49690 
.49738 
.49787 
.49835 
.49884 


.34504 
•34526 
.34548 
.34570 
.34592 


.52681 
.52732 
.52784 
.52835 
.52886 


•35833 
•35855 
•35877 
•35900 
.35922 


.55843 
.55897 
.55951 
.56005 
.56060 


•37181 
.37204 
.37226 
•37249 
.37272 


.59188 
.59245 
.59302 
.59360 
.59418 


5 
6 
7 
8 
9 


LO 
1 
2 
3 
4 


.33303 
.33325 
.33347 
.33368 
.33390 


.49933 
.49981 
.50030 
.50079 
.50128 


.34614 
•34636 
.34658 
•34680 
.34702 


.52938 
.52989 
.53041 
.53092 
.53144 


•35944 
.35967 
.35989 
.36011 
.36034 


.56114 
.56169 
.56223 
.56278 
.56332 


.37294 
•37317 
.37340 
.37362 
.37385 


•59475 
.59533 
•59590 
.59648 
.59706 


10 

11 
12 
13 
14 


5 
6 
7 
8 
9 


.33412 
.33434 
.33455 
.33477 
.33499 


.50177 
.50226 
.50275 
.50324 
.50373 


.34724 
.34746 
•34768 
.34790 
.34812 


.53196 
.53247 
.53299 
.53351 
.53403 


.36056 
•36078 
•36101 
•36123 
.36146 


.56387 
.56442 
.56497 
.56551 
.56606 


.37408 
.37430 
.37453 
.37476 
.37498 


.59764 
.59822 
.59880 
.59938 
.59996 


15 
16 
17 
18 
19 


50 

11 
!2 
!3 
>A 


.33520 
.33542 
.33564 
.33586 
.33607 


.50422 
.50471 
.50521 
.50570 
.50619 


.34834 
.34856 
.34878 
.34900 
.34923 


.53455 
.53507 
.53559 
.53611 
.53863 


•36168 
.36190 
.36213 
.36235 
.36258 


.56661 
.56716 
.56771 
.56826 
.56881 


.37521 
.37544 
.37567 
.37589 
.37612 


.60054 
.60112 
.60171 
.60229 
.60287 


30 

21 
22 
23 
24 


J5 
26 
27 
28 
29 


.33629 
.33651 
.33673 
.33694 
.33716 


.50669 
.50718 
.50767 
.50817 
.50868 


.34945 
.34967 
•34989 
.35011 
.35033 


.53715 
.53768 
.53820 
.53872 
.53924 


.36280 
•36302 
•36325 
•36347 
.36370 


.56937 
.56992 
.57047 
.57103 
.57158 


.37635 
.37658 
.37680 
.37703 
.37728 


.60346 
. 60404 
.60463 
.60521 
.60580 


25 
26 
27 
28 
29 


50 

$1 
J2 
$3 
$4 


.33738 

.33760 
.33782 
.33803- 
.33825 


.50916 
.50966 
.51015 
.51035 
.51115 


.35055 
.35077 
.35099 
.35122 
.35144 


.53977 
.54029 
.54082 
.54134 
.54187 


•36392 
•36415 
•36437 
•36460 
.36482 


.57213 
.57269 
.57324 
.57380 
.57436 


.37749 
.37771 
.37794 
.37817 
.37840 


.60639 
.60698 
•60756 
•60815 
•60874 


3Q| 

31 

32 
33 
34 


J5 
36 
57 
38 
39 


.33847 
.33889 
.33891 
.33912 
.33934 


.51165 
.51215 
.51265 
.51314 
.51364 


.35166 
.35188 
.35210 
.35232 
.35254 

.35277 
.35299 
•35321 
.35343 
.35365 


.54240 
.54292 
.54345 
.54398 
•54451 


.36504 
.36527 
.36549 
.36572 
.36594 


.57491 
.57547 
.57603 
.57659 
.57715 


.37862 
.37885 
.37908 
.37931 
.37954 

.37976 
.37999 
.38022 
.38045 
.38068 


.60933 
.60992 
.61051 
.61111 
•61170 

.61229 
.61288 
•61348 
.61407 
•61467 


33 

3e 

37 

J 


40 

11 
12 
i3 
44 


.33956 
.33978 
.34000 
•34022 
.34044 


.51415 
.51485 
.51515 
.51565 
.51615 


.54504 
.54557 
.54610 
.54663 
.54716 


.36617 
•36639 
•36662 
•36684 
.36707 


.57771 
.57827 
.57883 
.57939 
.57995 


43 


45 
46 
47 
48 
49 


•34065 
•34087 
•34109 
•34131 
•34153 


.51665 
.51716 
.51766 
.51817 
.51867 


.35388 

.35410 
•35432 
•35454 
.35476 


.54769 
..54822 
.54876 
.54929 
.54982 


.36729 
.36752 
.36775 
.36797 
.36820 


.58051 
.58108 
.58164 
.58221 
.58277 


•38091 
.38113 
.38136 
.38159 
.38182 


.61526 
.61586 
.61646 
.61705 
.61765 


451 

4e 

47 
4i 
41 


50 

51 
52 
53 
54 


•34175 
.34197 
.34219 
.34241 
.34262 


.51918 
.51968 
.52019 
.52069 
.52120 


•35499 
•35521 
•35543 
•35565 
.35588 


.55036 
.55089 
.55143 
.55196 
.55250 


•36842 
•36865 
•36887 
•36910 
.36932 


.58333 
.58390 
.58447 
.58503 
.58560 


•38205 
•38228 
•38251 
.38274 
•38296 


.61825 
•61885 
•61945 
.62005 
•62065 


5( 

51 

54 


55 
56 
57 
58 

59 


.34284 
.34306 
.34328 
.34350 

•34372 


.52171 
.52222 
.52273 
.52323 
.52374 


•35610 
•35632 
.35654 
•35677 
.35699 


.55303 
.55357 
.55411 
.55465 
.55518 


.36955 
.36978 
.37000 
.37023 
37045 


.58617 
.58674 
.58731 
.58788 
.58845 


•38319 
.38342 
.38365 
.38388 

.38411 


•62125 
•62185 
•62246 
•62306 
•62366 


55 
56 
57 
58 
59 


60 


.34394 


.52425 


•35721 


.55572 


•37068 


.58902 


.38434 


.62427 


60 



772 



TABLE X— NATURAL VERSED SINES AND EXTERNAL SECANTS 



53^ 



53^ 



54' 



55' 



1 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




\ 

3 

4 


.38434 
.38457 
.38480 
.38503 
.38526 

.38549 
.38571 
.38594 
.38617 
.38640 


.62427 
.62487 
.62548 
.62609 
62669 


.39819 
.39842 
.39865 
.39888 
.39911 


.66164 
.66228 
.66292 
.66357 
.66421 


.41221 
.41245 
.41269 
.41292 
•41316 


.70130 
.70198 
.70267 
.70335 
. 70403 


.42642 
.42666 
.42690 
•42714 
.42738 

.42762 
.42785 
.42809 
.42833 
•42857 


.74345 
.74417 
.74490 
.74562 
•74635 




1 
2 
3 
4 


5 
6 

\ 

9 


.62730 
.62791 
.62852 
.62913 
.62974 


.39935 
.39958 
.39981 
.40005 
.40028 


.66486 
.66550 
.66615 
.66679 
.66744 


.41339 
.41363 
.41386 
.41410 
.41433 


.70472 
.70540 
.70609 
.70677 
.70746 


.74708 
.74781 
.74854 
.74927 
.75000 


5 
6 
7 
8 

9 


10 

11 
12 
13 
14 


.38663 
.38686 
.38709 
.38732 
.38755 


.63035 
.63096 
.63157 
.63218 
.63279 


.40051 
.40074 
.40098 
.40121 
.40144 


.66809 
.66873 
•66938 
.67003 
.67068 

.67133 
.67199 
.67264 
.67329 
.67394 


.41457 
•41481 
•41504 
.41528 
.41551 


=70815 
.70884 
.70953 
.71022 
•71091 


•42881 
•42905 
•42929 
.42953 
•42976 


.75073 
.75146 
.75219 
.75293 
.75366 


10 

11 
12 
13 
14 


15 
16 
17 
18 
11- 


.38778 
.38801 
.38824 
.38847 
.38870 


.63341 
.63402 
.63464 
.63525 
.63587 


.40168 
.40191 
.40214 
.40237 
.40261 


.41575 
.41599 
.41622 
.41646 
.41670 


.71160 
.71229 
.71298 
.71368 
•71437 


.43000 
.43024 
•43048 
.43072 
.43096 


.75440 
.75513, 
.75587 
.75661 
.75734 


15 
16 
17 
18 
-19 


20 

21 
22 
23 
24 


.38893 
.38916 
.38939 
.38962 
.38985 


.63648 
.63710 
.63772 
.63834 
.63895 


.40284 
.40307 
.40331 
.40354 
.40378 

.40401 
.40424 
.40448 
.40471 
.40494 


.67460 
.67525 
.67591 
.67656 
.67722 


.41693 
.41717 
.41740 
.41764 
.41788 


.71506 
.71576 
.71646 
.71715 
.71785 


.43120 
•43144 
.43168 
.43132 
43216 


.75808 
.75882 
.75956 
.76031 
.76105 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.39009 
.39032 
•39055 
.39078 
.39101 


.63957 
.64019 
.64081 
.64144 
.64206 


.67788 
.67853 
.67919 
.67985 
.68051 


.41811 
.41835 
.41859 
.41882 
.41906 


.71855 
.71925 
.71995 
.72065 
•72135 


.43240 
•43264 
•43287 
■43311 
.43335 

.43359 
.43383 
.43407 
•43431 
•43455 


.76179 
.76253 
.76328 
.76402 
•76477 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.39124 
.39147 
.39170 
.39193 
.39216 


.64268 
.64330 
.64393 
.64455 
.64518 


.40518 
.40541 
.40565 
.40588 
.40611 


.68117 
.68183 
.68250 
.68316 
.68382 


.41930 
.41953 
.41977 
.42001 
.42024 


.72205 
.72275 
.72346 
.72416 
•72487 


.76552 
.76626 
.76701 
.76776 
.76851 


30 

31 
32 
33 
34 


35 
36 
37 
38 


.39239 
.39262 
.39286 
.39309 
.39332 


.64580 
.64643 
.64705 
.64768 
.64831 


.40635 
.40658 
.40682 
.40705 
.40728 


.68449 
.68515 
.68582 
.68648 
.68715 


.42048 
.42072 
.42096 
•42119 
.42143 


.72557 
.72628 
.72698 
.72769 
. 72840 


•43479 
.43503 
.43527 
.43551 
.43575 


.76926 
.77001 
.77077 
.77152 
•77227 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 

45 
46 
47 
48 
49 


.39355 
.39378 
.39401 
.39424 
.39447 


.64894 
.64957 
.65020 
.65083 
.65146 


.40752 
.40775 
.40799 
.40822 
.40846 


.68782 
.68848 
.68915 
.68982 
.69049 


.42167 
.42191 
.42214 
.42238 
.42262 


.72911 
.72982 
.73053 
.73124 
.73195 


.43599 
.43623 
•43647 
•43671 
•43695 


.77303 
.77378 
.77454 
.77530 
•77606 


40 

41 
42 
43 
44 


.39471 
.39494 
.39517 
.39540 
.39563 


.65209 
.65272 
.65336 
.65399 
.65462 


.40869 
.40893 
.40916 
.40939 
.40963 


.69116 
.69183 
.69250 
.69318 
.69385 


.42285 
.42309 
.42333 
.42357 
.42381 


.73267 
.73338 
.73409 
.73481 
.73552 


.43720 
.43744 
.43768 
.43792 
.43816 


.77681 
.77757 
.77833 
.77910 
•77986 


45 
46 
47 
48 
49 


60 

51 
52 
53 
54 


.39586 
.39610 
.39633 
.39656 
.39679 


.65526 
.65589 
.65653 
.65717 
.65780 


.40986 
.41010 
.41033 
.41057 
.41080 


.69452 
.69520 
.69587 
.69655 
.69723 


.42404 
.42428 
.42452 
.42476 
.42499 


.73624 
.73696 
.73768 
.73840 
.73911 


.43840 
.43864 
.43888 
.43912 
.43936 


.78062 
.78138 
.78215 
.78291 
.78368 


50 

51 
52 
53 
54 


55 
56 
57 
58 
.59- 


.39702 
.39726 
.39749 
.39772 
.39795 


.65844 
.65908 
.65972 
.66036 
.66100 


.41104 
.41127 
.41151 
.41174 
.41198 


.69790 
.69858 
.69926 
.69994 
.70062 


.42523 
.42547 
.42571 
.42595 
.42619 


.73983 
.74056 
.74128 
.74200 
.74272 


•43960 
.43984 
.44008 
•44032 
.44057 


.78445 
.78521 
.78598 
.78675 
•78752 


55 
56 
57 
58 
59 


60 


.39819 


.66164 


.41221 


.70130 


.42642 


.74345 


.44081 


.78829 


60 










77 


3 











TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
56° 57° 58° 59° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.44081 
.44105 
.44129 
.44153 
.44177 


.78829 
.78906 
.78984 
.79061 
.79138 


•45536 
•45560 
•45585 
.45609 
•45634 


.83608 
•83690 
.83773 
.83855 
•83938 


•47008 
•47033 
•47057 
•47082 
•47107 


.88708 
.88796 
.88884 
.88972 
•89060 


•48496 
■48521 
.48546 
.48571 
•48596 


•94160 
•94254 
•94349 
•94443 
•94537 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.44201 
.44225 
.44250 
.44274 
.44298 


.79216 
.79293 
.79371 
.79449 
.79527 


•45658 
•45683 
•45707 
•45731 
.45756 


•84020 
.84103 
.84186 
.84269 
•84352 


.47131 
.47156 
.47181 
.47206 
.47230 


.89148 
.89237 
•89325 
.89414 
•89503 


.48621 
.48646 
•48671 
•48696 
•48721 


•94632 
.94726 
.94821 
.94916 
.95011 


5 
6 
7 
8 
9 


10 

11 
12 
13 

14 


.44322 
.44346 
.44370 
.44395 
•44419 


.79604 
.79682 
.79761 
.79839 
.79917 


•45780 
•45805 
•45829 
•45854 
.45878 


•84435 
.84518 
.84601 
.84685 
.84768 

•84852 
.84935 
.85019 
.85103 
.85187 


•47255 
.47280 
.47304 
.47329 
.47354 


.89591 
.89680 
.89769 
.89858 
•89948 


•48746 
.48771 
.48796 
.48821 
.48846 


.•95106 
•95201 
•95296 
•95392 
.95487 


10 

11 

12 
13 

14 


15 
16 
17 
18 
19 


•44443 
•44467 
•44491 
•44516 
.4.454-n 


.79995 
.80074 
.80152 
.80231 
.80309 


.45903 
.45927 
.45951 
.45976 
•46000 


.47379 
.47403 
.47428 
.47453 
•47478 


.90037 
.90126 
.90216 
.90305 
.90395 


.48871 
.48896 
.48921 
.48946 
.48971 


.95583 
.95678 
.95774 
.95870 
.95966 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


•44564 
•44588 
•44612 
•44637 
•44661 


.80388 
. 80467 
.80546 
.80625 
.80704 


•46025 
•46049 
•46074 
•46098 
.46123 


•85271 
.85355 
•85439 
•85523 
.85608 


•47502 
•47527 
•47552 
.47577 
.47601 


.90485 
.90575 
.90665 
.90755 
.90845 


.48996 
.49021 
.49046 
.49071 
.49096 


.96062 
.96158 
.96255 
.96351 
.96448 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•44685 
.44709 
•44734 
.44758 
•44782 


.80783 
.80862 
.80942 
.81021 
.81101 


•46147 
•46172 
•46196 
•46221 
•46246 


•85692 
.85777 
.85861 
.85946 
.86031 


.47626 
.47651 
.47676 
.47701 
.47725 


.90935 
.91026 
.91116 
.91207 
.91297 


.49121 
.49146 
.49171 
.49196 
.49221 


.96544 
.96641 
.96738 
.96835 
.96932 


25 
26 
27 
28 
29' 


30 

31 
32 
33 
34 


•44806 
•44831 
•44855 
•44879 
•44903 


•81180 
.81260 
.81340 
.81419 
.81499 


•46270 
•46295 
•46319 
.46344 
.46368 


.86116 
.86201 
.86286 
.86371 
.86457 


.47750 
.47775 
•47800 
.47825 
.47849 


.91388 
.91479 
.91570 
.91661 
.91752 


.49246 
.49271 
.49296 
49321 
.49346 


.97029 
.97127 
.97224 
.97322 
•97420 


30 

3] 
32 
33 
34 


35 
36 
37 
38 
39 


.44928 
•44952 
•44976 
•45001 
•45025 


.81579 
.81659 
.81740 
.81820 
.81900 


.46393 
.46417 
.46442 
.46466 
.46491 


.86542 
.86627 
.86713 
.86799 
.86885 


.47874 
.47899 
.47924 
.47949 
.47974 


•91844 
•91935 
•92027 
•92118 
.92210 


.49372 
.49397 
.49422 
.49447 
.49472 


.97517 
.97615 
.97713 
.97811 
.97910 


35 
36 
37 
38 

39 


40 

41 
42 
43 

M 


•45049 
•45073 
•45098 
•45122 
•45146 


.81981 
.82061 
.82142 
.82222 
.82303 


.46516 
.46540 
.46565 
.46589 
.46614 


.86970 
.87056 
.87142 
.87229 
.87315 


.47998 
.48023 
.48048 
.48073 
,48098 


•92302 
.92394 
.92486 
.92578 
.92670 


.49497 
•49522 
.49547 
.49572 
.49597 


.98008 
.98107 
.98205 
.98304 
.98403 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•45171 
•45195 
•45219 
•45244 
•45268 


.82384 
.82465 
.82546 
.82627 
.82709 


.46639 
.46663 
•46688 
•46712 
•46737 


.87401 
.87488 
.87574 
.87661 
.87748 


.48123 
.48148 
.48172 
.48197 
•48222 


.92762 
.92855 
.92947 
.93040 
•93133 


.49623 
.49648 
.49673 
•49698 
•49723 


.98502 
.98601 
.98700 
.98799 
.98899 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•45292 
.45317 
•45341 
•45365 
.45390 


.82790 
.82871 
.82953 
.83034 
.83116 


•46762 
.46786 
•46811 
.46836 
.46860 


.87834 
.87921 
.88008 
•88095 
•88183 


•48247 
•48272 
.48297 
.48322 
.48347 


.93226 
.93319 
.93412 
.93505 
•93598 


•49748 
.49773 
•49799 
.49824 
.49849 


.98998 
.99098 
•99198 
•99298 
.99398 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•45414 
•45439 
•45463 
•45487 
•45512 


.83198 
.83280 
.83362 
.83444 
.83526 


•46885 
•46909 
•46934 
•46959 
•46983 


.88270 
.88357 
.88445 
.88532 
.88620 


.48372 
.48396 
.48421 
.48446 
•48471 


.93692 
.93785 
.93879 
.93973 
.94066 


.49874 
.49899 
.49924 
.49950 
•49975 


.99498 
.99598 
.99698 
.99799 
.99899 


55 
56 
57 
58 
59 


60 


45536 


.83608 


.47008 


.88708 


.48496 


.94160 


.50000 1.00000 


60 



774 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 





60° 


ei** 


63° 


63° 






Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


r 


.50000 


1.00000 


.51519 


1.06267 


.53053 


1.13005 


.54601 


1.20269 





1 


.50025 


1.00101 


.51544 


1.06375 


•53079 


1.13122 


.54627 


1.20395 


1 


2 


.50050 


1.00202 


.51570 


1.06483 


•53104 


1.13239 


.54653 


1.20521 


2 


3 


.50076 


1.00303 


•51595 


1.06592 


•53130 


1.13356 


.54679 


1. 20647 


3 


4 
5 


.50101 


1.00404 


•51621 


1.06701 
1.06809 


•53156 


1.13473 


.54705 


1.20773 


4 


.50126 


1.00505 


•51646 


•53181 


1.13590 


.54731 


11.209C0 


5 


6 


.50151 


1.00607 


•51672 


1.06918 


.53207 


1.13707 


.54757 


1.21026 


6 


7 


•50176 


1.00708 


•51697 


1.07027 


•53233 


1.13825 


.54782 


1.21153 


7 


8 


.50202 


1.00810 


•51723 


1.07137 


•53258 


1.13942 


•54808 


1.21280 


8 


9 


.50227 


1.00912 


.51748 


1.07246 
1.07356 


•53284 


i.i4ceo 


•54834 


1.214C7 


9 


10 


.50252 


1.01014 


.51774 


•53310 


1.14178 


•54860 


1.21535 


10 


11 


.50277 


1.01116 


.51799 


1.07465 


•53336 


1.14296 


•54886 


1.21662 


11 


12 


.50303 


1.01218 


.51825 


1.07575 


.53361 


1.14414 


•54912 


1.21790 


12 


]3 


.50328 


L. 01320 


.51850 


1.07685 


.53387 


1.14533 


•54938 


1.21918 


13 


14 


.50353 


1.01422 


.51876 


1.07795 


.53413 


1.14651 


•54964 


1.22045 


14 


15 


•50378 


L. 01525 


.51901 


1.07905 


.53439 


1.14770 


• 54990 


1.22174 


15 


16 


. 50404 


L. 01628 


•51927 


1.08015 


.53464 


1.14889 


•55016 


1.22302 


16 


17 


. 50429 


1-01730 


•51952 


1.08126 


.53490 


1.150C8 


• 55042 


1-22430 


17 


18 


.50454 


L. 01833 


•51978 


1.08236 


•53516 


1.15127 


•55068 


1-22559 


18 


19 


.50479 


1.01936 


.52003 


1.08347 


.53542 


1.15246 


55094 


1.2?688 


19 


20 


.50505 


1.02039 


•52029 


1.08458 


.53567 


1.15366 


55120 


1-22817 


20 


21 


•50530 


L. 02143 


•52054 


1.08569 


.53593 


1.15485 


55146 


1-22946 


21 


22 


•50555 


!• 02246 


.52080 


1.08680 


.53619 


1.15605 


•55172 


1-23075 


22 


23 


.50581 


!• 02349 


.52105 


1.08791 


.53645 


1.15725 


•55198 


1.23205 


23 


24 


.50606 


!• 02453 


.52131 


1.08903 


•53670 


1.15845 


.55224 


] -23334 


24 


25 


•50631 


[•02557 


•52156 


1.09014 


•53696 


1.15965 


•55250 


1-23464 


25 


26 


•50656 


L. 02661 


•52182 


1.09126 


•53722 


1.16085 


•55276 


1-23594 


26 


27 


.50682 


..02765 


.52207 


1.C9238 


.53748 


1.16206 


•55302 


1-23724 


27 


28 


.50707 


L. 02869 


.52233 


1.09350 


.53774 


1.16326 


•55328 


1.23855 


28 


29 


.50732 


L. 02973 


•52259 


1^C9462 


•53799 


1.16447 


•55354 


1. 23985 


29 


30 


•50758 . 


L. 03077 


.52284 


1.09574 


.53825 


1.16568 


•55380 


1-24116 


30 


31 


•50783 . 


[•03182 


•52310 


1.09686 


.53851 


1^ 16689 


•55406 


1-24247 


31 


32 


•50808 : 


.•03286 


•52335 


1-09799 


.53877 


1-16810 


•55432 


1-24378 


32 


33 


.50834 ] 


[•03391 


.52361 


1.09911 


.53903 


1^16932 


•55458 


1-24509 


33 


34 


•50859 ] 


[•03496 


•52386 


1.10024 


•53928 


1^17053 


•55484 


1-24640 


34 


35 


•50884 ] 


[•03601 


•52412 


1.10137 


•53954 


1-17175 


•55510 


1.24772 


35 


36 


.50910 : 


[•03706 


•52438 


1.10250 


.53980 


1^17297 


•55536 


1.24903 


36 


37 


.50935 ] 


[•03811 


•52463 


1.10363 


.54006 


1^17419 


•55563 


1.25035 


37 


38 


•50960 ] 


[•03916 


•52489 


1.10477 


•54032 


1.17541 


•55589 


1.25167 


38 


39 


•50986 ] 


L. 04022 


•52514 


1.10590 


.54058 


] •17663 


.FFP3 5 


1 .?5?ro 


.39 


40 


•51011 ] 


L. 04128 


•52540 


1.10704 


•54083 


1.17786 


•55641 


1.25432 


40 


41 


•51036 ] 


[•04233 


•52566 


1.10817 


.54109 


1.17909 


•55667 


1.25565 


41 


42 


.51062 ] 


L. 04339 


•52591 


1.10931 


.54135 


1.18031 


-55693 


1.25697 


42 


43 


.51087 ] 


[. 04445 


•52617 


1.11045 


-54161 


1.18154 


•55719 


1-25830 


43 


44 


.51113 ] 


L .04551 


•52642 


1.11159 


.54187 


1^18277 


•55745 


1-25963 


44 


45 


•51138 ] 


L. 04658 


•52668 


1.11274 


•54213 


1.18401 


•55771 


1-26097 


45 


46 


.51163 ] 


L. 04764 


•52694 


1.11388 


.54238 


1.18524 


•55797 


1-26230 


46 


47 


.51189 ] 


L. 04870 


•52719 


1.11503 


. 54264 


1.18648 


.55823 


1-26364 


47 


48 


.51214 ] 


L. 04977 


•52745 


1.11617 


.54290 


1.18772 


.55849 


1-26498 


48 


49 


.51239 ] 


L. 05084 


•52771 


1.11732 


•54316 


1.18895 


.55876 


1-26632 


49 


50 


.51265 ] 


L. 05191 


•52796 


1.11847 


. 54342 


1.19019 


.55902 


1-26766 


50 


51 


=51290 ] 


L. 05298 


•52822 


1.11963 


.54368 


1.19144 


.55928 


1.26900 


51 


52 


.51316 : 


L. 05405 


•52848 


1.12078 


.54394 


1.19268 


•55954 


1.27035 


62 


53 


.51341 ] 


L. 05512 


•52873 


1.12193 


. 54420 


1.19393 


•55980 


1.27169 


53 


54 


•51366 ] 


L. 05619 


•52899 


1.12309 


• 54446 


1.19517 


.56006 


1.27304 


64 


55 


•51392 ] 


L. 05727 


.52924 


1.12425 


.54471 


1^19642 


.56032 


1.27439 


55 


56 


.51417 ] 


L. 05835 


•52950 


1.12540 


•54497 


1.19767 


•56058 


1.27574 


56 


57 


•51443 ] 


L. 05942 


.52976 


1.12657 


•54523 


1.19892 


•5B084 


1.27710 


57 


58 


.51468 ] 


L. 06050 


.53001 


1^12773 


•54549 


1.20018 


.56111 


L. 27845 


58 


59 


•51494 ] 


L.06158 


.53097 


1 .]2889 


•54575 


1-20143 


56137 


[-27981 


5f 


60 


.51519 ] 


L. 06267 


•53053 


1.13005 


•54601 


1.20269 


.56163 : 


[.28117 


60 



775 



TABLE ^.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
64° 65° 66° 67° 



1 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.56163 
.56189 
.56215 
.56241 
.56267 


1.28117 
1.28253 
1.28390 
1.28526 
1.28663 


.57738 
.57765 
.57791 
.57817 
-57844 


1.36620 
1.36768 
1.36916 
1.37064 
1-37212 


-59326 
-59353 
.59379 
.59406 
-59433 


1.45859 
1.46020 
1.46181 
1.46342 
1-46504 


.60927 
•60954 
.60980 
.61007 
.61034 




55930 
56106 
56282 
56458 
56634 




1 

2 
3 

4: 


5 
6 
7 
8 
9 


.56294 
•56320 
.56346 
•56372 
•56398 


1.28800 
1.28937 
1.29074 
1-29211 
1.29349 


.57870 
.57896 
.57923 
-57949 
•57976 


1-37361 
1-37509 
1-37658 
1-37808 
1-37957 


.59459 
.59486 
.59512 
.59539 
-59566 


1-46665 
1.46827 
1-46989 
1-47152 
1-47314 


.61061 
.61088 
61114 
-61141 
-61168 




56811 
56988 
57165 
57342 
57520 


5 

6 
7 
8 
9 


10 

11 
12 
13 
14 


•56425 
•56451 
•56477 
•56503 
•56529 


1.29487 
1.29625 
1.29763 
1.29901 
1-30040 


•58002 
•58028 
58055 
•58081 
-58108 

-58134 
-58160 
-58187 
-58213 
•58240 


1.38107 
1-38256 
1-38406 
1.38556 
1.38707 


-59592 
-59619 
-59645 
-59672 
-59699 


1-47477 
1.47640 
1-47804 
1.47967 
1.48131 


-61195 
.61222 
-61248 
-61275 
-61302 




57698 
57876 
58054 
58233 
58412 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•56555 
•56582 
•56608 
•56634 
.56660 


1.30179 
1.30318 
1.30457 
1.30596 
1.30735 


1-38857 
1-39008 
1-39159 
1-39311 
1-39462 


-59725 
-59752 
-59779 
-59805 
59832 


1.48295 
1.48459 
1.48624 
1.48789 
1-48954 


•61329 
-61356 
-61383 
-61409 
-61436 




58591 
58771 
58950 
59130 
59311 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


.56687 
.56713 
.56739 
•56765 
.56791 


1.30875 
1-31015 
1.31155 
1.31295 
1.31436 


.58266 
.58293 
.58319 
.58345 
.58372 

58398 
.58425 
.58451 

58478 
.58504 


1-39614 
1.39766 
1.39918 
1.40070 
1.40222 


59859 

59885 

.59912 

.59938 

.59965 


1-49119 
1-49284 
1-49450 
1-49616 
1-49782 


.61463 
.61490 
.61517 
.61544 
-61570 




59491 
59672 
59853 
60035 
60217 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•56818 
•56844 
•568/0 
•56896 
•56923 


1.31576 
1.31717 
1.31858 
1^31999 
1^32140 


1.40375 
1.40528 
1.40681 
1.40835 
1.40988 


.59992 
.60018 
.60045 
-60072 
-60098 


1-49948 
1-50115 
1-50282 
1.50449 
1.50617 


•61597 
.61624 
.61651 
.61678 
.61705 




60399 
60581 
60763 
60946 
61129 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•56949 
•56975 
•57001 
•57028 
•57054 


1-32282 
1.32424 
1-32566 
1-32708 
1^32850 


.58531 

58557 

.58584 

.58610 

58837 


1.41142 
1.41296 
1.41450 
1.41605 
1.41760 


-60125 
-60152 
-60178 
-60205 
-60232 


1.50784 
1.50952 
1.51120. 
1.51289 
1.51457 

1.51626 
1.51795 
1.51965 
1.52134 
1.52304 


.61732 
.61759 
.61785 
-61812 
-61839 




61313 
61496 
61680 
61864 
62049 


30 

31 
3L 
33 
34 


35 
36 
37 
38 
39.. 


•57080 
•57106 
•57133 
•57159 
•57185 


1-32993 
1.33135 
1-33278 
1-33422 
1-33565 


-58663 
58690 
•58716 
.58743 
.58769 


1.41914 
1.42070 
1.42225 
1.42380 
i-42536 


-60259 
.60285 
.60312 
-60339 
-60365 


•61866 
-61893 
-61920 
-61947 
-61974 


1 


62234 
62419 
62604 
62790 
62976 


35 
36 
37 
38 
.39 


40 

41 
42 
43 
44 


•57212 
•57238 
•57264 
•57291 
•57317 


1-33708 
1-33852 
1-33998 
1-34140. 
1 • 34284 


.58796 
.58822 
.58849 
.58875 
.58902 


1.42692 
1-42848 
1.43005 
1.43162 
1-43318 


.60392 
-60419 

- 60445 
-60472 

- 60499 


1.52474 
1.52645 
1.52815 
1.52986 
1.53157 


-62001 
-62027 
- 62054 
.62081 
-62108 




63162 
63348 
63535 
63722 
63909 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•57343 
•57369 
•57396 
•57422 
•57448 


1^34429 
1^ 34573 
1-34718 
!• 34863 
1^35009 


58928 
.58955 
.58981 
.59008 
.59034 


1.43476 
1.43633 
1.43790 
1.43948 
1.44106 


.60526 
.60552 
.60579 
.60606 
-60633 


1.53329 
1-53500 
1-53672 
1-53845 
1-54017 


-62135 
-62162 
•62189 
•62216 
• 62243 




64097 
64285 
64473 
64662 
64851 


45 
46 

47 
48 
49 


50 

51 
52 
53 
54 


•57475 
•57501 
•57527 
•57554 
•57580 


1.35154 
1-35300 
1.35446 
1.35592 
1-35738 

1-35885 
1-36031 
1-36178 
1-36325 
1-36473 


.59061 
.59087 
.59114 
-59140 
.59167 


1.44264 
1.44423 
1.44582 
1.44741 
1-44900 


.60659 
-60686 
.60713 
. 60740 
-60766 


1-54190 
1-54363 
1-54536 
1-54709 
1 - 54883 


•62270 
•62297 
-62324 
-62351 
-62378 




65040 
65229 
65419 
65609 
65799 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


.57606 
.57633 
.57659 
.57685 
•57712 


-59194 
-59220 
-59247 
-59273 
.59300 


1-45059 
1.45219 
1.45378 
1.45539 
1-45699 


.60793 
- 60820 
.60847 
.60873 
-60900 


1-55057 
1-55231 
1-55405 
1-55580 
1.55755 


-62405 
-62431 
-62458 
-62485 
•62512 


1 


65989 
66180 
66371 
66563 
66755 


55 
56 
57 
58 
59 


60 


•57738 


1-36620 


-59326 


1.45859 


-60927 


1.55930 


.62539 


1 


66947 


60 



776 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
68° 69° 70° 71° 



/ 


Vers. 


Ex. sec. 


Vers. 

•64163 
•64190 
•64218 
• 64245 
•64272 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


•62539 
•62566 
•62593 
.62620 
•62647 




•66947 
•67139 
•67332 
•67525 
•67718 




79043 
79254 
79466 
79679 
79891 


•65798 
■65825 
•65853 
.65880 
.65907 


I 


92380 
92614 
92849 
93083 
93318 


• 67443 
•67471 
•67498 
.67526 
.67553 


2 

i 

2 
2 


07155 
07415 
07675 
07936 
08197 


O 

1 
2 
3 
4 


5 
6 
7 
8 
9 


•62674 
•62701 
.62728 
.62755 
•62782 




.67911 

.68105 

.68299 

68494 

68689 


•64299 
•64326 
•64353 
•64381 
. 64408 




80104 
80318 
80531 
80746 
8Q960 


.65935 
•65962 
.65989 
•66017 
• 66044 




93554 
93790 
94026 
94263 
94500 


.67581 
.67608 
.67636 
.67663 
•67691 


2 
2 
2 
2 
2 


08459 
08721 
08983 
09246 
09510 


5 
6 
7 
8 

9 


10 

11 
12 
13 
14 


•62809 
•62836 
•62863 
.62890 
•62917 




68884 
69079 
69275 
69471 
69667 


•64435 
•64462 
• 64489 
•64517 
. 64544 




81175 
81390 
81605 
81821 
82037 


•66071 
•66099 
•66126 
•66154 
•66181 




94737 
94975 
95213 
95452 
95691 


•67718 
•67746 
•67773 
•67801 
.67829 


2 
2 
2 
2 
2 


09774 
.10038 
.10303 
.10568 

10834 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•62944 
•62971 
•62998 
•63025 
•63052 




69864 
70061 
70258 
70455 
70653 


.64571 
.64598 
.64625 
.64653 
•64680 




82254 
82471 
82688 
82906 
83124 


.66208 
•66236 
•66263 
•66290 
66318 




95931 
96171 
96411 
96652 
96893 


.67856 
.67884 
.67911 
.67939 
•67966 


2 
2 
2 
2 
2 


11101 
11367 
11635 
11903 
12171 


15 
16 
17 
18 
-19 


30 

21 
22 
23 
24 


•63079 
•63106 
•63133 
•63161 
•63188 




70851 
71050 
71249 
71448 
71647 


.64707 
. 64734 
.64761 
. 64789 
•64816 




83342 
83561 
83780 
83999 
84219 


•66345 
•66373 
•66400 
.66427 
.66455 




97135 
97377 
97619 
97862 
98106 


•67994 
•68021 
•68049 
.68077 
•68104 


2 
2 
2 
2 
2 


12440 
12709 
12979 
13249 
13520 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•63215 
•63242 
•63269 
•63296 
.63323 




71847 
72047 
72247 
72448 
72649 


• 64843 
•64870 
.64893 
•64925 
•64952 




. 84439 
.84659 
.84880 
.85102 
85323 


•66482 
•66510 
66537 
•66564 
.66592 




98349 
98594 
98838 
99083 
99329 


•68132 
.68159 
•68187 
•68214 
. 68242 


2 
2 
2 
2 
2 


13791 
14063 
14335 
14608 
14881 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•63350 
•63377 
. 63404 
•63431 
.63458 




72850 
73052 
73254 
73456 
73659 


•64979 
•65007 
•65034 
•65061 
•65088 




.85545 

.85767 

.85990 

86213 

86437 


.66619 
•66647 
•66674 
•66702 
.66729 


2 
2 
2 


99574 
99821 
00067 
00315 
00562 


•68270 
68297 
68325 

•68352 
68380 


2 
2 
2 
2 
2 


15155 
15429 
15704 
15979 
16255 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.63485 
•63512 
•63539 
.63566 
.63594 




73862 
74065 
74269 
74473 
74677 


•65116 
•65143 
•65170 
•65197 
•65225 


L 


86661 
86885 
87109 
87334 
87560 


•66756 
•66784 
•66811 
•66839 
•66866 


2 
2 
2 
2 
2 


00810 
01059 
01308 
01557 
01807 


• 68408 
•68435 
•68463 
•68490 
•68518 


2 
2 
2 
2 

L 


16531 
16808 
17085 
17363 
17641 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•63621 
•63648 
•63675 
•63702 
•63729 




74881 
75086 
75292 
75497 
75703 


•65252 
.65279 
.65306 
•65334 
•65361 


1 


87785 
88011 
88238 
88465 
88692 


■66894 
•66921 
•66949 
•66976 
-67003 


2 
2 
2 
2 
2 


02057 
02308 
02559 
02810 
03062 


•68546 
•68573 
•68601 
•68628 
68656 


2 
2 
2 
2 
2 


17920 
18199 
18479 
18759 
19040 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


•63756 
•63783 
•63810 
•63838 
•63865 




75909 
76116 
76323 
76530 
76737 


•65388 
.65416 
.65443 
•65470 
•65497 


1 
1 
1 


88920 
89148 
89376 
89605 
89834 


•67031 
•67058 
•67086 
.67113 
•67141 


2 
2 
2 
2 
2 


03315 
03568 
03821 
04075 
04329 


68684 
•68711 
•68739 
•68767 
•68794 


2 
2 
2 
2 
2 


19322 
19604 
19886 
20169 
20453 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•63892 
•63919 
•63946 
•63973 
• 64000 




76945 
77154 
77362 
77571 
77780 


•65525 
•65552 
•65579 
.65607 
•65634 




90063 
90293 
90524 
90754 
90986 


•67168 
•67196 
•67223 
•67251 
•67278 


2 
2 
2 
2 
2 


04584 
04839 
05094 
05350 
05607 


.68822 
.68849 
.68877 
.68905 
•68932 


2 
2 
2 
2 
2 


20737 
21021 
21306 
21592 
21878 


50 

51 

52 
53 
54 


55 
56 
57 
58 
59 


•64027 
•64055 
•64082 
-64109 
•64136 




77990 
78200 
78410 
78621 
78832 


•65661 
•65689 
•65716 
•65743 
.65771 




91217 
91449 
91681 
91914 
92147 


.67306 
.67333 
.67361 
•67388 
•67416 


2 
2 
2 
2 
2 


05864 
06121 
06379 
06637 
06896 


.68960 
.68988 
.69015 
. 69043 
• 69071 


2 
2 
2 
2 
2 


22165 
22452 
22740 
23028 
23317 


55 
56 
57 
58 
59 


60 


• 64^3 


1 


. 79043 


.65798 


1 


92380 


.67443 


2. 


07155 


.69098 


2 


23607 


60 



777 



TABLE X.— NATURAL VEl4,SED STNES AND EXTERNAL SECANTS i 

73° 73° 74° 75° 



9 


Vers. 


Ex. sec. 


Vers. 


Ex. see. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




1 

2 
3 
4 


•69098 
.69126 
.69154 
•69181 
•69209 


2.23607 
2.23897 
2-24187 
2-24478 
2.24770 


.70763 
.70791 
•70818 
•70846 
•70874 


2.42030 
2.42356 
2.42683 
2.43010 
2.43337 


•72436 
•72464 
•72492 
•72520 
•72548 


2.62796 
2.63164 
2.63533 
2.63903 
2.64274 


.74118 
.74146 
.74174 
. 74202 
.74231 


2.86370 
2-86790 
2.87211 
2^87633 
2.88056 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.69237 
.69264 
.69292 
.69320 
.69347 


2.25062 
2.25355 
2.25648 
2^25942 
2.26237 


•70902 
•70930 
.70958 
•70985 
•71013 


2.43666 
2.43995 
2.44324 
2.44655 
2.44986 


.72576 
.72604 
•72632 
.72660 
.72688 

•72716 
•72744 
.72772 
.72800 
.72828 


2.64645 
2.65018 
2.65391 
2.65765 
2.66140 


.74259 
-74287 
-74315 
. 74343 
.74371 


2.88479 
2.88904 
2.89330 
2.89756 
2.90184 


5 

6 
7 
8 
9 


10 

11 
12 
13 
14 


•69375 
•69403 
•69430 
•69458 
.69486 


2.26531 
2^26827 
2.27123 
2.27420 
2.27717 


•71041 
•71069 
•71097 
•71125 
.71153 


2.45317 
2.45650 
2.45983 
2.46316 
2.46651 


2.66515 
2.66892 
2.67269 
2.67647 
2.68025 


-74399 
- 74427 
.74455 
. 74484 
.74512 


2.90613 
2.91042 
2-91473 
2.91904 
2.92337 


10 

11 
12 
13 
14 


15 

16 
17 
18 
19 


•69514 
.69541 
•69569 
•69597 
•69624 


2.28015 
2.28313 
2.28612 
2.28912 
2.29212 


.71180 
.71208 
.71236 
•71264 
.71292 


2.46986 
2.47321 
2.47658 
2^47995 
2.48333 


.72856 
•72884 
.72912 
.72940 
•72968 


2.68405 
2-68785 
2-69167 
2-69549 
2.69931 


. 74540 
-74568 
-74596 
.74624 
-74652 


2.92770 
2.93204 
2.93640 
2.94076 
2.94514 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


•69652 
.69680 
.69708 
•69735 
69763 


2.29512 
2.29814 
2-30115 
2.30418 
2.30721 


.71320 
•71348 
•71375 
•71403 
•71431 


2.48671 
2^49010 
2.49350 
2^49691 
2.50032 


.72996 
.73024 
.73052 
•73080 
.73108 


2.70315 
2-70700 
2-71085 
2-71471 
2-71858 


.74680 
•74709 
•74737 
.74765 
-74793 


2.94952 
2.95392 
2.95832 
2-96274 
2-96716 


30 

21i 
22- 
23 
24 


25 
26 
27 
28 
29 


.69791 
-69818 
.69846 
.69874 
•69902 


2.31024 
2.31328 
2.31633 
2.31939 
2-32244 


.71459 
.71487 
.71515 
.71543 
.71571 


2.50374 
2.50716 
2.51060 
2.51404 
2^51748 


.73136 
.73164 
•73192 
•73220 
•73248 


2-72246 
2-72635 
2-73024 
2.73414 
2-73806 


-74821 
. 74849 
-74878 
- 74906 
-74934 


2-97160 
2-97604 
2-98050 
2-98497 
2-98944 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•69929 
•69957 
•69985 
.70013 
.70040 


2-32551 
2-32858 
2.33166 
2.33474 
2-33783 


•71598 
•71626 
•71654 
.71682 
•71710 


2-52094 
2^52440 
2^52787 
2-53134 
2.53482 


•73276 
•73304 
•73332 
.73360 
.73388 


2-74198 
2-74591 
2-74984 
2-75379 
2-75775 


-74962 
•74990 
-75018 
-75047 
-75075 


2-99393 
2-99843 
3.00293 
3-00745 
3.01198 


30 

31 

32 
33 
34 


35 
36 
37 
38 
.39_ 


•70068 
•70096 
.70124 
•70151 
•70179 

•70207 
.70235 
•70263 
.70290 
.70318 


2.34092 
2.34403 
2.34713 
2.35025 
2-35336 


.71738 
•71766 
.71794 
•71822 
• 718^0,. 


2.53831 
2.54181 
2.54531 
2.54883 

2.55235 


•73416 
. 73444 
•73472 
.73500 
•73529 


2.76171 
2.76568 
2.76966 
2-77365 
2-77765 


-75103 
-75131 
•75159 
•75187 
.75216 


3-01652 
3-02107 
3-02563 
3.03020 
3.03479 


35 
36 
37 
38j 

3? 


40 

41 
42 
43 
44 


2.35649 
2.35962 
2-36276 
2.36590 
2-36905 


.71877 
•71905 
•71933 
.71961 
•71989 


2.55587 
2.55940 
2.56294 
2.56649 
2.57005 


•73557 
.73585 
•73613 
•73641 
.73669 


2-78166 
2-78568 
2-78970 
2-79374 
2^79778 


. 75244 
.75272 
•75300 
•75328 
•75356 


3.03938 
3.04398 
3.04860 
3.05322 
3.05786 


40^ 

41 

42 
43 
44 


45 
46 
47 
48 
49 


.70346 
.70374 
.70401 
• 70429 
.70457 


2^37221 
2^37537 
2^37854 
2^38171 
2-38489 


•72017 
•72045 
•72073 
.72101 
-72129 


2.57361 
2.57718 
2.58076 
2.58434 
2.58794 


.73897 
.73725 
.73753 
.73781 
.73809 


2^80183 
2^80589 
2-80996 
2.81404 
2.81813 


.75385 
.75413 
-75441 
.75469 
.75497 


3-06251 
3-06717 
3-07184 
3-07652 
3.08121 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.70485 
.70513 
.70540 
.70568 
.70596 


2-38808 
2-39128 
2.39448 
2.39768 
2.40089 


.72157 
•72185 
•72213 
•72241 
•72269 


2.59154 
2^59514 
2^59876 
2^60238 
2-60601 


•73837 
•73865 
•73893 
•73921 
.73950 


2.82223 
2.82633 
2.83045 
2.83457 
2.83871 


-75526 
-75554 
-75582 
.75610 
•75639 


3^08591 
3-09063 
3-09535 
3-10009 
3-10484 


50 

51 

52 
53 
54 


55 
56 
57 
58 
59 


.70624 
•70652 
•70679 
•70707 
•70735 


2.40411 
2.40734 
2.41057 
2.41381 
2-41705^ 


•72296 
.72324 
•72352 
.72380 
•72408 


2.60965 
2.61330 
2.61695 
2.62061 
2.62428 


.73978 
.74006 
• 74034 
•74062 
•74090 


2.84285 
2.84700 
2.85116 
2.85533 
2.85951 


•75667 
•75695 
.75723 
.75751 
.75780 


3-10960 
3-11437 
3-11915 
3-12394 
3-12875 


55 
56 
57 
58 
59 


60 


•70763 


2.42030 


.72436 


2.62796 


•74118 


2.86370 


.75808 


3-13357 


60 



778 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
76° n"" 78° 79° 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 





•75808 


3^13357 


•77505 


3-44541 


.79209 


3-80973 


-80919 


4-24C84 





1 


.75836 


3 •13839 


•77533 


3-45102 


-79237 


3-81633 


-80948 


4-24870 


1 


2 


•75864 


3 • 14323 


•77562 


3-45664 


-79266 


3-82294 


-80976 


4-25658 


2 


3 


•75892 


3 • 14809 


•77590 


3-46228 


-79294 


3-82956 


-81005 


4-26448 


3 


4 


•75921 
•75949 


3^15295 
3-15782 


•77618 


3^46793 


.79323 


3-83621 


-81033 


4-27241 


4 


5 


•77647 


3- 47360 


-79351 ,3-84288 


-81062 


4.28036 


5 


6 


•75977 


3-16271 


•77675 


3-47928 


-79380 3-84956 


-81090 


4.28833 


6 


7 


•76005 


3-16761 


•77703 


3-48498 


-79408 3-85627 


-81119 


4-29634 


7 


8 


•76034 


3-17252 


•77732 


3-49069 


-79437 3-86299 


-81148 


4-30436 


8 


9 


•76062 


3-17744 


-77760 


3-49642 


-79465 3-86973 


-81176 


4.31241 


9 


10 


•76090 


3-18238 


-77788 


3-50216 


-79493 3-87649 


-81205 


4-32049 


10 


11 


•76118 


3-18733 


•77817 


3-50791 


-79522 


3.88327 


-81233 


4.32859 


11 


12 


•76147 


3-19228 


-77845 


3-51368 


-79550 


3.89007 


-81262 


4.33671 


12 


13 


•76175 


3.19725 


.77874 


3-51947 


.79579 


3-89689 


-81290 


4.34486 


13 


14 


•76203 


3.20224 


•77902 


3.52527 


.79607 3.90373 


-81319 
-81348 


4-35304 


14 


15 


•76231 


3^20723 


-77930 


3-53109 


-79636 13-91058 


4.36124 


15 


16 


•76260 


3.21224 


-77959 


3.53692 


-79664 


3-91746 


-81376 


4.36947 


16 


17 


•76288 


3.21726 


-77987 


3.54277 


-79693 


3-92436 


-81405 


4-37772 


17 


18 


•76316 


8 • 22229 


-78015 


3.54863 


-79721 


3-93128 


.81433 


4-38600 


18 


19 


• 76344 


3 • 22734 


. 78044 


3-55451 


-79750 3.93821 


.81462 


4-39430 


19 


30 


•76373 


3 •23239 


-78072 


3.56041 


-79778 ;3-94517 


-81491 


4.40263 


30 


21 


•76401 


3.23746 


-78101 


3.56632 


-79807 ,3-95215 


-81519 


4-41099 


2. 


22 


• 76429 


3-24255 


-78129 


3-57224 


-79835 


3-95914 


-81548 


4-41937 


2^ 


23 


•76458 


3-24764 


.78157 


3.57819 


-79864 


3-96616 


-81576 


4-42778 


23 


24 


• 76486 


3-25275 


-78186 


3.58414 


-79892 


3-97320 


-81605 


4-43622 


24 


25 


•76514 


3-25787 


-78214 


3.59012 


-79921 


3-98025 


81633 


4-44468 


25 


26 


•76542 


3-26300 


. 78242 


3.59611 


-79949 


3-98733 


.81662 


4-45317 


26 


27 


•76571 


3.26814 


.78271 


3.60211 


-79978 


3-99443 


.81691 


4-46169 


27 


28 


.76599 


3-27330 


-78299 


3.60813 


-80006 


4-00155 


.81719 


4-47023 


28 


29 


•76627 


3-27847 


-78328 


3-61417 


■80035 '4-00869 


.81748 


4-47881 


29 


30 


•76655 


3-28366 


-78356 


3.62023 


.80063 j4. 01585 


.81776 


4-48740 


30 


31 


•76684 


3-28885 


-78384 


3.62630 


.80092 4.02303 


.81805 


4-49603 


31 


32 


•76712 


3-29406 


-78413 


3-63238 


-80120 '4-03024 


.81834 


4-50468 


32 


33 


• 76740 


3.29929 


-78441 


3-63849 


-80149 4-03746 


-81862 


4-51337 


33 


34 


76769 


3.30452 


-78470 


3.64461 


-80177 '4-04471 


.81891 


4-52208 


34 


35 


•76797 


3.30977 


•78498 


3.65074 


-80206 4-05197 


-81919 


4-53081 


35 


36 


•76825 


3.31503 


•78526 


3-65690 


-80234 4-05926 


-81948 


4-53958 


36 


37 


•76854 


3-32031 


•78555 


3-66307 


-80263 4-06657 


-81977 


4-54837 


37 


38 


•76882 


3-32560 


•78583 


3-66925 


.80291 14.07390 


-82005 


4-55720 


38 


3?., 


.76910 


3-33090 


•78612 


3.67545 


.80320 14.08525 


•82034 


4.56605 


39 


40 


.76938 


3.33622 


.78640 


3-68167 


.80348 14-08863 


-82063 


4-57493 


40 


41 


•76967 


3-34154 


•78669 


3.68791 


-80377 


4-09602 


-82091 


4-58383 


4i 


42 


•76995 


3-34689 


•78697 


3.69417 


.80405 


4-10344 


-82120 


4-59277 


42 


43 


•77023 


3-35224 


-78725 


3 . 70044 


.80434 


4-11088 


.82148 


4-60174 


43 


44 


•77052 


3-35761 


-78754 


3-70673 


.80462 


4-11835 
4-12583 


.82177 


4-61073 


44 


45 


•77080 


3-36299 


.78782 


3-71303 


.80491 


.82206 


4-61976 


45 


46 


•77108 


3-36839 


.78811 


3.71935 


-80520 


4.13334 


-82234 


4-62881 


46 


47 


•77137 


3-37380 


•78839 


3.72569 


-80548 


4-14087 


-82263 


4-63790 


4V 


48 


•77165 


3-37923 


.78868 


3.73205 


-80577 


4-14842 


-82292 


4-64701 


48 


49 


•77193 


3 •38466 


-78896 


3-73843 


-80605 


4.15599 


82320 


4.65616 


49 


50 


•77222 


3-39012 


•78924 


3-74482 


•80634 


4.16359 


-82349 


4.66533 


50 


51 


•77250 


3-39558 


-78953 


3-75123 


•80662 


4.17121 


-82377 


4-67454 


51 


52 


•77278 


3-40106 


-78981 


3.75766 


-80691 


4.17886 


-82406 


4-68377 


52 


53 


•77307 


3-40656 


.79010 


3.76411 


.80719 


4.18652 


.82435 


4-69304 


53 


54 


•77335 


3-41206 


•79038 


3-77057 


-80748 


4-19421 


-82463 


4-70234 


64 


55 


•77363 


3-41759 


.79067 


3-77705 


.80776 


4-20193 


.82492 


4-71166 


55 


56 


.77392 


3.42312 


.79095 


3.78355 


-80805 


4-20966 


.82521 


4-72102 


56 


57 


.77420 


3.42867 


.79123 


3.79007 


.80833 


4.21742 


.82549 


4-73041 


57 


58 


• 77448 


3-43424 


.79152 


3.79661 


.80862 


4.22521 


.82578 


4-73983 


58 


59 


•77477 


3-43982 


-79180 


3-80316 


-80891 


4.23301 


-82607 


4-74929 


59 


60 


.77505 


3.44541 


•79209 


3.80973 


-80919 


4.24084 


.82635 


4.75877 


60 



779 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. L 



80" 



81^ 



83' 



83' 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


.82635 
.82664 
.82692 
.82721 
.82750 


4.75877 
4.76829 
4.77784 
4.78742 
4.79703 


-84357 
.84385 
.84414 
. 34443 
-84471 


5.39245 
5-40422 
5-41602 
5-42787 
5-43977 


.86083 
.86112 
.86140 
-86169 
.86198 


6-18530 
6-20020 
6.21517 
6.-23019 
6-24529 


-87813 
.87842 
-87871 
-87900 
•87929 


7-20551 
7.22500 
7.24457 
7.26425 
7-28402 




1 
2 
3 

4 


5 
6 
7 
8 
9 


.82778 
.82807 
.82836 
•82864 
.82893 


4.80667 
4.81635 
4.82606 
4.83581 
4.84558 


-84500 
-84529 
-84558 
-84586 
-84615 


5-45171 
5-46369 
5-47572 
5-48779 
5.49991 


-86227 
-86256 
.86284 
.86313 
•86342 


6.26044 
6-27566 
6-29095 
6-30630 
6.32171 


-87957 
-87986 
-88015 
- 88044 
.88073 


7-30388 
7-32384 
7-34390 
7-36405 
7-38431 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


•82922 
•82950 
•82979 
.83008 
•83036 


4.85539 
4-86524 
4.87511 
4.88502 
4.89497 


-84644 
.84673 
.84701 
-84730 
-84759 


5.51208 
5-52429 
5-53655 
5-54886 
5.56121 


.86371 
•86400 
•86428 
•86457 
•86486 


6-33719 
6-35274 
6.36835 
6.38403 
6.39978 


.88102 
.88131 
.88160 
.88188 
•88217 


7.40466 
7-42511 
7-44566 
7-46632 
7.48707 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.83065 
.83094 
.83122 
.83151 
.83;i8p. 

•83208 
•83237 
•83266 
.83294 
.83323 


4.90495 
4.91496 
4.92501 
4-93509 
4.94521 

4.95536 
4.96555 
4-97577 
4.98603 
4.99633 


-84788 
-84816 
-84845 
.84874 
. 84903 

.84931 
.84960 
-84989 
-85018 
.85046 


5.57361 
5-58606 
5-59855 
5-61110 
5.62369 


.86515 
.86544 
.86573 
.86601 
.86630 


6.41560 
6.43148 
6.44743 
6.46346 
6.47955 


.88246 
.88275 
.88304 
.88333 
•88362 


7.50793 
7.52889 
7.54996 
7.57113 
7.59241 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


5-63633 
5-64902 
5-66176 
5-67454 
5-68738 


.86659 
.86688 
.86717 
.86746 
•86774 


6.49571 
6-51194 
6-52825 
6-54462 
6-56107 


•88391 
.88420 
. 88448 
.88477 
•88506 


7-61379 
7-63528 
7-65688 
7-67859 
7.70041 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.83352 
•83380 
.83409 
.83438 
•83467 


5-00666 
5-01703 
5-02743 
5-03787 
5.04834 


-85075 
.85104 
.85133 
•85162 
.85190 


5-70027 
5.71321 
5-72620 
5.73924 
5-75233 


.86803 
.86832 
.86861 
.86890 
.86919 


6.57759 
6.59418 
6-61085 
6-62759 
6 - 64441 


.88535 
.88564 
.88593 
.88622 
•88651 


7-72234 
7-74438 
7-76653 
7-78880 
7-81118 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.83495 
•83524 
•83553 
• 83581 
•83610 


5.05886 
5.06941 
5.08000 
5.09062 
5.10129 


.85219 
-85248 
-85277 
.85305 
.85334 


5.76547 
5.77866 
5.79191 
5.80521 
5-81856 


.86947 
.86976 
.87005 
.87034 
.87063 


6-66130 
6-67826 
6-69530 
6-71242 
6-72962 


-88680 
-88709 
-88737 
-88766 
.88795 


7-83367 
7-85628 
7-87901 
7-90186 
7-92482 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•83639 
•83667 
•83696 
•83725 
.83754 


5.11199 
5.12273 
5-13350 
5-14432 
5.15517 


.85363 
.85392 
.85420 
.85449 
.85478 


5-83196 
5.84542 
5-85893 
5-87250 
5-88612„ 

5-89979 
5.91352 
5.92731 
5.94115 
5.95505 


.87092 
.87120 
.87149 
.87178 
.87207 

.87236 
•87265 
•87294 
•87322 
.87351 


6-74689 
6-76424 
6-78167 
6-79918 
6-81677 


.88824 
•88853 
.88882 
.88911 
.88940 


7-94791 
7-97111 
7-99444 
8-01788 
8-04146 


35^ 

36 

37 

38 

39 


40 

41 
42 
43 
44 


•83782 
•83811 
•83840 
.83868 
•83897 


5.16607 
5.17700 
5.18797 
5-19898 
5-21004 


.85507 
.85536 
.85564 
.85593 
.85622 

.85651 
.85680 
.85708 
-85737 
-85766 


6-83443 
6-85218 
6-87001 
6-88792 
6-90592 


•88969 
•88998 
•89027 
.89055 
•89084 


8.06515 
8.08897 
8-11292 
8-13699 
8-16120 


40| 

41 

42^ 

43 

44 


45 
46 
47 
48 
49 


.83926 
.83954 
.83983 
.84012 
. 84041 


5-22113 
5-23226 
5-24343 
5-25464 
5-26590 


5.96900 
5.98301 
5.99708 
6.01120 
6.02538 


.87380 
.87409 
•87438 
•87467 
•87496 


6-92400 
6-94216 
6-96040 
6-97873 
6-99714 


•89113 
.89142 
.89171 
-89200 
-89229 


8-18553 
8-20999 
8.23459 
8.25931 
8.28417 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.84069 
•84098 
.84127 
.84155 
.841«4 


5-27719 
5.28853 
5-29991 
5-31133 
5.32279 

5.33429 
5.34584 
5.35743 
5-36906 
5.38073 


.85795 
.85823 
-85852 
-85881 
-85910 


6.03962 
6.05392 
6.06828 
6.08269 
6.09717 


-87524 
-87553 
-87582 
-87611 
•87640 


7-01565 
7-03423 
7-05291 
7.07167 
7.09052 


-89258 
-89287 
-89316 
-89345 
-89374 


8-30917 
8-33430 
8-35957 
8.38497 
8-41052 


50 

51 
52 
53 
54 


55 
56 
57 
58 
,59 


.84213 
.84242 
.84270 
.84299 
.84328 


.85939 
-85967 
-85996 
.86025 
-86054 


6-11171 
6-12630 
6-14096 
6.15568 
6-17046 


•87669 
-87698 
-87726 
-87755 
.87784 


7.10946 
7.12849 
7.14760 
7-16681 
7-18612 • 


.89403 
.89431 
.89460 
-89489 
•89518 


8-43620 
8-46203 
8-48800 
8-51411 
8.54037 


55 
56 
57 
58 
59 


60 


.84357 


5-39245 


.86083 


6.18530 


.87813 


7.20551 


.89547 


8.56677 


60 



780 



TABLE X.— NATURAL VERSED SINES AND 



84^ 



85^ 



EXTERNAL SECANTS. 
86° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 




1 

2 
3 
4 


•89547 
•89576 
•89605 
•89634 
•89663 


8.56677 
8.59332 
8.62002 
8.64687 
8.67387 




.91284 
.91313 
.91342 
-91371 
.91400 


10-47371 
10-51199 
10-55052 
10.58932 
10-62837 




-93024 
.93053 
.93082 
.93111 
-93140 


13-33559 
13-39547 
13.45586 
13.51676 
13-57817 




1 
2 
3 
4 


5 
6 
7 
8 
9 


•89692 
•89721 
•89750 
•89779 
.89808 


8.70103 
8 •72833 
8-75579 
8-78341 
8.81119 




91429 
91458 
91487 
91516 
91545 


10-66769 
10-70728 
10-74714 
10-78727 
10-82768 




-93169 
.93198 
.93227 
.93257 
-93286 


13.64011 
13.70258 
13-76558 
13-82913 
13-89323 


5 
6 
7 
8 

9 


10 

11 
' 12 
^ 13 

14 


•89836 
•89865 
•89894 
.89923 
.89952 


8-83912 
8-86722 
8-89547 
8-92389 
8.95248 




91574 
91603 
91632 
91661 
91690 


10-86837 
10-90934 
10-95060 
10-99214 
11.03397 




-93315 
-93344 
-93373 
-93402 
.93431 


13-95788 
14-02310 
14-08890 
14-15527 
14-22223 


10 

11 
12 
13 
14 


15 

16 

17 

. 18 

1 19 


•89981 
•90010 
•90039 
•90068 
.90097 


8.98123 
9-01015 
9-03923 
9-06849 
9.09792 




91719 
91748 
91777 
91806 
91835 


1-07610 
11.11852 
11.16125 
11.20427 
11.24761 




•93460 
•93489 
-93518 
-93547 
93576 


14-28979 
14-35795 
14^42672 
14^49611 
14.56614 


15 
16 
17 
18 
19 


30 

21 
22 

■: 23 

24 


.90126 
•90155 
•90184 
•90213 
•90242 


9-12752 
9-15730 
9-18725 
9-21739 
9-24770 




91864 
91893 
91922 
91951 
91980 


11^29125 
11.33521 
11.37948 
11.42408 
11.46900 




93605 
93634 
-93663 
93692 
93721 


14.63679 
14.70810 
14.78005 
14.85268 
14-92597 


20 

21 
22 
23 
24 


25 
26 
27 
i 28 
29 


•90271 
•90300 
•90329 
•90358 
.90386 


9-27819 
9-30887 
9-33973 
9-37077 
9-40201 




92009 
92038 
92067 
92096 
92125 


11.51424 
11.55982 
11.60572 
11.65197 
11.69856 




93750 
93779 
93808 
93837 
93866 


14.99995 
15-07462 
15.14999 
15-22607 
15.30287 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•90415 
•90444 
•90473 
•90502 
•90531 


9-43343 
9-46505 
9-49685 
9-52886 
9.56106 




92154 
92183 
92212 
92241 
92270 


11^74550 
11^79278 
11^84042 
11.88841 
U. 93677 




93895 
93924 
93953 
93982 
94011 


15-38041 
15-45869 
15-53772 
15-61751 
15.69808 


30 

31 
32 
33 
34 


35 

36 

, 37 

1 38 

,39 


•90560 
•90589 
•90618 
•90647 
.90676 


9-59346 
9-62605 
9-65885 
9-69186 
9.72507 




92299 
92328 
92357 
92386 
92415 


11-98549 
12-03458 
12^08404 
112.13388 
12-18411 




94040 
94069 
94098 
94127 
94156 


15-77944 
15-86159 
15-94456 
16-02835 
16-11297 


35 
36 
37 
38 
39 


, 40 

1 41 
42 
43 
44 


•90705 
•90734 
•90763 
•90792 
•90821 


9-75849 
9-79212 
9-82596 
9-86001 
9.89428 




92444 
92473 
92502 
92531 
92560 


12-23472 
12-28572 
12.33712 
12.38891 
12-44112 




94186 
94215 
94244 
94273 
94302 


16-19843 
16.28476 
16.37196 
16-46005 
16-54903 


40 

41 
42 
43 
44 


i 45 

! 46 

' 47 

48 

49 


•90850 
•90879 
•90908 
•90937 
.90966 


9.92877 

9.96348 

9-99841 

10-03356 

10.06894 




92589 
92618 
92647 
92676 
92705 


12-49373 
12-54676 
12-60021 
12.65408 
12-70838 




94331 
94360 
94389 
94418 
94447 


16-63893 
16-72975 
16-82152 
16-91424 
17-00794 


45 
46 
47 
48 

49 


50 

' 51 
52 
53 

, 54 


•90995 
•91024 
•91053 
.91082 
•91111 


10.10455 
10.14039 
10-17646 
10-21277 
10.24932 




92734 
92763 
92792 
92821 
92850 


12-76312 
12.81829 
12-87391 
12.92999 
12.98651 




94476 
94505 
94534 
94583 
94592 


17-10262 
17-19830 
17-29501 
17.39274 
17-49153 


50 

51 
52 
53 
54 


( 55, 

! 56 
57 
58 

, 59 


.91140 
•91169 
•91197 
•91226 
.91255 


10.28610 
10-32313 
10-36040 
10-39792 
10.43569 




92879 
92908 
92937 
92966 
92995 

93024 


13-04350 
13-10096 
13-15889 
13-21730 
13-27620 




94621 
94650 
94679 
94708 
94737 


17-59139 
17-69233 
17-79438 
17.89755 
18-00185 


55 
56 
57 
58 
_5Q 


60 

1 . 


.91284 


10.47371 


13.83559 




94766 


18.10732 


60 


! 








78 


1 











TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 



87' 



88° 



89' 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 





.94766 


18.10732 


.96510 


27.65371 


.98255 


P.S. 29869 





1 


.94795 


18.21397 


.96539 


27.89440 


-98284 


57.26976 


1 


2 


.94825 


18.32182 


.96568 


28.13917 


-98313 


58.27431 


2 


3 


.94854 


18.43088 


.96597 


28-38812 


.98342 


59.31411 


3 


4 


.94883 


18.54119 


. .96626 


28-64137 
28-89903 


.98371 


60.39105 


4 


5 


.94912 


18.65275 


.96655 


.98400 


61.50715 


5 


6 


.94941 


18-76560 


.96684 


29.16120 


.98429 


62.66460 


6 


7 


.94970 


18.87976 


.96714 


29.42802 


-98458 


63.86572 


7 


8 


.94999 


18.99524 


.96743 


29.69960 


-98487 


65.11304 


8 


9 


.95028 


19.11208 


.96772 


29.97607 


.98517 


66.40927 


9 


10 


.95057 


19.23028 


.96801 


30.25758 


-98546 


67.75736 


10 


11 


.95086 


19.34989 


.96830 


30.54425 


-98575 


69-16047 


11 


12 


.95115 


19.47093 


.96859 


30.83623 


.98604 


70-62205 


12 


13 


.95144 


19-59341 


.96888 


31-13366 


.98633 


72.14583 


13 


14 


.95173 


19.71737 


.96917 


31.43671 


.98662 


73.73586 


14 


15 


.95202 


19.84283 


.96946 


31-74554 


-98691 


75.39655 


15 


16 


.95231 


19.96982 


.96975 


32.06030 


.98720 


77.13274 


16 


17 


.95260 


20.09838 


.97004 


32.38118 


.98749 


78.94968 


17 


18 


.95289 


20.22852 


.97033 


32.70835 


.98778 


80.85315 


18 


l^ 


.95318 


20.36027 


.97062 


33-04199 


.98807 


82-84947 


19 


^0 


.95347 


20.49368 


.97092 


33.38232 


.98836 


84.94561 


20 


21 


.95377 


20.62876 


.97121 


33.72952 


.98866 


87-14924 


21 


22 


.95406 


20.76555 


.97150 


34.08380 


.98895 


89.46886 


22 


23 


.95435 


20.90409 


.97179 


34.44539 


.98924 


91.91387 


23 


24 


.95464 


21.04440 


.97208 


34-81452 


.98953 


94-49471 


24 


25 


.95493 


21.18653 


.97237 


35.19141 


.98982 


97-22303 


25 


26 


.95522 


21.33050 


.97266 


35-57633 


.99011 


100-1119 


26 


27 


.95551 


21.47635 


.97295 


35.96953 


.99040 


103-1757 


27 


28 


.95580 


21.62413 


.97324 


36-37127 


.99069 


106.4311 


28 


29 


.95609 


21-77386 


.97353 


36-78185 


.99098 


109-8966 


29 


30 


.95638 


21.92559 


.97382 


37-20155 


.99127 


113-5930 


30 


31 


.95667 


22.07935 


.97411 


37-63068 


.99156 


117-5444 


31 


32 


.95696 


22.23520 


.97440 


38.06957 


.99186 


121-7780 


32 


33 


.95725 


22.39316 


.97470 


38.51855 


.99215 


126-3253 


33 


34 


.95754 


22.55328 


.97499 


38-97797 


-99244 


131-2223 


34 


35 


.95783 


22.71563 


.97528 


39.44820 


.99278 


136.5111 


35 


36 


.95812 


22.88022 


.97557 


39.92963 


.99302 


142.2406 


36 


37 


.95842 


23.04712 


.97586 


40.42266 


.99331 


148.4684 


37 


38 


.95871 


23-21637 


.97615 


40.92772 


.99360 


155.2623 


38 


39 


.95900 


23-38802 


.97644 


41-44525 


.99389 


162-7033 


•39 


40 


.95929 


23.56212 


.97673 


41.97571 


.99418 


170-8883 


40 


41 


.95958 


23.73873 


.97702 


42.51961 


.99447 


179.9350 


41 


42 


.95987 


23.91790 


.97731 


43.07746 


.99476 


189-9868 


42 


43 


.96016 


24.09969 


.97760 


43.64980 


.99505 


201-2212 


43 


44 


.96045 


24-28414 


-97789 


44.23720 


.99535 


213.8600 


44 


45 


.96074 


24.47134 


.97819 


44.84026 


.99564 


228-1839 


45 


46 


.96103 


24.66132 


.97848 


45.45963 


.99593 


244-5540 


46 


47 


.96132 


24.85417 


.97877 


46.09596 


.99622 


263-4427 


47 


48 


.96161 


25.04994 


.97906 


46.74997 


.99651 


285-4795 


48 


49 


.96190 


25.24869 


.97935 


47.42241 


.99680 


311-5230 


49 


,50 


.96219 


25^45051 


.97964 


48.11406 


.99709 


342.7752 


50 


51 


.96243 


25-65546 


.97993 


48.82576 


.99738 


380.9723 


51 


52 


.96277 


25.86360 


.98022 


49.55840 


.99767 


428.7187 


52 


53 


.96307 


2ft. 07503 


.98051 


50.31290 


.99796 


490.1070 


53 


54 


.96336 
.96365 


26 r:69Cl 


.98080 


51-09027 


.99825 


571-9581 


54 


55 


2e^\>0804 


.98109 


51.89156 


.99855 


686-5496 


55 


56 


.96394 


26.V2978 


.98138 


52-71790 


.99884 


858-4369 


56 


57 


.96423 


26.95513 


.98168 


53-57046 


.99913 


1144-916 


57 


58 


.96452 


27.18417 


.98197 


54-45053 


.99942 


1717.874 


58 


.59-. 


,96481 


27-41700 


.98226 


55.35946 


.99971 


3436.747 


59 


60 


.96510 


27.65371 


.98255 


56.29869 


1.00000 


Infinite 


60 



782 



TABLE XI.— REDUCTION OF BAROMETER 


READING TO 32* 


F. 














[nches. 












Temp. 
























O 
Fahr. 


260 


26-5 


'27.0 


27.5 


28.0 


28.5 


29.0 


29.5 


30-0 


30.5 


310 


45 


-.039 


-.039 


-.040 


-.041 


-.042 


-.042 


-.043 


-.044 


-.045 


-.045 


-.046 


46 


.041 


.042 


.043 


.043 


.044 


.045 


.046 


.046 


.047 


.048 


.049 


47 


.043 


.044 


.045 


.046 


.047 


.048 


.048 


.049 


.050 


.051 


.052 


48 


.046 


.047 


.047 


.048 


.049 


.050 


.051 


.052 


.053 


.053 


.054 


49 


.048 


.049 


.050 


.051 


.052 


.052 


.054 


.054 


.055 


.056 


.057 


50 


.050 


.051 


.052 


.053 


.054 


.055 


.056 


.057 


.058 


.059 


.060 


51 


.053 


.054 


.055 


.056 


.057 


.058 


.059 


.060 


.061 


.062 


.063 


52 


.055 


.056 


.057 


.058 


.059 


.060 


.061 


.062 


.064 


.065 


.066 


53 


.057 


.058 


.060 


.061 


.062 


.063 


.064 


.065 


.066 


.067 


.068 


54 


.060 


.061 


.062 


.063 


.064 


.065 


.067 


.068 


.069 


.070 


.071 


55 


.062 


.063 


.064 


.065 


.066 


.068 


.069 


.070 


.071 


.073 


.074 


56 


.064 


.065 


.067 


.068 


.069 


.070 


.072 


.073 


.074 


.075 


.077 


57 


.067 


.068 


.069 


.070 


•072 


.073 


.075 


.076 


.077 


.078 


.080 


58 


.069 


.070 


.071 


.073 


.074 


.076 


.077 


.078 


.080 


.081 


.082 


59 


.072 


.073 


.074 


.075 


.077 


.078 


.080 


.081 


.083 


.084 


•085 


60 


.074 


.076 


.077 


.078 


.079 


.081 


.082 


.084 


.085 


.086 


.088 


61 


.076 


.077 


.079 


.080 


.082 


.083 


.085 


.086 


.088 


.089 


.091 


62 


.079 


.080 


.082 


.083 


.085 


.086 


.088 


.089 


.091 


.092 


.094 


63 


.081 


.082 


.084 


.085 


.087 


.088 


.090 


.091 


.093 


.095 


.096 


64 


.083 


.085 


.086 


.088 


.090 


.091 


.093 


.094 


.096 


.097 


.099 


65 


.086 


.087 


.089 


.090 


.092 


.093 


.095 


.097 


.099 


.100 


.102 


66 


.088 


.089 


.091 


.093 


.095 


.096 


.098 


.099 


.101 


.103 


.105 


67 


.090 


.092 


.094 


.095 


.097 


.099 


.101 


.102 


.104 


.106 


.108 


68 


.093 


.094 


.096 


.098 


.100 


.101 


.103 


.105 


.107 


.108 


.110 


69 


.095 


.097 


.099 


.100 


.102 


.104 


.106 


.107 


.110 


.111 


.113 


70 


.097 


.099 


.101 


.103 


.105 


.106 


.109 


.110 


.112 


.114 


.116 


71 


.100 


.101 


.103 


.105 


.107 


.109 


.111 


.113 


.115 


.117 


.119 


72 


.102 


.104 


.106 


.108 


.110 


.112 


.114 


.116 


.118 


.120 


.122 


I 73 


.104 


.106 


.108 


.110 


.112 


.114 


.116 


.118 


.120 


.122 


.124 


1 74 


.107 


.109 


.111 


.113 


.115 


.117 


.119 


.121 


.123 


.125 


.127 


1 75 


.109 


.111 


.113 


.115 


.117 


.119 


.122 


.124 


.126 


.128 


.130 


1 76 


.111 


.113 


.116 


.118 


.120 


.122 


.124 


.126 


.128 


.130 


.133 


77 


.114 


.116 


.118 


.120 


.122 


.124 


.127 


.129 


.131 


.133 


.136 


78 


.116 


.118 


.120 


.122 


.125 


.127 


.129 


.131 


.134 


.136 


.138 


79 


.118 


.120 


.123 


.125 


.127 


.129 


.132 


.134 


.137 


.139 


.141 


80 


.121 


.123 


.125 


.127 


.130 


.132 


.135 


.137 


.139 


.141 


.144 


81 


.123 


.125 


.128 


.130 


.132 


.134 


.137 


.139 


.142 


.144 


.147 


82 


.125 


.128 


.130 


.132 


.135 


.137 


.140 


.142 


.145 


.147 


.149 


83 


.128 


.130 


.133 


.135 


.138 


.140 


.142 


.145 


.147 


.149 


.152 


84 


.130 


.132 


.135 


.138 


.140 


.142 


.145 


.147 


.150 


.152 


.155 


85 


.132 


.134 


.137 


.140 


.143 


.145 


.148 


.150 


.153 


.155 


.158 


86 


.135 


.137 


.140 


.142 


.145 


.148 


.150 


.153 


.155 


.158 


.161 


j87 


.137 


.139 


.142 


.144 


.148 


.150 


.153 


.155 


.158 


.161 


.163 


1 88 


.139 


.142 


.145 


.147 


.150 


.152 


.155 


.158 


.161 


.163 


.166 


^89 


.142 


.144 


.147 


.150 


.153 


.155 


.158 


.161 


.164 


.166 


.169 


«0 


.144 


.147 


.150 


.153 


.155 


.158 


.161 


.164 


.166 


.169 


.172 


91 


-.146 


-.149 


-.152 


-.155 


-.158 


-.160 


-.163 


-.166 


-.169 


-.172 


-.175 


1 










7^ 


13 













TABLE XII.— BAROMETRIC ELEVATIONS.* 



1 



B 



Inches. 



20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
23 
23 
23 
23 
23 
23 
23 
23 



Feet. 

11.047 
10,911 
10,776 
10,642 
10,508 
10,375 
10,242 
10,110 
9,979 
9,848 
9,718 
9,589 
9,460 
9,332 
9.204 
9,077 
8,951 
8,825 
8,700 
8,575 
8,451 
8,327 
8,204 
8,082 
7,960 
7,838 
7,717 
7,597 
7.477 
7358 
7,239 
7,121 
7,004 
6,887 
6,770 
6,554 
6,538 
6,423 



Diff. for 
.01. 



Feet. 



-13 


6 


13 


5 


13 


4 


13 


4 


13 


3 


13 


3 


13 


2 


13 


1 


13 


1 


13 





12 


9 


12 


9 


12 


8 


12 


8 


12 


7 


12 


6 


12 


6 


12 


5 


12 


5 


12 


4 


12 


4 


12 


3 


12 


2 


12 


2 


12 


2 


12 


1 


12 





12 





11 


9 


11 


9 


11 


8 


11 


7 


11 


7 


11 


7 


11 


6 


11 


6 


-11 


5 



Inches. 



Feet. 

6,423 
6,308 
6,194 
6,080 
5,967 
5,854 
5,741 
5,629 
5,518 
5.407 
5,296 
5,186 
5,077 
4,968 
4,859 
4,751 
4,643 
4,535 
4,428 
4,321 
4,215 
4.109 
4,004 
3 899 
3,794 
3 690 
3,586 
3,483 
3,380 
3,277 
3,175 
3,073 
2972 
2-871 
2 770 
2,670 
2,570 
2.470 



DifF. for 
.01. 



Feet. 



-11 


5 


11 


4 


11 


4 


11 


3 


11 


3 


11 


3 


11 


2 


11 


1 


11 


1 


11 


1 


11 





10 


9 


10 


9 


10 


9 


10 


8 


10 


8 


10 


8 


10 


7 


10 


7 


10 


6 


10 


6 


10 


5 


10 


5 


10 


4 


10 


4 


10 


4 


10 


3 


10 


3 


10 


3 


10 


2 


10 


2 


10 


1 


10 


1 


10 


1 


10 





10 






-10 



Inches. 



Feet. 

2,470 

2,371 

2,272 

2.173 

2,075 

1,977 

1,880 

1,783 

1 686 

1,589 

1,493 

1,397 

1,302 

1.207 

1,112 

1,018 

924 

830 

736 

643 

550 

458 

366 

274 

182 

91 



-91 

181 

271 

361 

451 

540 

629 

717 

805 

-893 



Diff. for 
.01. 



Feet. 



8 
8 
8 
8 
-8 



* Compiled from Report of U. S. C. & G. Survey for 1881, App. 10 Table XL 

TABLE XIII.— COEFFICIENTS FOR CORRECTIONS FOR TEMPERATURE 

AND HUMIDITY.* 



t + t' 



0° 
10 
20 
30 
40 
50 
60 



c 


Diff. for 
1°. 


t + t' 


C 


Diff. for 
1°. 


t + t' 


C 


-.1024 
.0915 
.0806 
.0698 
.0592 
.0486 

-.0380 


10.9 
10.9 
10.8 
10.6 
10.6 
10.6 


60° 

70 

80 

90 

100 

110 

120 


-.0380 
.0273 
.0166 

-.0058 

+ .0049 
.0156 

+ .0262 


10 
10 
10 
10 
10 
10 


7 
7 
8 
7 
7 
6 


120^* 

130 

140 

150 

160 

170 

180 


+ .0262 
.0368 
.0472 
.0575 
.0677 
.0779 

+ .0879 



Diff. for 
1°. 



10.6 
10.4 
10.3 
10.2 
10.2 
10.0 



* Compiled from Report of U. S. C. & G. Survey for 1881, App. 10, Tables I, IV. 

784 



TABLE XIV. — USEFUL TRIGONOMETRICAL FORMULA. 



10 

11 
12 



sin a 



cosec 



1 tan a _ /\_ 

,eca~V'l+tan2 a V 



cos 2a 



1 



2 Vi-f cot2 a 

cos a tan a = V 1 — cos^ a = 2 sin ^a cos ^a 

1 + cos a 2 tan ^a ^ , 

n = TT7 — n — = vers a cot ^a. 

cot ia 1 + tan^ ^a 



cos a 



1 



cot a 



1 



^c ^ vi + cot2 a Vl + tan2 a 
= 1 — vers a = sin a cot a = v 1— sin^ a = 2 cos^ ^a — 1 
=sin a cot ia—l = cos^ ^a — sin^ ^a = 1 — 2 sin^ ^a. 



tan a 



1 



sin a sec a 



1 



cot a cos a cosec a V^cosec^ a — 1 
vers 2a cosec 2a = cot a — 2 cot 2a = sin a sec a 
sin 2a 



cot a 



1 + cos 2a 
1 c. 



exsec a cot ^a = 3xsec 2a cot 2a. 
3 a sin 2a 1 + cos 2a 



tan a sin a 1 — cos 2a sin 2a 
= v cosec2 a — 1 = cot ^a — cosec a. 
vers a = 1 — cos a = sin a tan ^a = 2 sin^ ^a = cos a exsec do 
exsec a = sec a — 1 = tan a tan ^a = vers a sec a. 



sin 



, _ / vers a _ sin a _ vers a cos ^a 
y 2 ~ 2 cos ia sin a 



/ 1 + cos a _ sin a _ sin a sin ^a 
\ 2 ""2 sin ^a vers a 



cos ^a 

tan ^a = vers a cosec a = cosec a — cot a 



tan a 



cot ^a 



1 + cos a 



sm a 

vers^a =l->/i(l + cos a). 

1 



cosec a 4- cot a = 



l+sec a 
tan a 



exsec a cosec a — cot a 



exsec ^a = 



v^Cl + cosa) 



-1. 



785 



TABLE XIV. — USEFUL TRIGONOMETRICAL FORMULA, 



sin 2a 



■■ 2 sin a cos a 



2 tan a 



1 + tan2 a 

cos 2a =cos2 a— sin2a=l — 2sin2a=2 cos^a— 1 
1 — tan2 a 



tan 2a = 



' 1 + tan2 a* 
2 tan a 



1 — tan2 a' 



cot 2a =:i cot a-^ tan a = ^^^' ^"^ ^l-tan^ a 

2 cot a 2 tan a 

vers 2a = 2 sin2 a = 1 - cos 2a = 2 sin a cos a tan a. 

exsec 2a = i^2_2£. ^ 2 tan2 g _ 2 sin2 g 

cot a 1 — tan2 g 1 — 2 sin2 a * 

sin (g ± 6) =sin a cos 6 ± cos a sin 6. 

cos (a ± 6) = cos g cos 6 T sin a sin 6. 

sin a + sin 6 = 2 sin ^(g + 6) cos i(a - 6). 

sin a -sin 6 =2 sin i(a-6) cos i(a + &), 

cos a + cos 6 = 2 cos ^(a + &) cos i-(a — 6). 

cos a — cos 6= — 2sin^(a + 6)sini^(a — 6). 



Call the sides of any triangle A,B, C, and the opposite angles a, 
andc. Calls = iU+5 + C). 

tan^(a — 6) = . , p tan ^(a + 6) = . , p Cot ^c. 

A +i5 A. +x> 



6. 



COS i(a — 6) 



sin ^(g — 6) 



inia = |/^ 



sin 



COS 



s-B)(s-C) 



1. , A(^-^ ) 



vers a ■■ 



2(s-B)(s-C) 
BC 



Area =V8(s-A)(«-B)(8-C)=^2 



sin b sin c 
2 sin a 



786 



TABLE XV. — USEFUL FORMULA AND CONSTANTS. 



Circumference of a circle (radius = r) = 27rr. 

Area of a circle = n-r*. 

Area of sector (length of arc = Z) = ^Ir, 

*' " " (angle of arc = a°) = '^-nr^ 

360 • 

Area of segment (chord = c, mid. ord. = m) = ^cm (approx.). 
Area of a circle to radius 1 ^ 

Circumference of a circle to diameter 1 )■ = n- = 3.1415927 

Surface of -a sphere to diameter 1 j 

Volume of a sphere to radius 1 = 47r -r- 3 = 4.1887902 

r degrees = 57.2957795 

Arc equal to radius expressed in -{ minutes = 3437.7467708 

I seconds = 206264.8062471 

Length of arc of 1°, radius unity 0.01745329 

Sine of one second = 0.0000048481 

Cubic inches in United States standard gallon = 231 

Weight of one cubic foot of water at maximum density (therm. 

39°.8 F., barom. 30'0 62.379 

Weight of one cubic foot of water at ordinary temperature (therm. 

62^F.) 62.321 

Acceleration due to gravity at latitude of New York in feet per 

square second 32.15945 

Feet in one metre 3 280869 

Metres in one foot 0.304797 

787 



Logarithm. 



0.4971499 

. 622 0886 
1.7581226 
3.5362739 
5.3144251 
8.2418774 
4.6855749 
2.363 6120 

1.795 0384 

1.794 6349 

1.507 3086 
0.515 9889 
9.484 0111 



TABLE XVI. — SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


1 


1 


1 


1.0000000 


1.0000000 


1.000000000 


2 


4 


8 


1.4142136 


1.2599210 


.500000000 


3 


9 


27 


1.7320508 


1.4422496 


.333333333 


4 


16 


64 


2.0000000 


1.5874011 


.250000000 


5 


25 


125 


2.2360680 


1.7099759 


.200000000 


6 


36 


216 


2.4494897 


1.8171206 


.166666667 


7 


49 


343 


2.6457513 


1.9129312 


.142857143 


8 


64 


512 


2.8284271 


2.0000000 


.125000000 


9 


81 


729 


3.0000000 


2.0800837 


.111111111 


10 


100 


1000 


3.1622777 


2.1544347 


. 100000000 


11 


121 


1331 


3.3166248 


2.2239801 


.090909091 


12 


144 


1728 


3.4641016 


2.2894286 


.083333333 


13 


169 


2197 


3.6055513 


2.3513347 


.076923077 


14 


196 


2744 


3.7416574 


2.4101422 


.071428571 


15 


225 


3375 


3.8729833 


2.4662121 


.066666667 


16 


256 


4096 


4.0000000 


2.5198421 


.062500000 


17 


289 


4913 


4.1231056 


2.5712816 


.058823529 


18 


324 


5832 


4.2426407 


2.6207414 


.055555556 


19 


361 


6859 


4.3588989 


2.6684016 


.052631579 


20 


400 


8000 


4.4721360 


2.7144177 


.050000000 


21 


441 


9261 


4.5825757 


2.7589243 


.047619048 


22 


484 


10648 


4.6904158 


2.8020393 


.045454545 


23 


529 


12167 


4.7958315 


2.8438670 


.043478261 


24 


576 


13824 


4.8989795 


2.8844991 


.041666667 


25 


625 


15625 


5.0000000 


2.9240177 


.040000000 


26 


676 


17576 


5.0990195 


2.9624960 


.038461538 


27 


729 


19683 


5.1961524 


3.0000000 


.037037037 


28 


784 


21952 


5.2915026 


3.0365889 


.035714286 


29 


841 


24389 


5.3851648 


3.0723168 


.034482759 


30 


900 


27000 


5.4772256 


3.1072325 


.033333333 


31 


961 


29791 


5.5677644 


3.1413806 


.032258065 


32 


1024 


32768 


5.6568542 


3.1748021 


.031250000 


33 


1089 


35937 


5.7445626 


3.2075343 


.030303030 


34 


1156 


39304 


5.8309519 


3.2396118 


.029411765 


35 


1225 


42875 


5.9160798 


3.2710663 


.028571429 


38 


1296 


46656 


6.0000000 


3.3019272 


.027777778 


37 


1369 


50653 


6.0827625 


3.3322218 


.027027027 


38 


1444 


54872 


6.1644140 


3.3619754 


.026315789 


39 


1521 


59319 


6.2449980 


3.3912114 


.025641026 


40 


1600 


64000 


6.3245553 


3.4199519 


.025000000 


41 


1681 


68921 


6.4031242 


3.4482172 


.024390244 


42 


1764 


74088 


'6.4807407 


3.4760266 


.023809524 


43 


1849 


79507 


6.5574385 


3.5033981 


.023255814 


44 


1936 


85184 


6.6332496 


3.5303483 


.022727273 


45 


2025 


91125 


6.7082039 


3.5568933 


.022222222 


46 


2116 


97336 


6.7823300 


3.5830479 


.021739130 


47 


2209 


103823 


6.8556546 


3.6088261 


.021276600 


48 


2304 


110592 


6.9282032 


3.6342411 


.020833333 


. 49 


2401 


117649 


7.0000000 


3.6593057 


.020408163 


50 


2500 


125000 


7.0710678 


3.6840314 


.020000000 


51 


2601 


132651 


7.1414284 


3.7084298 


.019607843 


52 


2704 


140608 


7.2111026 


3.7325111 


.019230769 


53 


2809 


148877 


7.2801099 


3.7562858 


.018867925 


54 


2916 


157464 


7.3484692 


3.7797631 


.018518519 


55 


3025 


166375 


7.4161985 


3.8029525 


.018181818 


56 


3136 


175616 


7.4833148 


3.8258624 


.017857143 


57 


3249 


185193 


7.5498344 


3.8485011 


.017543860 


58 


3364 


195112 


7.6157731 


3.8708766 


.017241379 


59 


3481 


205379 


7.6811457 


3.8929965 


.016949153 


60 


3600 


216000 


7.7459667 


3.9148676 


.016666667 



788 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


61 


3721 


226981 


7.8102497 


3.9364972 


.016393443 


62 


3844 


238328 


7.8740079 


3.9578915 


.016129032 


63 


3969 


250047 


7.9372539 


3.9790571 


.015873016 


64 


4096 


262144 


8.0000000 


4.0000000 


.015625000 


65 


4225 


274625 


8-0622577 


4.0207256 


.015384615 . 


66 


4356 


287496 


8.1240384 


4.0412401 


.015151515 


67 


4489 


300763 


8.1853528 


4.0615480 


.014925373 


68 


4624 


314432 


8.2462113 


4.0816551 


.014705882 


69 


4761 


328509 


8.3066239 


4.1015661 


.014492754 


70 


4900 


343000 


8.3666003 


4.1212853 


.014285714 


71 


5041 


357911 


8.4261498 


4.1408178 


.014084507 


72 


5184 


373248 


8.4852814 


4.1601676 


.013888889 


73 


5329 


389017 


8.5440037 


4.1793390 


.013698630 


74 


5476 


405224 


8.6023253 


4.1983364 


.013513514 


75 


5625 


421875 


8.6602540 


4.2171633 


.013333333 


76 


5776 


438976 


8.71779.79 


4.2358236 


.013157895 


77 


5929 


456533 


8.7749644 


4.2543210 


.012987013 


78 


6084 


474552 


8.8317609 


4.2726586 


.012820513 


79 


6241 


493039 


8.8881944 


4.2908404 


.012658228 


80 


6400 


512000 


8.9442719 


4.3088695 


.012500000 


81 


6561 


531441 


9.0000000 


4.3267487 


.012345679 


82 


6724 


551368 


9.0553851 


4.3444815 


.012195122 


83 


6889 


571787 


9.1104336 


4.3620707 


.012048193 . 


84 


7056 


592704 


9.1651514 


4.3795191 


.011904762 


85 


7225 


614125 


9.2195445 


4.3968296 


.011764706 


86 


7396 


636056 


9.2736185 


4-4140049 


.011627907 


87 


7569 


658503 


9.3273791 


4.4310476 


.011494253 


88 


7744 


681472 


9.3808315 


4.4479602 


.011363636 


89 


7921 


704969 


9.4339811 


4.4647451 


.011235955 


90 


8100 


729000 


9.4868330 


4-4814047 


.011111111 


91 


8281 


753571 


9.5393920 


4.4979414 


.010989011 


92 


8464 


778688 


9.5916630 


4.5143574 


.010869565 


93 


8649 


804357 


9.6436508 


4.5306549 


.010752688 


94 


8836 


830584 


9.6953597 


4.5468359 


.010638298 


95 


9025 


857375 


9.7467943 


4.5629026 


.010526316 


96 


9216 


884736 


9.7979590 


4.5788570 


.010416667 


97 


9409 


912673 


9.8488578 


4.5947009 


.010309278 


98 


9604 


941192 


9.8994949 


4.6104363 


.010204082 


99 


9801 


970299 


9.9498744 


4.6260650 


.010101010 


100 


10000 


1000000 


10.0000000 


4-6415888 


.010000000 


101 


10201 


1030301 


10.0498756 


4.6570095 


.009900990 


102 


10404 


1061208 


10.0995049 


4.6723287 


.009803922 


103 


10609 


1092727 


10.1488918 


4.6875482 


.009708738 


104 


10816 


1124864 


10.1980390 


4.7026694 


.009615385 


105 


11025 


1157625 


10.2469508 


4.7176940 


.009523810 


106 


11236 


1191016 


10.2956301 


4.7326235 


.009433962 


107 


11449 


1225043 


10.3440804 


4.7474594 


.009345794 


108 


11664 


1259712 


10.3923048 


4.7622032 


.009259259 


109 


11881 


1295029 


10.4403065 


4.7768562 


.009174312 


110 


12100 


1331000 


10.4880885 


4-7914199 


.009090909 


111 


12321 


1367631 


10.5356538 


4.8058955 


.009009009 


112 


12544 


1404928 


10.5830052 


4.8202845 


.008928571 


, 113 


12769 


1442897 


10.6301458 


4.8345881 


.008849558 


\ 114 


12996 


1481544 


10.6770783 


4.8488076 


.008771930 


115 


13225 


1520875 


10.7238053 


4.8629442 


.008695652 


116 


13456 


1560896 


10.7703296 


4.8769990 


.008620690 


117 


13689 


1601613 


10.8166538- 


4.8909732 


.008547009 


118 


13924 


1643032 


10.8627805 


4.9048681 


.008474576 


119 


14161 


1685159 


10.9087121 


4.9186847 


.008403361 


130 


14400 


1728000 


10.9544512 


4.9324242 


.008333333 








789 







TABLE XVI. SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. I 


leciprocaK 


121 
122 
123 
124 
125 


14641 
14884 
15129 
15376 
15625 


1771561 
1815848 
1860867 
1906624 
1953125 


11.0000000 
11.0453610 
11.0905365 
11.1355287 
11.1803399 


4. 
4 
4 
4 
5 


9460874 
9596757 
9731898 
9866310 
0000000 


008264463 
008196721 
008130081 
008064516 
008000000 


126 
127 
128 
129 
130 


15876 
16129 
16384 
16641 
16900 


2000376 
2048383 
2097152 
2146689 
2197000 


11.2249722 
11.2694277 
11.3137085 
11.3578167 
11.4017543 


5 
5 
5 
5 
5 


0132979 
0265257 
0396842 
0527743 
0657970 


007936508 
007874016 
007812500 
007751938 
007692308 


131 
132 
133 
134 
135 


17161 
17424 
17689 
17956 
18225 


2248091 
2299968 
2352637 
2406104 
2460375 


11.4455231 
11-4891253 
11.5325626 
11.5758369 
11.6189500 


5 
5 
5 
5 
5 


0787531 
0916434 
1044687 
1172299 
1299278 


007653588 
.007575758 
.007518797 
.007462687 

007407407 


136 
137 
138 
139 
140 


18496 
18769 
19044 
19321 
19600 


2515456 
2571353 
2628072 
2685619 
2744000 


11.6619038 
11.7046999 
11-7473401 
11.7898261 
11.8321596 


5 
5 
5 
5 
5 


1425632 
1551367 
1676493 
1801015 
1924941 


007352941 
007299270 
007246377 
007194245 
007142857 


141 
142 
143 
144 
145 


19881 
20164 
20449 
20736 
21025 


2803221 
2863288 
2924207 
2985984 
3048625 


11.8743421 
11.9163753 
11.9582607 
12-0000000 
12-0415946 


5 
5 
5 
5 
5 


2048279 
2171034 
2293215 
2414828 
2535879 


007092199 
007042254 
006993007 
006944444 
•006896552 


146 
147 
148 
149 
150 


21316 
21609 
21904 
22201 
22500 


3112136 
3176523 
3241792 
3307949 
3375000 


12-0830460 
12-1243557 
12-1655251 
12-2065556 
12.2474487 


5 
5 
5 
5 
5 


2656374 
2776321 
2895725 
3014592 
3132928 


.006849315 
.006802721 
.006756757 
.006711409 
•006666667 


151 
152 
153 
154 
155 


22801 
23104 
23409 
23716 
24025 


3442951 
3511808 
3581577 
3652264 
3723875 


12.2882057 
12-3288280 
12-3693169 
12-4096736 
12-4498996 


5 
5 
5 
5 
5 


3250740 
3368033 
3484812 
3601084 
3716854 


.006622517 
.006578947 
.006535948 
•006493506 
•006451613 


156 
157 
158 
159 
160 


24336 
24649 
24964 
25281 
25600 


3796416 
3869893 
3944312 
4019679 
4096000 


12-4899960 
12-5299641 
12-5698051 
12-6095202 
12-6491106 


5 
5 
5 
5 
5 


3832126 
3946907 
4061202 
.4175015 
4288352 


.006410256 
.006369427 
.006329114 
.006289308 
•006250000 


161 
162 
163 
164 
165 


25921 
26244 
26569 . 
26896 
27225 


4173281 
4251528 
4330747 
4410944 
4492125 


12-6885775 
12-7279221 
12-7671453 
12-8062485 
12-8452326 


5 
5 
5 
5 
5 


4401218 
4513618 
4625556 
4737037 
-4848066 


.006211180 
.006172840 
.006134969 
.006097561 
•006060606 


166 
167 
168 
169 
170 


27556 
27889 
28224 
28561 
28900 


4574296 
4657463 
4741632 
4826809 
4913000 


12-8840987 
12-9228480 
12-9614814 
13-0000000 
13-0384048 


5 
5 
5 
5 
5 


-4958647 
•5068784 
-5178484 
-5287748 
-5396583 


.006024096 
.005988024 
.005952381 
.005917160 
•005882353 


171 
172 
173 
174 
175 


29241 
29584 
29929 
30278 
30625 


5000211 
5088448 
5177717 
5268024 
5359375 


13-0766968 
13-1148770 
13-1529464 
13-1909060 
13-2287566 


5 
5 
5 
5 
5 


•5504991 
5612978 
-5720546 
-5827702 
•5934447 


.005847953 

.005813953 

•005780347 

005747126 

005714286 


176 
177 
178 
179 
180 


30976 
31329 
31684 
32041 
32400 


5451776 
5545233' 
5639752 
5735339 
5832000 


13-2664992 
13-3041347 
13-3416641 
13-3790882 
13-4164079 


5 
5 
5 
5 
5 


6040787 
6146724 
6252263 
6357408 
6462162 


.005681818 
005649718 
005617978 
005586592 
005555556 



790 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


181 
182 
183 
184 
185 


32761 
33124 
33489 
33856 

34225 


5929741 
6028568 
6128487 
6229504 
6331625 


13.4536240 
13-4907376 
13-5277493 
13-5646600 
13-6014705 


5.6566528 
5.6670511 
5.6774114 
5.6877340 
5-6980192 


•005524862 
•005494505 
•005464481 
.005434783 
-00540540'! 


186 
187 
188 
189 
190 


34596 
34969 
35344 
35721 
36100 


6434856 
6539203 
6644672 
6751269 
6859000 


13-6381817 
13-6747943 
13.7113092 
13.7477271 
13-7840488 


5-7082675 
5.7184791 
5.7286543 
5-7387936 
5-7488971 


•005376344 
•005347594 
•005319149 
•005291005 
.0052631 58 


191 
192 
193 
194 
195 


36481 
36864 
37249 
37636 
38025 


6967871 
7077888 
7189057 
7301384 
7414875 


13-8202750 
13-8564065 
13-8924440 
13-9283883 
13-9642400 


5-7589652 
5-7689982 
5-7789966 
5-7889604 
5-7988900 


•005235602 
.005208333 
•005181347 
.005154639 
.005128205 


196 
197 
198 
199 
300 


38416 
38809 
39204 
39601 
40000 


7529536 
7645373 
7762392 
7880599 
8000000 


14-0000000 
14-0356688 
14-0712473 
14-1067360 
14-1421356 


5-8087857 
5-8186479 
5-8284767 
5-8382725 
5.8480355 


.005102041 
•005076142 
.005050505 
.005025126 
-005000000 


201 
202 
203 
204 
205 


40401 
40804 
41209 
41616 
42025 


8120601 
8242408 
8365427 
8489664 
8615125 


14-1774469 
14-2126704 
14-2478068 
14-2828569 
14-3178211 


5-8577660 
5-8674643 
5-8771307 
5-8867653 
5-8963685 


.004975124 
.004950495 
.004926108 
.004901961 
.004878049 


208 
207 
208 
209 
310 


42436 
42849 
43264 
43681 
44100 


8741816 
8869743 
8998912 
9129329 
9261000 


14-3527001 
14-3874946 
14-4222051 
14-4568323 
14-4913767 


5-9059406 
5-9154817 
5-9249921 
5-9344721 
5-9439220 


.004854369 
.004830918 
.004807692 
.004784689 
.004761905 


211 
212 
213 
214 
215 


44521 
44944 
45369 
45796 
46225 


9393931 
9528128 
9663597 
9800344 
9938375 


14-5258390 
14-5602198 
14-5945195 
14-6287388 
14-6628783 


5-9533418 
5-9627320 
5-9720926 
5-9814240 
5. 9907264 


.004739336 
.004716981 
.004694836 
.004672897 » 
.004651163 


216 
217 
218 
219 
220 


46656 
47089 
47524 
47961 
48400 


10077696 
10218313 
10360232 
10503459 
10648000 


14-6969385 
14-7309199 
14-7648231 
14-7986486 
U.R328970 


6-0000000 
6-0092450 
6-0184617 
6-0276502 
R. 0368107 


.004629630 
.004608295 
.004587156 
.004566210 
.004545455 


221 
222 
223 
224 
225 


48841 
49284 
49729 
50176 
50625 


10793861 
10941048 
11089567 
11239424 
11390625 


14-8660687 
14-8996644 
14-9331845 
14-9666295 
15-0000000 


6-0459435 
6-0550489 
6-0641270 
6-0731779 
6-0822020 


.004524887 
.004504505 
.004484305 
.004464286 
. 004444444 


226 
227 
228 
229 
280 


51076 
51529 
51984 
52441 
52900 


11543176 
11697083 
11852352 
12008989 
12167000 


15-0332964 
15.0665192 
15-0096689 
15.1327460 
15-1657509 


6-0911994 
6.1001702 
6.1091147 
6.1180332 
6.1269257 


.004424779 
.004405286 
.004385965 
.004366812 
.004347826 


231 

232 

) 233 

^ 234 

235 


53361 
53824 
54289 
54756 
55225 


12326391 
12487168 
12649337 
12812904 
12977875 


15-1986842 
15.2315462 
15.2643375 
15.2970585 
15.3297097 


6.1357924 
6.1446337 
6.1534495 
6.1622401 
6.1710058 


.004329004 
.004310345 
.004291845 
.004273504 
.004255319 


236 
237 
238 

\ 239 

j.240 

4 


55696 
56169 
56644 
57121 
57600 


13144256 
13312053 
13481272 
13651919 
13824000 


15.3622915 
15.3948043 
15.4272486 
15.4596248 
15.4919334 


6.1797466 
6.1884628 
6.1971544 
6.2058218 
6.2144650 


.004237288 
.004219409 
.004201681 
.004184100 
.004166667 



791 



TABLE XVI. — SQUARES, CUBES, SQUARE ROOTS, 



^ 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


241 
242 
243 
244 
245 


58081 
58564 
59049 
59536 
60025 


13997521 
14172488 
14348907 
14526784 
14706125 


15. 
15. 
15. 
15. 
15. 


5241747 
5563492 
5884573 
6204994 
6524758 


6 
6 
6 
6 
6 


2230843 
2316797 
2402515 
2487998 
2573248 


.004149378 
.004132231 
•004115226 
.004098361 
.004081633 


246 
247 
248 
249 
250 


60516 
61009 
61504 
62001 
62500 


14886936 
15069223 
15252992 
15438249 
15625000 


15. 
15. 
15. 
15. 
15. 


6843871 
7162336 
7480157 
7797338 
8113883 


6 
6 
6 
6 
6 


2658266 
2743054 
2827613 
2911946 
2996053 


•004065041 
•004048583 
.004032258 
.004016064 
.004000000 


251 
252 
253 
254 
255 


63001 
63504 
64009 
64516 
65025 


15813251 
16003008 
16194277 
16387064 
16581375 


15. 
15. 
15. 
15. 
15 


8429795 
8745079 
9059737 
9373775 
9687194 


6 
6 
6 
6 
6 


3079935 
3163596 
3247035 
3330256 
3413257 


.003984064 
•003968254 
•003952569 
•003937008 
.003921569 


256 
257 
258 
259 
260 


65536 
66049 
66564 
67081 
67600 


16777216 
16974593 
17173512 
17373979 
17576000 


16 

16 

16 

16. 

16 


0000000 
0312195 
0623784 
0934769 
1245155 


6 
6 
6 
6 


3496042 
3578611 
3660968 
3743111 
3825043 


.003906250 
.003891051 
.003875969 
.003861004 
•003846154 


261 
262 
263 
264 
265 


68121 
68644 
69169 
69696 
70225 


17779581 
17984728 
18191447 
18399744 
18609625 


16. 

16 

16 

16 

16 


1554944 
1864141 
2172747 
2480768 
2788206 


6 
6 
6 
6 
6 


3906765 
3988279 
4069585 
4150687 
4231583 


.003831418 
•003816794 
•003802281 
•003787879 
.003773585 


266 
267 
268 
269 
270 


70756 
71289 
71824 
72361 
72900 


18821096 
19034163 
19248832 
19465109 
19683000 


16 
16 
16 
16 
16 


3095064 
3401346 
3707055 
4012195 
4316767 


6 
6 
6 
6 
6 


4312276 
4392767 
4473057 
4553148 
4633041 


.003759398 
.003745318 
•003731343 
•003717472 
.003703704 


271 
272 
273 
274 
275 


73441 
73984 
74529 
75076 
75625 


19902511 
20123648 
20346417 
20570824 
20796875 


16 
16 
16 
16 
16 


4620776 
4924225 
5227116 
5529454 
5831240 


6 
6 
6 
6 
6 


4712736 
4792236 
4871541 
4950653 
5029572 


.003690037 
.003676471 t 
•003663004 1 
•003649635 1 
•003636364 


276 
277 
278 
279 
280 


76176 
76729 
77284 
77841 
78400 


21024576 
21253933 
21484952 
21717639 
21952000 


16 
16 
16 
16 
16 


6132477 
6433170 
6733320 
7032931 
7332005 


6 
6 
6 
6 
6 


5108300 
.5186839 
•5265189 
•5343351 
•5421326 


•003623188 
•003610108 
•003597122 
•003584229 
•003571429 


281 
282 
283 
284 
285 


78961 
79524 
80089 
80656 
81225 


22188041 
22425768 
22665187 
22906304 
23149125 


16 
16 
16 
16 
16 


7630546 
7928556 
8226038 
8522995 
8819430 


6 
6 
6 
6 
6 


.5499116 
.5576722 
.5654144 
.5731385 
.5808443 


.003558719 
.003546099 
.003533569 
.003521127 
.003508772 


286 
287 
288 
289 
290 


81796 
82369 
82944 
83521 
84100 


23393656 
23639903 
23887872 
24137569 
24389000 


16 
16 
16 
17 
17 


9115345 

9410743 

•9705627 

.0000000 

.0293864 


6 
6 
6 
6 
6 


•5885323 
•5962023 
.6038545 
.6114890 
•6191060 


.003496503 
.003484321 
•003472222 
.003460208 
.003448276 


291 
292 
293 
294 
295 


84681 
85264 
85849 
86436 
87025 


24642171 
24897088 
25153757 
25412184 
25672375 


17 
17 
17 
17 
17 


.0587221 
.0880075 
.1172428 
.1464282 
.1755640 


6 
6 
6 
6 
6 


.6267054 
•6342874 
•6418522 
•6493998 
•6569302 


.003436426 ' 

.003424658 

.003412969 

.003401361 

.003389831 


296 
297 
298 
299 
300 


87616 
88209 
88804 
89401 
90000 


25934336 
26198073 
26463592 
26730899 
27000000 


17 
17 
17 
17 
17 


.2046505 
•2336879 
.2626765 
.2916165 
.3205081 


6 
6 
6 
6 
6 


•6644437 
•6719403 
•6794200 
•6868831 
.6943295 


•003378378 | 

•003367003 

•003355705 

.003344482 

.003333333 



792 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


301 
302 
303 
304 
305 


90601 
91204 
91809 
92416 
93025 


27270901 
27543608 
27818127 
28094464 
28372625 


17.3493516 
17.3781472 
17.4068952 
17.4355958 
17.4642492 


6.7017593 
6.7091729 
6.7165700 
6.7239508 
6.7313155 


.003322259 
.003311258 
.003300330 
.003289474 
.003278689 


306 
307 
308 
309 
310 


93636 
94249 
• 94864 
95481 
96100 


28652616 
28934443 
29218112 
29503629 
29791000 


17.4928557 
17.5214155 
17.5499288 
17.5783958 
17.6068169 


6.7386641 
6.7459967 
6.7533134 
6.7606143 
6.7678995 


.003267974 
.003257329 
.003246753 
.003236246 
.003225806 


311 
312 
313 
314 
315 


96721 
97344 
97969 
98596 
99225 


30080231 
30371328 
30664297 
30959144 
31255875 


17.6351921 
17.6635217 
17.6918060 
17.7200451 
17.7482393 


6.7751690 
6.7824229 
6.7896613 
6.7968844 
6.8040921 


.003215434 
.003205128 
.003194888 
.003184713 
.003174603 


316 
317 
318 
319 
330 


99856 
100489 
101124 
101761 
102400 


31554496 
31855013 
32157432 
32461759 
32768000 


17.7763888 
17.8044938 
17.8325545 
17.8605711 
17.8885438 


6.8112847 
6-8184620 
6.8256242 
6.8327714 
6.8399037 


.003164557 
•003154574 
.003144654 
.003134796 
.003125000 _ 


321 
322 
323 
324 
325 


103041 
103684 
104329 
104976 
105625 


33076161 
33386248 
33698267 
34012224 
34328125 


17.9164729 
17.9443584 
17.9722008 
18.0000000 
18.0277564 


6.8470213 
6.8541240 
6.8612120 
6-8682855 
6-8753443 


.003115265 
.003105590 
.003095975 
.003086420 
.003076923 


326 
327 
328 
329 
330 


106276 
106929 
107584 
108241 
108900 


34645976 
34965783 
35287552 
35611289 
35937000 


18.0554701 
18.0831413 
18.1107703 
18.1383571 
18.1659021 


6. 8823888 
6.8894188 
6.8964345 
6.9034359 
6.9104232 


.003067485 
.003058104 
.003048780 
.003039514 
-003030303 


331 
332 
333 
334 
335 


109561 
110224 
110889 
111556 
112225 


36264691 
36594368 
36926037 
37259704 
37595375 


18.1934054 
18.2208672 
18.2482876 
18.2756669 
18.3030052 


6.9173964 
6.9243556 
6.9313008 
6.9382321 
6. 9451496 


.003021148 
.003012048 
.003003003 
.002994012 
.002985075 


336 
337 
338 
339 
340 


112896 
113569 
114244 
114921 
115600 


37933056 
38272753 
38614472 
38958219 
39304000 


18.3303028 
18.3575598 
18.3847763 
18.4119526 
18.4390889 


6.9520533 
6.0589434 
6.9658198 
6.9726826 
6.9795321 


.002976190 
.002967359 
.002958580 
.002949853 
.002941176 


341 
342 
343 
344 
345 


116281 
116964 
117649 
118336 
119025 


39651821 
40001688 
40353607 
40707534 
41063625 


18.4661853 
18.4932420 
18.5202592 
18.5472370 
18.5741756 


6.9863681 
6.9931906 
7.0000000 
7.0067962 
7.0135791 


.002932551 
.002923977 
.002915452 
.002906977 
.002898551 


346 
347 
348 
349 
350 


119716 
120409 
121104 
121801 
122500 


41421736 
41781923 
42144192 
42508549 
42875000 


18.6010752 
18.6279360 
18.6547581 
18.6815417 
18.7082869 


7.0203490 
7.0271058 
7.0338497 
7.0405806 
7.0472987 


.002890173 
.002881844 
.002873563 
.002865330 
.002857143 


351 
352 
353 
354 
355 


123201 
123904 
124609 
125316 
126025 


43243551 
43614208 
43986977 
44361864 
44738875 


18.7349940 
18.7616630 
18.7882942 
18.8148877 
18.8414437 


7.0540041 
7.0606967 
7-0673767 
7.0740440 
70806988 


.002849003 
.002840909 
.002832861 
.002824859 
.002816901 


356 
357 
358 
359 
360 


126736 
127449 
• 128164 
128881 
129600 


45118016 
45499293 
45882712 
46268279 
46656000 


18.8679623 
18.8944436 
18.9208879 
18.9472953 
18.9736660 


7.0873411 
7.0939709 
7-1005885 
7-10719?7 
7. 1137:^6 


.002808989 
.002801120 
.002793296 
.002785515 
.002777778 



798 



TABLE XVI. — SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


361 
362 
363 
364 
365 


130321 
131044 
131769 
132496 
133225 


47045881 
47437928 
47832147 
48228544 
48627125 


19.0000000 
19.0262976 
19.0525589 
19.0787840 
19.1049732 


7.1203674 
7.1269360 
7.1334925 
7.1400370 
7.1465695 


.002770083 
.002762431 
.002754821 
.002747253 
.002739726 


366 
367 
368 
369 
370 


133956 
134689 
135424 
136161 
136900 


49027896 
49430863 
49836032 
50243409 
50653000 


19.1311265 
19.1572441 
19.1833261 
19.2093727 
19.2353841 


7.1530901 
7.1595988 
7.1660957 
7.1725809 
7.1790544 


.002732240 
.002724796 
.002717391 
.002710027 
.002702703 


371 
372 
373 
374 
375 


137641 
138384 
139129 
139876 
140625 


51064811 
51478848 
51895117 
52313624 
52734375 


19.2613603 
19.2873015 
19.3132079 
19.3390796 
19.3649167 


7.1855162 
7.1919663 
7.1984050 
7.2048322 
7.2112479 


.002695418 
.002688172 
.002680965 
.002673797 
.002666667 


376 
377 
378 
379 
380 


141376 
142129 
142884 
143641 
144400 


53157376 
53582633 
•54010152 
54439939 
54872000 


19.3907194 
19.4164878 
19.4422221 
19.4679223 
19.4935887 


7.2176522 
7.2240450 
7.2304268 
7.2367972 
7.2431565 


.002659574 
.002652520 
.002645503 
.002638522 
.002631579 


381 
382 
383 
384 
. 385 


145161 
145924 
148689 
147456 
148225 


55306341 
55742968 
56181887 
56623104 
57066625 


19.5192213 
19.5448203 
19.5703858 
19.5959179 
19.6214169 


7.2495045 
7.2558415 
7.2621675 
7.2684824 
7.2747864 


.002624672 
.002617801 
.002610966 
.002604167 
.002597403 


386 
387 
388 
389 
390 


148996 
149769 
150544 
151321 
152100 


57512456 
57960603 
58411072 
58863869 
59319000 


19.6468827 
19.6723156 
19.6977156 
19.7230829 
19.7484177 


7.2810794 
7.2873617 
7.2936330 
7.2998936 
7.3061436 


.002590674 
.002583979 
.002577320 
.002570694 
.002564103 


391 
392 
393 
394 
395 


152881 
153664 
154449 
155236 
156025 


59776471 
60236288 
60698457 
61162984 
61629875 


19.7737199 
19.7989899 
19.8242276 
19.8494332 
19.8746069 


7.3123828 
7.3186114 
7.3248295 
7.3310369 
7.3372339 


.002557545 
.002551020 
.002544529 
.002538071 
.002531646 


396 
397 
398 
399 
400 


156816 
157609 
158404 
159201 
160000 


62099136 
62570773 
63044792 
63521199 
64000000 


19.8997487 
19.9248588 
19.9499373 
19.9749844 
20.0000000 


7.3434205 
7.3495966 
7.3557624 
7.3619178 
7.3680630 


.002525253 
.002518892 
.002512563 
.002506266 
.002500000 


401 
402 
403 
404 
405 


160801 
161604 
162409 
163216 
164025 


64481201 
64964808 
65450827 
.65939264 
66430125 


20.0249844 
20.0499377 
20.0748599 
20.0997512 
20.1246118 


7.3741979 
7.3803227 
7.3864373 
7.3925418 
7.3986363 


.002493766 
•002487562 
.002481390 
.002475248 
.002469136 


406 
407 
408 
409 
410 


164836 
165649 
166464 
167281 
168100 


66923416 
67419143 
67917312 
68417929 
68921000 


20.1494417 
20.1742410 
20.1990099 
20.2237484 
20.248.4567 


7.4047206 
7.4107950 
7.4168595 
7.4229142 
7.4289589 


.002463054 
.002457002 
.002450980 
.002444988 
.002439024 


411 
412 
413 
414 
415 


168921 
169744 
170569 
171396 
172225 


69426531 
69934528 
70444997 
70957944 
71473375 


20.2731349 
20.2977831 
20.3224014 
20.3469899 
20.3715488 


7.4349938 
7.4410189 
7.4470342 
7.4530399 
7.4590359 


.002433090 
.002427184 
.002421308 
.002415459 
.002409639 


416 
417 
418 
419 
430 


173056 
173889 
174724 
175561 
176400 


71991296 
72511713 
73034632 
73560059 
74088000 


20.3960781 
20.4205779 
20.4450483 
20.4694895 
20.4939015 


7.4650223 
7.4709991 
7.4769664 
7.4829242 
7.4888724 


.002403846 
.002398082 
.002392344 
.002386635 
.00238C^?a 



794 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


121 

422 
423 
424 
425 


177241 
178084 
178929 
179776 
180625 


74618461 
75151448 
75686967 
76225024 
76765625 


20.5182845 
20.5426386 
20.5669638 
20-5912603 
20-6155281 


7.4948113 
7.5007406 • 
7.5066607 
7.5125715 
7-5184730 


.CG2375297 
.002369668 
.CG2364066 
•002358491 
.002852941 


426 
427 
428 
429 
430 


181476 
182329 
183184 
184041 
184900 


77308776 
77854483 
78402752 
78953589 
79507000 


20-6397674 
20-6639783 
20-6881609 
20-7123152 
20-7364414 


7.5243652 
7-5302482 
7-5361221 
7-5419867 
7-5478423 


•002347418 
.002341920 
•002336449 
•002331002 
-rC2325581 


431 
432 
433 
434 
435 


185761 
186624 
187489 
188356 
189225 


80062991 
80621568 
81182737 
81746504 
82312875 


20.7605395 
20.7846097 
20-8086520 
20-8326667 
20-8566536 


7-5536888 
7-5595263 
7.5653548 
7-5711743 
7-5769849 


•C02320186 
•002314815 
•C02309469 
.002304147 
-002298851 


436 
437 
438 
439 
440 


190096 
190969 
191844 
192721 
193600 


82881856 
83453453 
84027672 
84604519 
85184000 


20.8806130 
20.9045450 
20-9284495 
20.9523268 
20.9761770 


7-5827865 
7-5885793 
7-5943633 
7-6001385 
7. 6059049 


.C02293578 
.002288330 
.C02283105 
.CC2277904 
.fC2272727^ 


441 
442 
443 
444 
445 


194481 
195364 
196249 
197136 
198025 


85766121 
86350888 
86938307 
87528384 
88121125 


21.0000000 
21.0237960 
21.0475652 
21.0713075 
21-0950231 


7-6116626 
7-6174116 
7-6231519 
7-6288837 
7-6346067 


.CC2267574 
.002262443 
•002257336 
•002252252 
-002247191 . 


446 
447 
448 
449 
450 


198916 
199809 
200704 
201601 
202500 


88716536 
89314623 
89915392 
90518849 
91125000 


21.1187121 
21.1423745 
21.1660105 
21.1896201 
21-2132034 


7-6403213 
7-6460272 
7-6517247 
7-6574138 
7-6630943 


-CC2242152 
.002237136 
•002232143 
•002227171 
-002222222 


451 
452 
453 
454 
455 


203401 
204304 
205209 
206116 
207025 


91733851 
92345408 
92959677 
93576664 
94196375 


21-2367606 
21.2602916 
21-2837967 
21-3072758 
21-3307290 


7-6687665 
7-6744303 
7-6800857 
7-6857328 
7.6913717 


.002217295 
.002212389 
-002207506 
.002202643 
•002197802 


456 
457 
458 
459 
, 460 


207936 
208849 
209764 
210681 
211600 


94818816 
95443993 
96071912 
96702579 
97336000 


21.3541565 
21.3775583 
21.4009346 
21.4242853 
21-4476106 


7-6970023 
7-7026246 ^ 
7-7082388 
7-7138448 
7-7194426 


-002192982 
-002188184 
.002183406 
.002178649 
.002] 73913 


461 

462 
463 
464 
465 


212521 
213444 
214369 
215296 
216225 


97972181 
98611128 
99252847 
99897344 
100544625 


21.4709106 
21.4941853 
21.5174348 
21.5406592 
21-5638587 


7-7250325 
7-7306141 
7-7361877 
7.7417532 
7-7473109 


•002169197 
•002164502 
•002159827 
•002155172 
-002150538 


466 
467 
468 
469 
470 


217156 
218089 
219024 
219961 
220900 


101194696 
101847563 
102503232 
103161709 
103823000 


21-5870331 
21.6101828 
21-6333077 
21-6564078 
21-6794834 


7-7528606 
7.7584023 
7.7639361 
7.7694620 
7-7749801 


•002145923 
•002141328 
•002136752 
.002132196 
-002127660 


471 
472 
473 
474 
475 


221841 
222784 
223729 
224676 
225625 


104487111 
105154048 
105823817 
106496424 
107171875 


21.7025344 
21.7255610 
21.7485632 
21.7715411 
21-7944947 


7.7804904 
7.7859928 
7.7914875 
7.7969745 
7-8024538 


.CC2123142 
.002118644 
.002114165 
.002109705 
-002105263 


476 
477 
478 
479 
480 


226576 
227529 
228484 
229441 
230400 


107850176 
108531333 
109215352 
109902239 
110592000 


21.8174242 
21-8403297 
21.8632111 
21.8860686 
21.9089023 


7.8079254 
7.8133892 
7.8188456 
7.8242942 
7.8297353 


•002100840 
•002096436 
•002092050 
.002087683 
.002083333 



795 



yL 



TABLE XVI. SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


481 
482 
483 
484 
485 


231361 
232324 
233289 
234256 
235225 


111284641 
111980168 
112678587 
113379904 
114084125 


21.9317122 
21.9544984 
• 21.9772610 
22.0000000 
22.0227155 


7.8351688 
7.8405949 
7.8460134 

7.8514244 
7.8568281 


.002079002 
.002074689 
.002070393 
.002066116 
.002061856^ 


486 
487 
488 
489 
490 


236196 
237169 
238144 
239121 
240100 


114791256 
115501303 
116214272 
116930169 
117649000 


22.0454077 
22.0680765 
22.0907220 
22.1133444 
22.1359436 


7.8622242 
7.8676130 
7.8729944 
7.8783684 
7.8837352 


.002057613 
.002053388 
.002049180 
.002044990 
.002040816 


491 
492 
493 
494 
495 


241081 
242064 
243049 
244036 
245025 


118370771 
119095488 
119823157 
120553784 
121287375 


22.1585198 
22.1810730 
22.2036033 
22.2261108 
22.2485955 


7.8890946 
7.8944468 
7.8997917 
7.9051294 
7.9104599 


.002036660 
.002032520 
.002028398 
.002024291 
.002020202 


496 
497 
498 
499 
500 


246016 
247009 
248004 
249001 
250000 


122023936 
122763473 
123505992 
124251499 
125000000 


22.2710575 
22.2934968 
22.3159136 
22.3383079 
22.3606798 


7.9157832 
7.9210994 
7-9264085 
7.9317104 
7.9370053 


.002016129 
.002012072 
.002008032 
.002004008 
.002000000 


501 
502 
503 
504 
505 


251001 
252004 
253009 
254016 
255025 


125751501 
126506008 
127263527 
128024064 
128787625 


22.3830293 
22.4053565 
22.4276615 
22.4499443 
22.4722051 


7.9422931 
7.9475739 
7.9528477 
7.9581144 
7.9633743 


.001996008 
.001992032 
.001988072 
.001984127 
.001980198 


506 
507 
508 
509 
510 


256036 
257049 
258064 
259081 
260100 


129554216 
130323843 
131096512 
131872229 
132651000 


22.4944438 
22.5166605 
22.5388553 
22.5610283 
22.5831796 


7.9686271 
7.9738731 
7.9791122 
7.9843444 
7.9895697 


.001976285 
.001972387 
.001968504 
.001964637 
.001960784 


511 
512 
513 
514 
515 


261121 
262144 
263169 
264196 
265225 


133432831 
134217728 
135005697 
135796744 
136590875 


22.6053091 
22.6274170 
22.6495033 
22.6715681 
22.6936114 


7.9947883 
8.0000000 
8.0052049 
8.0104032 
8.0155946 


.001956947 
.001953125 
.001949318 
.001945525 
.001941748 


516 
517 
518 
519 
530 


266256 
267289 
268324 
269361 
270400 


137388096 
138188413 
138991832 
139798359 
140608000 


22.7156334 
22.7376340 
22.7596134 
22.7815715 
22.8035085 


8.0207794 
8.0259574 
8.0311287 
8.0362935 
8.0414515 


.001937984 
.001934236 
.001930502 
.001926782 
.001923077 ^ 


521 
522 
523 
524 
525 


271441 
272484 
273529 
274576 
275625 


141420761 
142236648 
143055667 
143877824 
144703125 


22.8254244 
22.8473193 
22.8691933 
22.8910463 
22.9128785 


8.0466030 
8.0517479 
8.0568862 
8.0620180 
8.0671432 


.001919386 
.001915709 
.001912046 
.001908397 
.001904762 


526 
527 
528 
529 
530 


276676 
277729 
278784 
279841 
280900 


145531576 
146363183 
147197952 
148035889 
148877000 


22.9346899 
22.9564806 
22.9782506 
23.0000000 
23.0217289 


8.0722620 
8.0773743 
8.0824800 
8.0875794 
8.0926723 


.001901141 
.001897533 
.001893939 
.001890359 
.001886792 


531 
532 
533 
534 
535 


281961 
283024 
284089 
285156 
286225 


149721291 
150568768 
151419437 
152273304 
153130375 


23.0434372 
23.0651252 
23.0867928 
23.1084400 
23.1300670 


8.0977589 
8.1028390 
8.1079128 
8.1129803 
8.1180414 


.001883239 

.001879699 
.001876173 
.001872659 
.001869159 


536 
537 
538 
539 
540 


287296 
. 288369 
289444 
290521 
291600 


153990656 
154854153 
155720872 
156590819 
157464000 


23.1516738 
23.1732605 
23.1948270 
23.2163735 
23.2379001 


8.1230962 
8.1281447 
8.1331870 
8.1382230 
8.1432529 


.001865672 
.001862197 
.001858786 
.001855288 
.001851852 



796 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots 


Cube Roots 


Rec procals. 


541 
542 
543 
544 

545 


292681 
293764 
294849 
295936 
297025 


158340421 
159220088 
160103007 
160989184 
161878625 


23.2594067 
23.2808935 
23.3023604 
23.3238076 
23.3452351 


8.1482765 
8.1532939 
8.1583051 
8.1633102 
8.1683092 


.001848429 
.001845018 
.001841621 
.001838235 
.001834862 


546 
547 
548 
549 
550 


298116 
299209 
300304 
301401 
302500 


162771336 
163667323 
164566592 
165469149 
166375000 


23.3666429 
23.3880311 
23.4093998 
23.4307490 
23.4520788 


8.1733020 
8.1782888 
8-1832695 
8.1882441 
8.1932127 


.001831502 
•001828154 
.001824818 
.001821494 
.001818182 


551 
552 
553 
554 
555 


303601 
304704 
305809 
306916 
308025 


167284151 
168196608 
169112377 
170031464 
170953875 


23.4733892 
23.4946802 
23.5159520 
23.5372046 
23-5584380 


8.1981753 
8.2031319 . 
8.2080825 
8.2130271 
8-2179657 


.001814882 
.001811594 
.001808318 
.001805054 
-001801802 


556 
557 
558 
559 
560 


309136 
310249 
311364 
312481 
313600 


171879616 
172808693 
173741112 
174676879 
175616000 


23. 5786522 
23.6008474 
23.6220236 
23.6431808 
23.6643191 


8 2228985 
8-2278254 
8-2327463 
8. 2376614 
8-2425706 


.001798561 
.001795332 
.001792115 
.001788909 
-001785714 


561 
562 
563 
564 
■ 565 


314721 
315844 
316969 
318096 
319225 


176558481 
177504328 
178453547 
179406144 
180362125 


23.6854386 
23-7065392 
23-7276210 
23-7486842 
23-7697286 


8-2474740 
8-2523715 
8-2572633 
8-2621492 
8-2670294 


.001782531 
.001779359 • 
.001776199 
.001773050 
.001769912 


566 
567 
568 
569 
570 


320356 
321489 
322624 
323761 
324900 


181321496 
182284263 
183250432 
184220009 
185193000 


23-7907545 
23-8117618 
23.8327506 
23-8537209 
23-8746728 


8^2719039 
8-2767726 
8-2816355 
8=2864928 
8-2913444 


.001766784 
.001763668 
.001760563 
^001757469 
.001754386 


571 
572 
573 
574 
575 


326041 
327184 
328329 
329476 
330625 


186169411 
187149248 
188132517 
189119224 
190109375 


23-8956063 
23-9165215 
23-9374184 
23-9582971 
23.9791576 


8.2961903 
8-3010304 
8.3058651 
8.3106941 
8. 3155175 


.001751313 
■ .001748252 
.001745201 
-001742160 
.001739130 


576 
577 
578 
579 

580 


331776 
332929 
334084 
335241 
3J?«Ann 


191102976 
192100033 
193100552 
194104539 
lP5n20n0 


24.0000000 
24-0208243 
24-0416306 
24-0624188 

9/1 .rpST PPT 


8.3203353 
8-3251475 
8-3299542 
8.3347553 
P .^?P5509 


.001736111 
.001733102 
.001730104 
.001727116 
.001724138 


581 
582 
583 
584 
585 


337561 
338724 
339889 
341056 
342225 


196122941 
197137368 
198155287 
199176704 
200201625 


24.1039416 
24.1246762 
24.1453929 
24.1660919 
24.] 867732 


8-3443410 
8.3491256 
8.3539047 
8.3586784 
P- 3634466 


.001721170 
.001718213 
.001715266 
.001712329 
.001709402 


586 
587 
588 
589 
590 


343396 
344569 
345744 
346921 
348100 


201230056 
202262003 
203297472 
204336469 
205379000 


24.2074369 
24-2280829 
24.2487113 
24.2693222 
24.2899156 


8.3682095 
8-3729668 
8-3777188 
8-3824653 
8-3872065 


.001706485 
.001703578 
.001700680 
.001697793 
.001694915 


591 
592 
593 
594 
595 


349281 

350464- 

351649 

352836 

354025 


206425071 
207474688 
208527857 
209584584 
2T0R44875 


24-3104916 
24.3310501 
24 3515913 
24-3721152 
24-3926218 


8.3919423 
8-3966729 
8-4013981 
8-4061180 
8. 4108326 


.001692047 
.001689189 
.001686341 
.001683502 
.001680672 


596 
597 
598 
599 
600 


355216 
356409 
357604 
358801 
360000 


211708736 
212776173 
213847192 
214921799 
216000000 


24-4131112 
24.4335834 
24.4540385 
24-4744765 
24.4948974 


8-4155419 
8-4202460 
8-4249448 
8.4296383 
8-4343267 


.001677852 
.001675042 
.001672241 
.001669449 
.001666667 



797 



TABLE XVI. SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


601 
602 
603 
604 
605 


361201 
362404 
363609 
364816 
366025 


217081801 
218167208 
219256227 
220348864 
221445125 


24.5153013 
24.5356883 
24.5560583 
24.5764115 
24.5967478 


8.4390098 
8.4436877 
8.4483605 
8.4530281 
8.4576906 


.001663894 
.001661130 
.001658375 
.001655629 
•001652893 ^ 


606 
607 
608 
609 
610 


367236 
368449 
369664 
370881 
372100 


222545016 
223648543 
224755712 
225866529 
226981000 


24.6170673 
24.6373700 
24.6576560 
24.6779254 
24.6981781 


8.4623479 
8.4670001 
8.4716471 
8.4762892 
8.4809261 


.001650165 
.001647446 
.001644737 
.001642036 
.001639344 


611 
612 
613 
614 
615 


373321 
374544 
375769 
376996 
378225 


228099131 
229220928 
230346397 
231475544 
232608375 


24.7184142 
24.7386338 
24.7588368 
24.7790234 
24.7991935 


8.4855579 
8.4901848 
8.4948065 
8.4994233 
8.5040350 


.001636661 
.001633987 
.001631321 
.001628664 
.001626016 


616 
617 
618 
619 
620 


379456 
380689 
381924 
383161 
384400 


233744896 
234885113 
236029032 
237176659 
238328000 


24.8193473 
24.8394847 
24.8596058 
24.8797106 
24.8997992 


8.5086417 
8.5132435 
8.5178403 
8.5224321 
8.5270189 


.001623377 
.001620746 
.001618123 
.001615509 
.001612903 


621 
622 
623 
624 
625 


385641 
386884 
388129 
389376 
390625 


239483061 
240641848 
241804367 
242970624 
244140625 


24.9198716 
24.9399278 
24.9599679 
24.9799920 
25.0000000 


8.5316009 
8.5361780 
8.5407501 
8.5453173 
8.5498797 


.001610306 
.001607717 
.001605136 
.001602564 
.001600000 


626 
627 
628 
629 
630 


391876 
393129 
394384 
395641 
396900 


245314376 
246491883 
247673152 
248858189 
250047000 


25.0199920 
25.0399681 
25.0599282 
25.0798724 
25.0998008 


8.5544372 
8.5589899 
8.5635377 
8.5680807 
8.5726189 


.001597444 
.001594896 
.001592357 
.001589825 
.001587302 


631 
632 
633 
634 
635 


398161 
399424 
400689 
401956 
403225 


251239591 
252435968 
253636137 
254840104 
256047875 


25.1197134 
25.1396102 
25.1594913 
25.1793566 
25.1992063 


8.5771523 
8.5816809 
8.5862047 
8.5907238 
8.5952380 


.001584786 
.001582278 
.001579779 
.001577287 
.001574803 


636 
637 
638 
639 
640 


404496 
405769 
407044 
408321 
409600 


257259456 
258474853 
259694072 
260917119 
262144000 


25.2190404 
25.2388589 
25.2586619 
25.2784493 
25.2982213 


8.5997476 
8.6042525 
8.6087526 
8.6132480 
8.6177388 


.001572327 
.001569859 
.001567398 
.001564945 
.001562500 


641 
642 
643 
644 
645 


410881 
412164 
413449 
414736 
416025 


263374721 
264609288 
265847707 
267089984 
268336125 


25.3179778 
25.3377189 
25.3574447 
25.3771551 
25.3968502 


8.6222248 
8.6267063 
8.6311830 
8.6356551 
8.6401226 


.001560062 
.001557632 
.001555210 
.001552795 
.001550388 


646 
647 
648 
649 
650 


417316 
418609 
419904 
421201 
422500 


269586136 
270840023 
272097792 
273359449 
274625000 


25.4165301 
25.4361947 
25.4558441 
25.4754784 
25.4950976 


8.6445855 
8.6490437 
8.6534974 
8.6579465 
8.6623911 


.001547988 
.001545595 
.001543210 
.001540832 
.001538462 


651 
652 
653 
654 
655 


423801 
425104 
426409 
427716 
429025 


275894451 
277167808 
278445077 
279726264 
281011375 


25.5147016 
25.5342907 
25.5538647 
25.5734237 
25.5929678 


8.6668310 
8.6712665 
8.6756974 
8.6801237 
8.6845456 


.001536098 
.001533742 
.001531394 
.001529052 
.001526718 


656 
657 
658 
659 
660 


430336 
431649 
432964 
434281 
435600 


282300416 
283593393 
284890312 
286191179 
287496000 


25.6124969 
25.6320112 
25.6515107 
25.6709953 
25.6904652 


8.6889630 
8.6933759 
8.6977843 
8.7021882 
8.7065877 


.001524390 
.001522070 
.001519757 
.001517451 
.001515152 



798 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


661 


436921 


288804781 


25.7099203 


8.7109827 


.001512859 


662 


438244 


290117528 


25.7293607 


8.7153734 


.001510574 


663 


439569 


291434247 


25.7487864 


8.7197596 


.001508296 


664 


440896 


292754944 


25.7681975 


8.7241414 


.001506024 


665 


442225 


294079625 


25.7875939 


8-7285187 


.001503759 


666 


443556 


295408296 


25.8069758 


8.7328918 


.001501502 


667 


444889 


296740963 


25.8263431 


8.7372604 


.001499250 


668 


446224 


298077632 


25.8456960 


8.7416246 


.001497006 


669 


447561 


299418309 


25.8650343 


8.7459846 


.001494768 


^ 670 


448900 


300763000 


25.8843582 


8.7503401 


.001492537 , 


671 


450241 


302111711 


25.9036677 


8.7546913 


.001490313 


672 


451584 


303464448 


25.9229628 


8.7590383 


.001488095 


673 


452929 


304821217 


25.9422435 


8.7633809 


.001485884 


674 


454276 


306182024 


25.9615100 


8.7677192 


.001483680 


675 


455625 


307546875 


25.9807621 


8.7720532 


-001481481 


676 


456976 


308915776 


26.0000000 


8.7763830 


.001479290 


677 


458329 


310288733 


26.0192237 


8.7807084 


.001477105 


678 


459684 


311665752 


26.0384331 


8.7850296 


.001474926 


679 


461041 


313046839 


26.0576284 


8.7893466 


.001472754 


680 


462400 


314432000 


26-0768096 


8-7936593 


-001470588 


681 


463761 


315821241 


26.0959767 


R. 7979679 


.001468429 


682 


465124 


317214568 


26.1151297 1 


8.8022721 


.001466276 


683 


466489 


318611987 


26.134268? 


8.8065722 


.001464129 


684 


467856 


320013504 


26.15339S-J 
26.17250^.V 


8.8108681 


.001461988 


685 


469225 


321419125 


8.8151598 


.001459854 


686 


470596 


322828856 


26.19160.7 


8.8194474 


.001457726 


687 


471969 


324242703 


26.2106848 


8.8237307 


.001455604 


688 


473344 


325660672 


26.22975a 


8.8280099 


.001453488 


689 


474721 


327082769 


26.2488095 


8.8322850 


.001451379 


690 


476100 


328509000 


26.2678Uil 


8.8365559 


.001449275 


691 


477481 


329939371 


26.286'»:?89 


8.8408227 


.001447178 


692 


478864 


331373888 


26.3058929 
26.3248932 


8.8450854 


.001445087 


693 


480249 


332812557 


8.8493440 


.001443001 


694 


481636 


334255384 


26.3408797 


8.8535985 


.001440922 


695 


483025 


S35702375 


26.3628527 


8.8578489 


.001438849 


696 


484416 


337153536 


26.3818119 


8.8620952 


.001436782 


697 


485809 


338608873 


26.4007576 


8.8663375 


.001434720 


698 


487204 


340068392 


26.4196896 


8.8705757 


.001432665 


699 


488601 


341532099 


26.4386081 


8-8748099 


.001430615 


700 


490000 


343000000 


26.4575131 


8-8790400 


.001428571 


701 


491401 


344472101 


26.4764046 


8.8832661 


.001426534 


702 


492804 


345948408 


26.4952826 


8.8874882 


.001424501 


703 


494209 


347428927 


26.5141472 


8.8917063 


.001422475 


704 


495616 


348913664 


26.5329983 


8.8959204 


.001420455 


705 


497025 


350402625 


26-5518361 


8.9001304 


.001418440 


1 706 


498436 


351895816 


26-5706605 


8.9043366 


.001416431 


1 707 


499849 


353393243 


26.5894716 


8.9085387 


.001414427 


1 708 


501264 


354894912 


26.6082694 


8.9127369 


.001412429 


709 


502681 


356400829 


26.6270539 


8.9169311 


.001410437 


710 


504100 


357911000 


26.6458252 


8.9211214 


.001408451 


; 711 


505521 


359425431 


26.6645833 


8.9253078 


.001406470 


' 712 


506944 


360944128 


26.6833281 


8.9294902 


.001404494 


j 713 


508369 


362467097 


26.7020598 


8.9336687 


.001402525 


714 


509796 


363994344 


26.7207784 


8.9378433 


.001400560 


715 


511225 


365525875 


26-7394839 


8.9420140 


.001398501 


716 


512656 


367061696 


26.7581763 


8.9461809 


.001396648 


717 


514089 


368601813 


26.7768557 


8.9503438 


.001394700 


718 


515524 


370146232 


26.7955220 


8.9545029 


.001392758 


719 


516961 


371694959 


26.8141754 


8.9586581 


.001390821 


730 


518400 


373248000 


26.8328157 


8.9628095 


.001388889 








799 







TABLE XVI. — SQUARES, . CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


721 
722 
723 
724 
725 


519841 
521284 
522729 
524176 
525625 


374805361 
376367048 
377933067 
379503424 
381078125 


26.8514432 
26.8700577 
26.8886593 
26.9072481 
26.9258240 


8.9669570 
8.9711007 
8.9752406 
8.9793766 
8.9835089 


.001386963 | 
.001385042 \ 
.001383126 
.001381215 
.001379310 


726 
121 
728 
729 
730 


527076 
528529 
529984 
531441 
532900 


382657176 
384240583 
385828352 
387420489 
389017000 


26-9443872 
26.9629375 
26.9814751 
27.0000000 
27.0185122 


8-9876373 
8-9917620 
8-9958829 
9-0000000 
9.0041134 


.001377410 
.001375516 
.001373626 
.001371742 
-001369863 


731 
732 
733 
734 
735 


534361 
535824 
537289 
538756 
540225 


390617891 
392223168 
393832837 
395446904 
397065375 


27.0370117 
27.0554985 
27.0739727 
27.0924344 
27.1108834 


9-0082229 
9-0123288 
9.0164309 
9.0205293 
9.0246239 


.001367989 
.001366120 
.001364256 
.001362398 
-001360544 


736 
IZl 
738 
739 
740 


541696 
543169 
544644 
546121 
547600 


398688256 
400315553 
401947272 
403583419 
405224000 


27.1293199 
27.1477439 
27.1661554 
27.1845544 
27.2029410 


9.0287149 
9-0328021 
9-0368857 
9-0409655 
9-0450419 


.001358696 
.001356852 
.001355014 
.001353180 
-001351351 


741 
742 
743 
744 
745 


549081 
550564 
552049 
553536 
555025 


406869021 
408518488 
410172407 
411830784 
413493625 


27.2213152 
27.2396769 
27.2580263 
27.2763634 
27.2946881 


9-0491142 
9-0531831 
9-0572482 
9-0613098 
9-0653677 


.001349528 
.001347709 
.001345895 
.001344086 
.001342282 


746 
747 
748 
749 
750 


556516 
558009 
559504 
561001 
562500 


415160936 
416832723 
418508992 
420189749 
421875000 


27.3130006 
27.3313007 
27.3495887 
27.3678644 
27.3861279 


9-0694220 
9.0734726 
9.0775197 
9.0815631 
9.0856030 


.001340483 
.001338688 
•001336898 
.001335113 
-001333333 


751 
752 
753 
754 
755 


564001 
565504 
567009 
568516 
570025 


423564751 
425259008 
426957777 
428661064 
430368875 


27.4043792 
27.4226184 
27.4408455 
27.4590604 
27.4772633 


9.0896392 
9.0936719 
9-0977010 
9-1017265 
9-1057485 


-001331558 
.001329787 
.001328021 
.001326260 
-001324503 


756 
757 
758 
759 
760 


571536 
573049 
574564 
576081 
577600 


432081216 
433798093 
435519512 
437245479 
438976000 


27.4954542 
27.5136330 
27-5317998 
27.5499546 
27.5680975 


9-1097669 
9-1137818 
9-1177931 
9-1218010 
9.1258053 


.001322751 
.001321004 
.001319261 
.001317523 
-001315789 


761 
762 
763 
764 
765 


579121 
580644 
582169 
583696 
585225 


440711081 
442450728 
444194947 
445943744 
447697125 


27.5862284 
27.6043475 
27.6224546 
27-6405499 
27.6586334 


9.1298061 
9.1338034 
9-1377971 
9-1417874 
9.1457742 


.001314060 
.001312336 
.001310616 
.001308901 
-001307190 


766 
767 
768 
769 
770 


586756 
588289 
589824 
591361 
592900 


449455096 
451217663 
452984832 
454756609 
456533000 


27-6767050 
27-6947648 
27-7128129 
27-7308492 
27.7488739 


9.1497576 
9-1537375 
9-1577139 
9-1616869 
9-1656565 


.001305483 
.001303781 
.001302083 
.001300390 
-001298701 


771 
772 
773 
774 
775 


594441 
595984 
597529 
599076 
600625 


458314011 
460099648 
461889917 
463684824 
465484375 


27-76688.68 
27-7848880 
27-8028775 
27.8208555 
27.8388218 


9-1696225 
9-1735852 
9-1775445 
9-1815003 
9-1854527 


.001297017 
.001295337 
.001293661 
.001291990 
.001290323 


776 
111 

lis 

119 
780 


602176 
603729 
605284 
606841 
608400 


467288576 
469097433 
470910952 
472729139 
474552000 


27.8567766 
27.8747197 
27.8926514 
27.9105715 
27-9284801 


9.1894018 
9.1933474 
9-1972897 
9-2012286 
9-2051641 


.001288660 
.001287001 
.001285347 
.001283697 
.001282051 



800 



CUBE ROOTSj AND RECIPROCALS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube RootSo 


Reciprocals. 


781 
782 
783 
784 
785 


609961 
611524 
613089 
614656 
616225 


476379541 
478211768 
480048687 
481890304 
483736625 


27.9463772 
27.9642629 
27.9821372 
28.0000000 
28.0178515 


9.2090962 
9.2130250 
9.2169505 
9.2208726 
9.2247914 


.001280410 
.001278772 
.001277139 
.001275510 
-001273885 


786 
787 
788 
789 
790 


617796 
619369 
620944 
622521 
624100 


485587656 
487443403 
489303872 
491169069 
493039000 


28.0356915 
28.0535203 
28.0713377 
28.0891438 
28.1069386 


9.2287068 
9.2326189 
9.2365277 
9.2404333 
9-2443355 


.001272265 
.C01270648 
.001269036 
.001267427 
.001265823 


791 
792 
793 
794 
795 


625681 
627264 
628849 
630436 
632025 


494913671 
496793088 
498677257 
500566184 
502459875 


28.1247222 
28.1424946 
28-1602557 
28.1780056 
28.1957444 


9.2482344 
9.2521300 
9.2560224 
9.2599114 
9-2637973 


•001264223 
.001262626 
.001261034 
.001259446 
.001257862 


796 
797 
798 
799 
800 


633616 
635209 
636804 
638401 
640000 


504358336 
506261573 
508169592 
510082399 
512000000 


28.2134720 
28.2311884 
28.2488938 
28-2665881 

28.2842712 


9-2676798 
9.2715592 
9-2754352 
9.2793081 
9-2831777 


.001256281 
.001254705 
.001253133 
.001251564 
-001250000 


801 
802 
803 
804 
805 


641601 
643204 
644809 
646416 
648025 


513922401 
515849608 
517781627 
519718464 
521660125 


28.3019434 
28.3196045 
28.3372546 
28.3548938 
28.3725219 


9.2870440 
9.2909072 
9.2947671 
9.2986239 
9-3024775 


.001248439 
.001246883 
.001245330 
.001243781 
.001242236 


806 
807 
808 
809 
810 


649636 
651249 
652864 
654481 
656100 


523606616 
525557943 
527514112 
529475129 
531441000 


28.3901391- 

28-4077454 

28-4253408 

28.4429253 

28.4604989 


9.3063278 
9.3101750 
9.3140190 
9.3178599 
9-3216975 


.001240695 
.001239157 
.001237624 
.001236094 
•001234568 


811 
812 
813 
814 
815 


657721 
659344 
660969 
662596 
664225 


533411731 
535387328 
537367797 
539353144 
541343375 


28.4780617 
28.4956137 
28.5131549 
28.5306852 
28.5482048 


9-3255320 
9-3293634 
9-3331916 
9.3370167 
9-3408386 


.001233046 
.001231527 
.001230012 
.001228501 
.001226994 


816 
817 
818 
819 
820 


665856 
667489 
669124 
670761 
672400 


543338496 
545338513 
547343432 
549353259 
551368000 


28.5657137 
28.5832119 
28.6006993 
28.6181760 
98.6356421 


9-3446575 
9-3484731 
9-3522857 
9-3560952 
9-3599016 


.001225490 
.001223990 
.001222494 
.001221001 
.001219512 


821 
822 
823 
824 
825 


674041 
675684 
677329 
678976 
680625 


553387661 
555412248 
557441767 
559476224 
561515625 


28.6530976 
28.6705424 
28.6879766 
28.7054002 
28.7228132 


9-3637049 
9-3675051 
9-3713022 
9-3750963 
9-3788873 


.001218027 
=001216545 
.001215067 
.001213592 
.001212121 


826 
827 
828 
829 
830 


682276 
683929 
685584 
687241 
688900 


563559976 
565609283 
567663552 
569722789 
571787000 


28.7402157 
28.7576077 
28.7749891 
28.7923601 
28.8097206 


9-3826752 
9-3864600 
9-3902419 
9-3940206 
9-3977964 


.001210654 
.001209190 
.001207729 
.001206273 
.001204819 


831 
832 
833 
834 
835 


690561 
692224 
693889 
695556 
697225 


573856191 
575930368 
578009537 
580093704 
582182875 


28.8270706 
28.8444102 
28.8617394 
28.8790582 
28.8963666 


9.4015691 
9-4053387 
9.4091054 
9.4128690 
9.4166297 


.001203369 
.001201923 
.001200480 
.001199041 
-001197605 


836 
837 
838 
839 
840 


698896 
7Q0569 
702244 
703921 
705600 


584277056 
586376253 
588480472 
590589719 
592704000 


28.9136646 
28. 9309523 
28.9482297 
28.9654967 
28.9827535 


9.4203873 
9.4241420 
9.4278936 
9.4316423 
9.4353880 


.001196172 
.001194743 
.001193317 
.001191895 
.001190476 



801 



TABLE XVI. — SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. . 


841 


707281 


594823321 


29.0000000 


9-4391307 


.001189061 1 


842 


708964 


596947688 


29.0172363 


9-4428704 


-001187648 f 


843 


710649 


599077107 


29.0344623 


9-4466072 


.001186240 


844 


712336 


601211584 


29.0516781 


9.4503410 


.001184834 


845 


714025 


603351125 


29.0688837 


9.4540719 


-001183432 


846 


715716 


605495736 


29.0860791 


9-4577999 


.001182033 


847 


717409 


607645423 


29.1032644 


9-4615249 


.001180638 


848 


719104 


609800192 


29.1204396 


9-4652470 


.001179245 


849 


720801 


611960049 


29.1376046 


9-4689661 


.001177856 


^, 850 


722500 


614125000 


29.1547595 


9.4726824 


-001176471 


851 


724201 


616295051 


29.1719043 


9-4763957 


.001175088 


852 


725904 


618470208 


29.1890390 


9-4801061 


.001173709 


853 


727609 


620650477 


29.2061637 


9-4838136 


.001172333 


854 


729316 


622835864 


29.2232784 


9-4875182 


.001170960 


. 855 


731025 


625026375 


29.2403830 


9.4912200 


-001169591 


856 


732736 


627222016 


29.2574777 


9.4949188 


.001168224 


857 


734449 


629422793 


29.2745623 


9.4986147 


.001166861 


858 


736164 


631628712 


29.2916370 


9.5023078 


.001165501 


859 


737881 


633839779 


29.3087018 


9.5059980 


.001164144 


. 860 


739600 


636056000 


29.3257566 


9.5096854 


-001162791 


861 


741321 


638277381 


29.3428015 


9.5133699 


-001161440 


862 


743044 


640503928 


29.3598365 


9.5170515 


.001160093 


863 


744769 


642735647 


29.3768616 


9-5207303 


-001158749 


864 


746496 


644972544 


29.3938769 


9-5244063 


.001157407 


865 


748225 


647214625 


29.4108823 


9.5280794 


.001156069 


866 


749956 


649461896 


29.4278779 


9.5317497 


-001154734 


867 


751689 


651714363 


9.4448637 


9-5354172 


-001153403 . 


868 


753424 


653972032 


29-4618397 


9-5390818 


-001152074 


869 


755161 


656234909 


29-4788059 


9-5427437 


-001150748 


870 


756900 


658503000 


29.4957624 


9-5464027 


.001149425 


871 


758641 


660776311 


29.5127091 


9.5500589 


-001148106 


872 


760384 


663054848 


29.5296461 


9-5537123 


.001146789 


873 


762129 


665338617 


29-5465734 


9-5573630 


-001145475 


874 


763876 


667627624 


29.5634910 


9.5610108 


.001144165 


875 


765625 


669921875 


29.5803989 


9.5646559 


-001142857 


876 


767376 


672221376 


29.5972972 


9.5682982 


.001141553 


877 


769129 


674526133 


29.6141858 


9-5719377 


.001140251 


878 


770884 


676836152 


29-6310648 


9-5755745 


.001138952 


879 


772641 


679151439 


29-6479342 


9-5792085 


.001137656 


880 


774400 


681472000 


29.6647939 


9.5828397 


-001136364 


881 


776161 


683797841 


29.6816442 


9-5864682 


.001135074 


882 


777924 


686128968 


29.6984848 


9-5900939 


.001133787 


883 


779689 


688465387 


29.7153159 


9-5937169 


•001132503 


884 


781456 


690807104 


29-7321375 


9-5973373 


.001131222 


885 


783225 


693154125 


29.7489496 


9.6009548 


.001129944 


886 


784996 


695506456 


29-7657521 


9-6045696 


-001128668 


887 


786769 


697864103 


29-7825452 


9-6081817 


.001127396 


888 


788544 


700227072 


29.7993289 


9-6117911 


-001126126 


889 


790321 


702595369 


29-8161030 


9-6153977 


-001124859 


890 


792100 


704969000 


29-8328678 


9.6190017 


-001123596 


891 


793881 


707347971 


29.8496231 


9.6226030 


.001122334 


892 


795664 


709732288 


29.8663690 


9.6262016 


.001121076 


893 


797449 


712121957 


29.8831056 


9.6297975 


.001119821 


894 


799236 


714516984 


29.8998328 


9.6333907 


.001118568 


895 


801025 


716917375 


29.9165506 


9.6369812 


-001117318 


896 


802816 


719323136 


29.9332591 


9.6405690 


.001116071 


897 


804609 


721734273 


29.9499583 


9.6441542 


.001114827 


898 


806404 


724150792 


29.9666481 


9.6477367 


.001113586 . 
.001112347 J 


899 


808201 


726572699 


29.9833287 


9.6513166 


900 


810000 


729000000 


30-0000000 


9.6548938 


.001111111 1 



802 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Sqfuares. 


Cubes. 


Square RootSo 


Cube Roots. 


Reciprocals. 


901 
902 
903 
904 
905 


811801 
813604 
815409 
817216 
819025 


731432701 
733870808 
736314327 
738763264 
741217625 


30.0166620 
30.0333148 
30-0499584 
30.0665928 
30.0832179 


9-6584684 
9-6620403 
9=6656096 
9-6691762 
9-6727403 


.001109878 
.001108647 
.001107420 
.001106195 
.001104972 


906 
907 
908 
909 
910 


820836 
822649 
824464 
826281 
828100 


743677416 
746142643 
748613312 
751089429 
753571000 


30.0998339 
30ai64407 
30.1330383 
30.1496269 
30.1662063 


9-6763017 
9-6798604 
9-6834166 
9-6869701 
9-6905211 


.001103753 
.001102536 
.001101322 
.001100110 
.001098901 


911 
912 
913 
914 
915 


829921 
831744 
833569 
835396 
837225 


756058031 
758550528 
761048497 
763551944 
766060875 


30.1827765 
30.1993377 
30.2158899 
30.2324329 
30.2489669 


9-6940694 
9-6976151 
9-7011583 
9-7046989 
9-7082369 


.001097695 
.001096491 
.001095290 
.001094092 
.001092896 


916 
917 
918 
919 
920 


839056 
840889 
842724 
844561 
846400 


768575296 
771095213 
773620632 
776151559 
778688000 


30-2654919 
30.2820079 
30.2985148 
30.3150128 
30-3315018 


9-7117723 
9-7153051 
9-7188354 
9-7223631 
9-7258883 


.001091703 
.001090513 
.001089325 
.001088139 
-001086957 


921 
922 
923 
924 
925 


848241 
850084 
851929 
853776 
855625 


781229961 
783777448 
786330467 
788889024 
791453125 


30-3479818 
30.3644529 
30-3809151 
30-3973683 
30-4138127 


9-7294109 
9-7329309 
9-7364484 
9-7399634 
9-7434758 


.001085776 
.001084599 
.001083423 
.001082251 
-001081081 


926 
927 
928 
929 
930 


857476 
859329 
861184 
863041 
864900 


794022776 
796597983 
799178752 
801765089 
804357000 


30-4302481 
30-4466747 
30-4630924 
30-4795013 
30-4959014 


9-7469857 
9-7504930 
9-7539979 
9-7575002 
9-7610001 


.001079914 
.001078749 
.001077586 
.001076426 
.001075269 


931 
932 
933 
934 
935 


866761 
868624 
870489 
872356 
874225 


806954491 
809557568 
812166237 
814780504 
817400375 


30-5122926 
30-5286750 
30-5450487 
30-5614136 
30-5777697 


9-7644974 
9-7679922 
9-7714845 
9.7749743 
9.7784616 


-001074114 
.001072961 
.001071811 
.001070664 
-001069519 


936 
937 
938 
939 
940 


876096 
877969 
879844 
881721 
883600 


820025856 
822656953 
825293672 
827936019 
830584000 


30-5941171 
30-6104557 
30-6267857 
30-6431069 
30.6594194 


9.7819466 
9=7854288 
9-7889087 
9-7923861 
9.7958611 


.001068376 
.001067236 
.001068098 
.001064963 
.001063830 


941 
942 
943 
944 
945 


885481 
887364 
889249 
891136 
893025 


833237621 
835896888 
838561807 
841232384 
843908625 


30-6757233 
30-6920185 
30-7083051 
30-7245830 
30-7408523 


9-7993336 
9-8028036 
9-8062711 
9-8097362 
9-8131989 


.001062699 
.001061571 
.001060445 
.001059322 
.001058201 


946 
947 
948 
949 
950 


894916 
896809 
898704 
900601 
902500 


846590536 
849278123 
851971392 
854670349 
857375000 


30-7571130 
30-7733651 
30-7896086 
30-8058436 
30-8220700 


9-8166591 
9-8201169 
9.8235723 
9-8270252 
9 8304757 


.001057082 
.001055966 
.001054852 
.001053741 
.001052632 


951 
952 
953 
954 
955 


904401 
906304 
908209 
910116 
912025 


860085351 
862801408 
865523177 
868250664 
870983875 


30-8382879 
30.8544972 
30-8706981 
30-8868904 
30-9030743 


9-8339238 
9.8373695 
9.8408127 
9.8442536 
9-8476920 


.001051525 
.001050420 
.001049318 
.001048218 
-001047120 


956 
957 
958 
959 
960 


913936 
915849 
917764 
919681 
921600 


873722816 
876467493 
879217912 
881974079 
884736000 


30-9192497 
30.9354186 
30.9515751 
30.9677251 
30-9838668 


9.8511280 
9-8545617 
9-8579929 
9.8614218 
9.8648483 


.001046025 
.001044932 
.001043841 
.001042753 
.001041667 



803 



TABLE XVI 


. SQUARES 


1 
, CUBES, SQUARE ROOTS, ETC. 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


RecipFocals. 


961 
962 
963 
964 
965 


923521 
925444 
927369 
929296 
931225 


887503681 
890277128 
893056347 
895841344 
898632125 


31.0000000 
31.0161248 
31.0322413 
31.0483494 
31.0644491 


9.8682724 
9.8716941 
9.8751135 
9.8785305 
9.8819451 


.001040583 
.0010^9501 1 
.001038422 
.001037344 1 
.001036269 


966 
967 
968 
969 
970 


933156 
935089 
937024 
938961 
940900 


901428696 
904231063 
907039232 
909853209 
912673000 


31.0805405 
31.0966236 
31.1126984 
31.1287648 
31.1448230 


9.8853574 
9.8887673 
9.8921749 
9.8955801 
9.8989830 


.001035197 1 

.001034126 

.001033058 

.001031992 

-001030928 


971 
972 
973 
974 
975 


942841 
944784 
946729 
948676 
950625 


915498611 
918330048 
921167317 
924010424 
926859375 


31.1608729 
31.1769145 
31.1929479 
31.2089731 
31.2249900 


9.9023835 
9.9057817 
9.9091776 
9.9125712 
9.9159624 


.001029866 
.001028807 
.001027749 
.001026694 
.001025641 


976 
977 
978 
979 
. 980 


952576 
954529 
956484 
958441 
960400 


929714176 
932574833 
935441352 
938313739 
941192000 


31.2409987 
31.2569992 
31.2729915 
31.2889757 
31.3049517 


9.9193513 
9.9227379 
9.9261222 
9.9295042 
9.9328839 


.001024590 
.001023541 
.001022495 
.001021450 
.001020408 


981 
982 
983 
984 
985 


962361 
964324 
966289 
968256 
970225 


944076141 
946966168 
949862087 
952763904 
955671625 


31.3209195 
31.3368792 
31.3528308 
31.3687743 
31.3847097 


9.9362613 
9.9396363 
9.9430092 
9.9463797 
9.9497479 


.001019368 
.001018330 
.001017294 
.001016260 
.001015228 


986 
987 
988 
989 
. 990 


972196 
974169 
976144 
978121 
980100 


958585256 
961504803 
964430272 
967361669 
970299000 


31.4006369 
31.4165561 
31.4324673 
31.4483704 
31.4642654 


9.9531138 
9.9564775 
9.9598389 
9-9631981 
9-9665549 


.0010141991 
.0010131711 
.001012146! 
.001011122 1 
.001010101 1 


991 
992 
993 
994 
995 


982081 
984064 
986049 
988036 
990025 


973242271 
976191488 
979146657 
982107784 
985074875 


31.4801525 
31.4960315 
31.5119025 
31.5277655 
31.5436206 


9-9699095 
9-9732619 
9-9766120 
9-9799599 
9.9833055 


.0010090821 
.001008065 * 
.001007049 
.001006036 
.001005025 


996 
997 
998 
999 
1000 


992016 
994009 
996004 
998001 
1000000 


988047936 
991026973 
994011992 
997002999 
1000000000 


31.5594677 
31.5753068 
31.5911380 
31.6069613 
31.6227766 


9-9866488 
9-9899900 
9-9933289 
9-9966656 
10.0000000 


.001004016 
.001003009 
.001002004 
.001001001 
-001000000 


1001 
1002 
1003 
1004 
1005 


1002001 
1004004 
1006009 
1008016 
1010025 


1003003001 
1006012008 
1009027027 
1012048064 
1015075125 


31.6385840 
31.6543836 
31.6701752 
31.6859590 
31.7017349 


10-0033322 
10-0066622 
10-0099899 
10-0133155 
10-C166389 


.0009990010 
.0009980040 
.0009970090 
.0009960159 
.0009950249 


1006 
1007 
1008 
1009 
1010 


1012036 
1014G49 
1016064 
1018081 
1020100 


1018108216 
1021147343 
1024192512 
1027243729 
1030301000 


31.7175030 
31.7332633 
31.7490157 
31.7647603 
31.7804972 


10-0199601 
10-0232791 
10-0265958 
10-0299104 
10.0332228 


.0009940358 
.0009930487 
.0009920635 
.0009910803 
.0009900990 


1011 
1012 
1013 
1014 
1015 


1022121 
1024144 
1026169 
1028196 
1030225 


1033364331 
1036433728 
1039509197 
1042590744 
. 1045678375 


31.7962262 
31.8119474 
31.8276609 
31.8433666 
31.8590646 


10-0365330 
10-0398410 
10-0431469 
10-0464506 
10.0497521 


.0009891197 
.0009881423 
.0009871668 
.0009861933 
.0009852217 


1016 
1017 
1018 
1019 
1030 


1032256 
1034289 
1036324 
1038361 
1040400 


1048772096 
1051871913 
1054977832 
1058089859 
1061208000 


31.8747549 
31.8904374 
31.9061123 
31.9217794 
31.9374388 


10-0530514 
10.0563485 
10-0596435 

10-0629304 
10-0662271 


.0009842520 
.0009832842 
.0009823183 

o 00^08.1 SS'^S 
.0009803922 



804 



TABLE XVII. CUBIC YARDS PER 100 FEET OF LEVEL 

SECTIONS. SLOPE 1:1. 



Depth, 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


d 


12 feet. 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


48 


56 


63 


70 


78 


107 


115 


122 


2 


104 


119 


133 


148 


163 


222 


237 


252 


3 


167 


189 


211 


233 


256 


344 


367 


389 


4 


237 


267 


296 


326 


356 


474 


504 


533 


5 


315 


352 


389 


426 


463 


611 


648 


685 


6 


400 


444 


489 


533 


578 


756 


800 


844 


7 


493 


544 


596 


648 


700 


907 


959 


1011 


8 


593 


652 


711 


770 


830 


1067 


1126 


1185 


9 


700 


767 


833 


900 


967 


1233 


1300 


1367 


10 


815 


889 


963 


1037 


1111 


1407 


1481 


1556 


11 


937 


1019 


1100 


1181 


1263 


1589 


1670 


1752 


12 


1067 


1156 


1244 


1333 


1422 


1778 


1867 


1956 


13 


1204 


1300 


1396 


1493 


1589 


1974 


2070 


2167 


14 


1348 


1452 


1556 


1659 


1763 


2178 


2281 


23fi5 


15 


1500 


1611 


1722 


1833 


1944 


2389 


2500 


2611 


16 


1659 


1778 


. 1896 


2015 


2133 


2607 


2726 


2844 


17 


1826 


1952 


2078 


2204 


2330 


2833 


2959 


3085 


18 


2000 


2133 


2267 


2400 


2533 


3067 


3200 


3333 


19 


2181 


2322 


2463 


2604 


2744 


3307 


3448 


3589 


20 


2370 


2519 


2667 


2815 


2963 


3556 


3704 


3852 


21 


2567 


2722 


2878 


3033 


3189 


3811 


3967 


4122 


22 


2770 


2933 


3096 


3259 


3422 


4074 


4237 


4400 


'23 


2981 


3152 


3322 


3493 


3663 


4344 


4515 


4685 


24 


3200 


3378 


3556 


3733 


3911 


4622 


4800 


4978 


25 


3426 


3611 


3796 


3981 


4167 


4907 


5093 


5278 


26 


3659 


3852 


4044 


4237 


4430 


5200 


5393 


5585 


27 


3900 


4100 


4300 


4500 


4700 


5500 


5700 


5900 


28 


4148 


4356 


4563 


4770 


4978 


5807 


6015 


6222 


29 


4404 


4619 


4833 


5048 


5263 


6122 


6337 


6552 


80 


4667 


4889 


5111 


5333 


5556 


6444 


6667 


6889 


81 


4937 


5167 


5396 


5626 


5856 


6774 


7004 


7233 


32 


5215 


5452 


5689 


5926 


6163 


7111 


7348 


7585 


33 


5500 


5744 


5989 


6233 


6478 


7456 


7700 


7944 


34 


5793 


6044 


6296 


6548 


6800 


7807 


8059 


8311 


35 


6093 


6352 


6611 


6870 


7130 


8167 


8426 


8685 


86 


6400 


6667 


6933 


7200 


7467 


8533 


8800 


9067 


&7 


6715 


6989 


7263 


7537 


7811 


8907 


9181 


9456 


38 


7037 


7319 


7600 


7881 


8163 


9289 


9570 


9852 


89 


7367 


7656 


7944 


8233 


8522 


9678 


9967 


10256 


40 


7704 


8000 


8296 


8593 


8889 


10074 


10370 


10667 


41 


8048 


8352 


8656 


8959 


9263 


10478 


10781 


11085 


42 


8400 


8711 


9022 


9333 


9644 


10889 


11200 


11511 


43 


8759 


9078 


9396 


9715 


10033 


11307 


11626 


11944 


44 


9126 


9452 


9778 


10104 


10430 


11733 


12059 


12385 


45 


9500 


9833 


10167 


10500 


10833 


12167 


12500 


12833 


46 


9881 


10222 


10563 


10904 


11244 


12607 


12948 


13289 


47 


10270 


10619 


10967 


11315 


11663 


13056 


13404 


13752 


48 


10667 


11022 


11378 


11733 


12089 


13511 


13867 


14222 


49 


11070 


11433 


11796 


12159 


12522 


13974 


14337 


14700 


50 


11481 


11852 


12222 


12593 


12963 


14444 


14815 


15185 


51 


11900 


12278 


12656 


13033 


13411 


14922 


15300 


15678 


52 


12326 


12711 


13096 


13481 


13867 


15407 


15793 


16178 


53 


12759 


13152 


13544 


13937 


14330 


15900 


16293 


16685 


54 


13200 


13600 


14000 


14400 


14800 


16400 


16800 


17200 


55 


13648 


14056 


14463 


14870 


15278 


16907 


17315 


17722 


56 


14104 


14519 


14933 


15348 


15763 


17422 


17837 


18252 


57 


14567 


14989 


15411 


15833 


16256 


17944 


18367 


18789 


58 


15037 


15467 


15896 


16326 


16756 


18474 


18904 


19333 


59 


15515 


15952 


16389 


16826 


17263 


19011 


19448 


19885 


60 


16000 


16444 


16889 


17333 


17778 


19556 


20000 


20444 



805 



^ 



TABLE XVII. — CUBIC YARDS PER 100 FEET OF LEVEL 
SECTIONS. SLOPE 1.5 : 1. 



Depth 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


12 feet. 
50 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


57 


65 


72 


~ scT 


109 


117 


124 


2 


111 


126 


141 


156 


170 


230 


244 


259 


S 


183 


206 


228 


250 


272 


361 


383 


406 


4 


267 


296 


326 


356 


385 


504 


533 


563 


5 


361 


398 


435 


472 


509 


657 


694 


731 


6 


467 


511 


556 


600 


644 


822 


867 


911 


? 


583 


635 


687 


739 


791 


998 


1050 


1102 


8 


711 


770 


830 


889 


948 


1185 


1244 


1304 


G 


850 


917 


983 


1050 


1117 


1383 


1450 


1517 


10 


1000 


1074 


1148 


1222 


1296 


1593 


1667 


1741 


11 


1161 


1243 


1324 


1406 


* 1487 


1813 


1894 


1976 


12 


1333 


1422 


1511 


1600 


1689 


2044 


2133 


2222 


13 


1517 


1613 


1709 


1806 


1902 


2287 


2383 


24180 


14 


1711 


1815 
2028 


1919 


2022 


2126 


2541 


2644 


2748 


15 


1917 


2139 


2250 


2361 


2806 


2917 


3028 


16 


2133 


2252 


2370 


2489 


2607 


3081 


3200 


3319 


17 


2361 


2487 


2613 


2739 


2865 


3369 


3494 


3620 


:l8 


2600 


2733 


2867 


3000 


3133 


3667 


3800 


3933 


19 


2850 


2991 


3131 


3272 


3413 


3976 


4117 


4257 


20 


3111 


3259 


3407 


3556 


3704 


4296 


4444 


4593 


21 


8383 


3539 


3694 


3850 


4006 


4628 


4783 


4939 


22 


3667 


3830 


3993 


4156 


4319 


4970 


5133 


5296 


23 


3961 


4131 


4302 


4472 


4642 


5324 


5494 


5665 


24 


4267 


4444 


4622 


4800 


4978 


5689 


5867 


6044 


26 


4583 


4769 


4954 


5139 


5324 


6065 


6250 


6435 


26 


4911 


5104 


5296 


5489 


5681 


6452 


6644 


6837 


2? 


5250 


5450 


5650 


5850 


6050 


6850 


7050 


7250 


28 


5600 


5807 


6015 


6222 


6430 


7259 


7467 


7674 


29 


5961 


6176 


6391 


6606 


6820 


7680 


7894 


8109 


80 


6333 


6556 


6778 


7000 


7222 


8111 


8333 


8556 


81 


6717 


6946 


7176 


7406 


7635 


8554 


8783 


9013 


32 


7111 


7348 


7585 


7822 


8059 


9007 


9244 


9481 


83 


7517 


7761 


8006 


8250 


8494 


9472 


9717 


9961 


84 


7933 


8185 


8437 


8689 


8941 


9948 


10200 


10452 


85 


8361 


8620 


8880 


913S 


9398 


10435 


10694 


10954 


86 


8800 


9067 


9333 


9600 


9867 


10933 


11200 


11467 


87 


9250 


9524 


9798 


10072 


10346 


11443 


11717 


11991 


88 


9711 


9993 


10274 


10556 


10837 


11963 


12244 


12526 


89 


10183 


10472 


10761 


11050 


11339 


12494 


12783 


13072 


40 


10667 


10963 


11259 


11556 


11852 


13037 


13333 


13630 


41 


11161 


11465 


11769 


12072 


12376 


13591 


13894 


14198 


42 


11667 


11978 


12289 


12600 


12911 


14156 


14467 


14778 


43 


12183 


12502 


12820 


13139 


13457 


14731 


15050 


15369 


44 


12711 


13037 


13363 


13689 


14015 


15319 


15644 


15970 


45 


13250 


13588 


13917 


14250 


14583 


15917 


16250 


16583 


46 


13800 


14141 


14481. 


14822 


15163 


16526 


16867 


17207 


47 


14361 


14709 


15057 


15406 


15754 


17146 


17494 


17843 


48 


14933 


15289 


15644 


16000 


16356 


17778 


18133 


18489 


49 


15517 


15880 


16243 


16606 


16969 


18420 


18783 


19146 


50 


16111 


16481 


16852 


17222 


17593 


19074 


19444 


19815 


51 


16717 


17094 


17472 


17850 


18228 


19739 


20117 


20494 


62 


17333 


17719 


18104 


18489 


18874 


20415 


20800 


21185 


53 


17961 


18354 


18746 


19139 


19531 


21102 


21494 


21887 


54 


18600 


19000 


19400 


19800 


20200 


21800 


22200 


22600 


55 


19250 


19657 


20065 


20472 


20880 


22509 


22917 


23324 


56 


19911 


20326 


20741 


21156 


21570 


23230 


23644 


24059 


57 


20583 


21006 


21428 


21850 


22272 


23961 


24383 


24805 


58 


21267 


21696 


22126 


22556 


22985 


24704 


25133 


25563 


59 


21961 


22398 


22835 


23272 


23709 


25457 


25894 


26331 


60 


22667 


23111 


23556 


24000 


24444 


26222 


26667 


27111 



806 



TABLE XVII. CORRECTIVE PERCENTAGE FACTORS FOE 

TABLES OF LEVEL SECTIONS. 

To be applied when cross-sections are not level. See § 95. 

Side slope = 1.5:1 or /? = 33°4r. 



Trans- 


6=12 feet 


6=20 feet 


6=30 feet 


verse 








surface 


and d= 


and d— 


and d= 


slope. 








^o 


Per- 


10 


20 


50 


10 


20 


50 


10 


20 


50 


a 


cent 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 






% 


% 


% 


/o 


% 


% 


% 


% 


% 


5 


9 


1.9 


1.8 


1.8 


2.1 


1.8 


1.8 


2.3 


2.0 


1.8 


10 


18 


8.2 


7.7 


7.5 


9.0 


8.0 


7.6 


10.0 


8.4 


7.7 


15 


27 


21 


20 


19 


23 


21 


20 


28 


22 


20 


20 


36 


4^ 


44 


43 


51 


45 


44 


57 


48 


44 


30 


57 


327 


324 


317 


358 


336 


321 


400 


354 


326 



Side slope = 1 : 1 or /? = 45°. 



Trans- 


6=12 feet 


6=20 feet 


6=30 feet 


verse 








Burface 


and d= 


and d= 


and d= 


elope. 








_,o 


Per- 


10 


20 


50 


10 


20 


50 


10 


20 


50 


or 


cent 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 


feet. 






% 


% 


% 


% 


% 


% 


% 


% 


% 


5 


P 


0.9 


0.8 


0.8 


1.0 


0.9 


0.8 


1.2 


0.9 


0.8 


10 


18 


3.7 


3.4 


3.2 


4.3 


3.6 


3.3 


5.0 


4.0 


3.4 


15 


27 


9.0 


8.2 


7.8 


10.3 


8.7 


8.0 


12.1 


9.5 


8.2 


20 


36 


18 


16 


15 


20 


17 


16 


24 


19 


16 


30 


57 


58 


53 


50 


67 


56 


51 


78 


61 


53 



807 



m 






c 



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808 



INDEX. 

Numbers refer to sections except where specifically marked pages (p.). 

Abandonment of existing track 534, c 

Abutments for trestles 176 

Accelerated motion, application of laws to movement of trains 514 

Acceleration-speed curves 462 

Accidents, danger of, due to curvature 507 

Accuracy of earthwork computations 125 

numerical example 117 

tunnel surveying 197 

Additional business; methods of securing (or losing) it 532 

Adhesion of wheels and rails 421, 422 

Adjustments of dumpy level — Appendi'x pp. 603, 604 

instruments, general principles — Appendix p. 596 

transit — Appendix pp. 597-601 

wye level — Appendix pp. 601-603 

Advance signals, in block signaling 391 

Advantages of re-location of old lines 533 

tie-plates 286 

Air-brakes 424, 425 

Air resistance — see Atmospheric resistance. 

Allowance for shrinkage of earthwork 128 

Alternating current, used in signaling 394 

American locomotives, frame 401 

equalizing levers 412 ' 

wheel base 400 

Rwy. Eng. Assoc, formula for train resistance 439 

system of tunnel excavation 205 

Aneroid barometer, use in reconnoissance leveling 7 

Angle-bars, angles and dimensions of standard designs — Table XXIV p. 319 

cost 447, d 

efficiency of 280 

for various weights of rail — Table XXXII p. 496 

number per mile of track — Table XXXV p. 497 

standard 284 

Angle of slope in earthwork 90 

Annual charge against a tie, at 5% interest — Table XVIII p. 808 

Antiseptics 38 

Appliances, medical, surgical 37 

Apprehension of danger, effect on travel 508, c 

ARCH CULVERTS ^226, 227 

design , 226 

example 227 

809 



810 INDEX. 

Area of culverts, computation. 212, 217 

A. S. C. E. standard rail sections 267 

Ash pits 363 

Asphyxiation, treatment ' 42 

Assistant engines — see Pusher engines and Pusher grades. 

Atmospheric resistance, train 430 

Atlantic locomotives, wheel base 400 

Austrian system of tunnel excavation 205 

Automatic air-brakes , . . . 425 

signaling, track circuit 394 

stokers 407 

Averaging end areas, volume of prismoid computed by 101 

Axle, effect of parallelism 396 

effect of rigid wheels on , r 395 

radial, possibilities of 397 

size of standard M.C.B 420 

" Bad order " tracks 378 

Balance of grades for unequal traffic . . 529-531 

determination of relative traffic. . 531 

general principle 529 

' theoretical balance 530 

Baldwin Locomotive Works formula for train resistance 438 

BALLAST— Chap. VII. 

cost.. 238, 447, a 

cross-sections ; 233 

laying 237 

materials 232 

proper depth , 236 

Banjo signals, in block signaling 392 

Barometer, reduction of readings to 32° F. — Table XI p. 783 

use of aneroid in reconnoissance leveling 7 

Barometric elevations — Table XII ' p. 784 

coefficients for corrections for temperatures and 

humidity — Table XIII p. 784 

Beams, strength of stringers considered as . . 190 

Bearings, compass, use as check on deflections 20, 21 

in preliminary surveys 11 

Beds, camping 33 

Belgian system of tunnel excavation 205 

Belpaire fire-box 403 

Blasting 149-155 

use in loosening earth 138 

Bleeding, treatment , 40 

BLOCK SIGNALING— Chap. XIV. 

*• Body tracks " , 378 

Boiler compounds 323 

for locomotive 402- 404 

Boiler-power of locomotives, relation to tractive and cylinder power. . . 414 
Bolts — see Track bolts. 

Bonds of railroads, security and profits. 469 

Borrow-pits, earthwork 120 

Bowls (or pots) as rail supports 239, 263 



INDEX. 811 

Box-cars, size and capacity 416 

Box culverts 222-224 

old-rail 224 

stone 223 

wooden 222 

Bracing for trestles 174, 175 

design 193 

Brakes — see Train-brakes. 

Brake resistances 434 

Bridge joints (rail) 282 

spirals 5 

warning 375 

Bridges and culverts, cost of repairs and renewals — Table XLI.. pp. 540, 541 

Bridges of standard dimensions for small spans 230 

in block signaling 392 

Bridges, trestles, and culverts on railroads, cost 446 

Broken-stone ballast 232 

Burnettizing (chloride-of-zinc process) for preserving timber 251 

Burnt clay ballast 232 

Capital, railroad, classification of 469 

returns on 469, 470 

Caps (trestle) , design 192 

Car mileage, nature and cost — Table XLI, pp. 540, 541, and 495 

Cars 416-420 

brake-beams 418 

capacity and size 416 

cost of renewals and repairs — Table XLI pp. 540, 541 

draft gear 419 

gauge of wheel and form of wheel-tread 420 

stresses in car frames 417 

truck frames 418 

use of metal 418 

wheels, kinetic energy of 435 

Cars and horses, use in earthwork 140, e 

and locomotives, use in earthwork 140, / 

Carts and horses, use in earthwork 140, a 

Cattle guards 228 

passes 229 

Center of gravity of side-hill sections, earthwork 123 

Central angle of a curve 51 

Centrifugal force, counteracted by superelevation of outer rail 71, 72 

of connecting-rod, etc., of locomotive 413 

Chairs as supports for double-headed rails 267 

Chats for ballast 232 

Chemical composition of rails 273, 274 

purification of water 321 

Chert for ballast 232 

Cinders for ballast 232 

Circular lead rails for switches 304 

Classification of excavated material 156 

railroads 234 

Cleaning, mechanical, locomotive boilers 320 



812 INDEX. 

Clearance card in permissive block signaling 388 

spaces in locomotives 410 

Clearing and grubbing for railroads, cost 444 

Clothing, surveying parties 35 

Coal consumption in locomotives 407 

effect of increasing rate 457 

^ varying quality 458 

per car-mile 407 

Coaling stations — see Locomotive coaling stations. 

Columbia locomotives, wheel base 400 

Compass, use of, in preliminary surveys 11 

Competitive traffic 498 et seq. 

Competitive rates, equality, regardless of distance 499 

Compensation for curvature 510, 511 

rate 511 

reasons 510 

rules for 511 

Compensators in block signaling 393 

Compound curves 67-70 

modifications of location 69 

nature and use 67 

mutual relations of the parts 68 

Compound sections, earthwork 91 

Computation of earthwork 101-128 

approximate, from profiles 126 

using a slide rule 106 

Concrete pipe culverts 221 

Conducting transportation, cost of 489-495 

Coning wheels, effect 397 

Connecting curve from a curved track to the inside 310 

from a curved track to the outside 309 

from a straight track 308 

Consolidation locomotives, equalizing levers 412 

frame 401 

wheel-base 400 

Constants, numerical, in common use — Table XV p. 787 

Construction of tunnels 203-208 

Contours, obtained by cross-sectioning 12 

Contractor's profit, earthwork 147 

Control points, in general route for a railroad 2 

Cooking utensils, camping 29 

Corbels for trestles 178 

Cost of ballast 238 

of blasting 155 

of chemical treatment of timber 256 

of earthwork , . , r 137 et seq. 

of framed-timber trestles 184 

of metal ties. 262 

of pile trestles 168 

COST OF RAILROAD^.— Chap. XVII. 

detailed estimate 451 

Cost of raUs.. 278 



INDEX. 813 

Cost t)f station buildings 329 

of ties ' 248 

of treating wooden ties 256 

of tunneling 209 

Counterbalancing for locomotives 413 

Crawford's formula for train resistance 438 

Creosoting for preserving timber 250 

Cross-country routes — reconnoissance 4 

Crossings, one straight, one curved track 316 

two curved tracks , 317 

numerical example 317 

two straight tracks 315 

Cross-over between two parallel curved tracks, straight connecting 

curve 312 

straight tracks 311 

Cross-sectioning, for earthwork computations 98 

for preliminary surveys 12 

irregular sections for earthwork computations 118 

Cross-sections of ballast 233, 235 

of tunnels 198 

Cross-ties — see Ties. 

Crown-bars in locomotive fire-box 403 

Cubic yards per 100 feet of level sections — Table XVII pp. 805-807 

CULVERTS ANJ> MINOR BRIDGES.— Chap. VI. 

Culverts, arch 226, 227 

area of waterway 212-217 

iron-pipe .' 220 

old-rail 224 

reinforced-concrete 225 

stone box 223 

tile-pipe 221 

wooden box 222 

CURVATURE.— Chap. XXII. 

compensation for 510, 511 

correction for, in earthwork computations.. . 121-124 

danger of accident due to 507 

effect on travel 508 

extremes of sharp 512 

general objections 506 

of existing track, determination. 65 

proper rate of compensation 511 

Curve, elements of a 1° 53 

location by deflections 55 

by middle ordinates 59 

by offsets from long chord 60 

by tangential offsets 58 

by two transits 57 

notation, alinement curves 50 

resistance of trains 395, 396, 433 

Curves, compound, — see Compound curves. 

elements of 51 

instrumental work in location. . 55 



814 INDEX. 

Curves, limitations in location 64 

method of computing length 49 

metric 47 

modifications of location 63 

mutual relations of elements ^ . , , 52 

obstacles to location 62 

simple, method of designation 46 

transition — see Transition curves. 

use and value of other methods of location (not using a transit) 61 
vertical — see Vertical curves. 
Cylinder power of locomotives, relation to boiler and tractive power. . 414 

Dating nails, for marking ties 247 

Deflecting rods for operating block signals 393 

Deflections for a transition curve 78 

Degree of a curve 46 

Design of culverts 211 et seq. 

framed trestles 185-193 

bracing I93 

caps and sills 192 

floor stringers 190 

posts 191 

nutlocks 295 

pile trestles 165 

tie-plates 287 

track bolts 294 

tunnels 202 

distinctive systems 205 

Development, definition 5 

example, with map 5 

methods of reducing grade 5 

Disadvantages of re-location of old lines 534 

Diseases, medicines 41 

DISTANCE.— Chap. XXI. 

effect of change on business done 505 

on division of through rates 500 

justification of decrease to save time 504 

relation to rates and expenses 496 

Distant signals in block signaling 390 

Ditches to drain roadbed 94 

Dividends actually paid on railroad stock 469 

Double-ender locomotives, wheel-base 400 

Double-track, distance between centers 92 

Draft gear 419 

*• continuous" 419 

Drainage of roadbed, value of 94, 95 

Drains in tunnels 202 

Draw-bar pull, locomotives 456 

Draw-bars 419 

Drifting, locomotives, relation to speed curves 464 

Drilling holes for blasting 150, 151 

Drinking water, camping parties 39 

Driving-wheels of locomotives 40(f 



INDEX. 815 

Driving-wheels of locomotives, section of 413 

Drop tests for train resistance 437 

Drowning, treatment 42 

Durability of metal ties 259 

rails 275, 276 

wooden ties 242 

Dynamometer tests of train resistance 436 

Earnings of railroads, estimation of 473 

per mile of road 473 

EARTHWORK.— Chap. III. 

Earthwork computations, accuracy 125 

appi:oximate computations from profiles 126 

level sections, approximate volume 102 

numerical example 103 

probable error 116 

relation of actual volume to numerical results 96 

simple approximations 101 

Earthwork, cost 137 et seq., 445 

limit of free haul 136 

method of computing haul 130 et seq. 

shrinkage 127 

surveys 96-100 

Eccentricity of center of gravity of earthwork cross-section 122 

Economics, railroad, nature and limitations 478 

of ties 240 

of treated ties 257 

Efficiency, loss, in steam pressure 410 

Electric shock, treatment 42 

Elements of a 1° curve 53 

simple curve 51 

transition curves — Table IV pp. 621-623 

Embankments, method of formation 129 

usual form of cross-section 88 

Empirical formulae for culvert area 214 

accuracy required 217 

value 215 

Engine-houses for locomotives 341-355 

doors 342 

drop pits 348 

electric lighting 351 

engine pits 345 

floors 347 

form 341 

heating 349 

hoists 354 

length 343 

materials of construction 344 

piping 352 

smokejacks 346 

tools 353 

turntables 355 

window lighting 350 



816 



INDEX. 



Engineering, proportionate and actual cost, in railroad construction. . . 442 

Engineering News formula for pile-driving 163 

for train resistance 438 

Engineer's duties in locating a railroad 479 

Engine-houses for locomotives 341-355 

Enginemen, basis of wages 490 

English system of tunnel excavation : . 205 

Enlargement of tunnel headings 204 

Entrained water in steam 410 

Equalizing-levers on locomotives 412 

Equivalent sections in earthwork, determination of area 104 

Estimation of probable volume of traffic and of probable growth 473 

Evaporation per pound of fuel — Table XXXVI, and 452 

Excavation, usual form of cross-section 88 

Exhaust-steam, effect of back-pressure 410 

Expansion of rails 271 

Explosives, amount used 152 

firing 154 

tamping 153 

use in blasting 149 

Expenditure of money for railroad purposes, general principles 477 

External distance, simple curve 51 

table of, for a 1° curve — Table II pp. 616-618 

Facilities, traffic, effect of increase 475 

Factors of safety, design of timber trestles 189 

Failures of rail joints 283 

Fastenings for metal cross-ties 261 

Fences • 366-371 

braces 369 

concrete posts 370 

construction details 371 

posts 368 

types 367 

wire fences 366 

Field work for locating a simple curve 56 

a spiral 80 

Fire-box of locomotive 403 

area of grate 404 

Fire-brick arches in locomotive fire-box 403 

Fire protection on trestles 182 

Fixed charges, nature and ratio to total disbursements 480 

Flanges of wheels, form . 420 

Flanging locomotive driving-wheels, effect 398 

Floor systems for trestling 177-184 

Foaming and priming, in locomotive boilers 322 

Formation of embankments, earthwork 127-129 

railroad corporations, method 468 

Formulse for pile-driving 163 

required area of culverts 214 

train resistance 438 

trigonometrical — Table XIV pp. 785, 786 

useful, and constants — Table XV p. 787 



INDEX. 817 

Forney's formula for train resistance 438 

Fouling point of a siding 394 

Foundations for framed trestles 173 

Fractures, bone, treatment 43 

FRAMED TRESTLES 169-193 

abutments 176 

bracing 174, 175 

cost 184 

design 169, 185-193 

foundations 173 

Joints 170 

multiple story construction 171 

span 172 

Frame of locomotive, construction 401 

Free haul of earthwork, limit of ^ 136 

Freight houses 330-339 

dimensions 332 

doors 335 

fire risk 331 

floors 334 

lighting 337 

platforms 333 

ramps 339 

roofs 336 

scales 338 

two types, in-bound, out-bound 330 

Freight yards 378-383 

general principles 379 

minor yards 381 

relation of yard to main track 380 

track scales 382 

transfer cranes , 382 

French system of tunnel excavation 205 

Friction, laws of, as applied to braking trains 422 

Frogs, diagrammatic design 297 

for switches 297, 298 

to find frog number 298 

trigonometrical functions — Table III pp. 619, 620 

Fuel for locomotives, cost of 491, and Table XLI. . pp. 540, 541 

pumps and engines, cost — Table XXVIII p. 374 

Gauge of wheels, form of wheel-tread 420 

German system of tunnel excavation 205 

GRADE.— Chap. XXIII. 

(see Pusher grades. Ruling grades.) 

accelerated motion of trains on 514 

distinction between ruling and minor grades 513 

effect on tractive power of locomotives 461 

in tunnels 199 

line, change in, based on mass diagram 135 

resistance of 432 

starting resistance at stations, reduction 537 

undulatory, advantages, disadvantages, and safe limits 518 



818 INDEX. 

Grade, virtual 515 

use, value, and misuse 517 

Grade resistance of trains 432 

Gravel ballast , 232 

Gravity tests of train resistance 437 

Grate area of locomotives , 404-409 

ratio to total heating surface 409 

Gravity, effect on trains on grades 432 

tests of train resistance 437 

Ground levers for switches 301 

Growth of railroad traffic 473 

affected by increase of facilities 475 

Guard rails for switches 303 

for trestles 179 

Guides around curves and angles (signaling mechanism) 393 

Gumbo, used for ballast 232 

Gunpowder pile-drivers 162 

Hand-brakes 423 

Haul of earthwork, computation of length 130 e« seq. 

cost 140, 148 

limit of profitable 148 

method, depending on distance hauled 141 

Headings in tunnels 203 

Heating surface in locomotives 409 

Henderson's formula for train resistance 438 

Hoosac Tunnel, surveys for 194, 197 

Hump yards 379 

I-beam bridges, standard 230 

IMPROVEMENT OF OLD LINES.— Chap. XXIV. 

classification 532 

Inertia resistances 435 

Insect bites, treatment 44 

Instrumental work in locating simple curves 56 

spirals 80 

Interest on cost of railroads during construction 449 

Iron pipe culverts 220 

Irregular prismoids, volume 108 

numerical example 109 

sections in earthwork, computation of area 107 

Joints, framed trestles 170 

rail 279-285 

Journar friction of axles 431, h 

Kinetic energy of trains '. 514 

Kyanizing (bichloride-of-mercury or corrosive sublimate process) for 

preserving timber 252 

Land and Land damages, cost 443 

Lateral bracing for trestles 175 

Length of rails 270 

a simple curve 49 

a spiral 81. 83 

Level, dumpy, adjascmento of — Appendix p. 603 

wye, adjustments oi— Appeiidix , , , , , p. 601 



INDEX. 819 

ft 

Leveling, location surveyg . 20 

Level sections, volume of prismoids surveyed as 102 

numerical example 103 

Life of locomotives 415 

Limitations in location of track 64 

of maximum curvature 512 

Lining of tunnels 200 

Loading earthwork, cost 139 

of trestles 188 

Local traffic, definition and distinction from through 498 

Location of stations at distance from business centers, effect 476 

Location Surveys — paper location 18 

surveying methods 20 

Locomotive coaling stations 356-359 

coal conveyors 359 

coaling trestles 358 

hand shoveling 356 

locomotive crane 357 

rating 467 

resistances 429 

Locomotives, cost of renewals and repairs 488 

general structure 401-414 

life of 415 

resistance — Table XXIX p. 374 

types permissible on sharp curvature 509, b 

Logarithmic sines and tangents of small angles — Table VI ... . pp. 644 646 
sines, cosines, tangents, and cotangents — Table VII. pp. 647-691 
versed sines and external secants — Table VIII ... pp. 692-736 

Logarithms of numbers — Table V pp. 624 643 

Long chords for a 1° curve — Table II pp. 616 618 

of a simple curve 51 

Longitudinal bracing of a trestle 174 

Longitudinals (rails) 239, 264 

Loop— see Spiral. 

Loosening earthwork, cost 138 

Loss in traffic due to lack of facilities 476 

Lundie's formula for train resistance 438 

Maintenance of equipment, as affected by pusher engines 528 

cost of 488 

Maintenance of way as affected by pusher engines 528 

cost of 485-487 

Mallet locomotives, wheel-base 400 

Map chest, for field parties 31 

Maps, use of, in reconnoissance 1, 6 

Mass curve, area 133 

properties 132 

diagram, effect of change of grade line 135 

haul of earthwork 131 

value 134 

Mathematical design of switches 304-312 

Measurements, location surveys. ' 21 

Mechanism of brakes 423-425 



820 ^ INDEX, 

Medical and surgical treatment 36-45 

METAL TIES — see Ties, metal 258-263 

Metric curves 47 

Middle ordinate of a simple curve 51 

Mileage, car 495 

locomotives, average annual , 415 

Mikado locomotive, power of one typical engine under various con- 
ditions 465 

wheel-base 400 

Minor openings in roadbed 228-230 

Minor stations, rooms required, construction 329 

MISCELLANEOUS STRUCTURES AND BUILDINGS.— Chap. XII. 

Modifications in location, compound curves 69 

simple curves 63 

Mogul locomotives, wheel-base 400 

Monopoly, extent to which a railroad may be such 471 

Mountain routes — reconnoissance 5 

" Mud '» ballast 232 

sills, trestle foundations 173, 6 

Multiple story construction for trestles 171 

Myer's formula for culvert area 214 

Natural sines, cosines, tangents, and cotangents — Table IX . . . pp. 737- 759 

versed sines and external secants — Table X pp. 760-782 

Non-competitive traffic, definition 498 

effect of variations in distance 502, 503 

extent of monopoly 471 

Notes — form for cross-sectioning 12 

location surveys 21 

reconnoissance 7 

Number of a frog, to find 298 

of trains per day, probable 474 

Nut-locks, design 295 

N. Y. Central formula for train resistance 438 

Obstacles to location of trackwork 62 

Obstructed curve, in curve location 62, c 

Odometer, use in reconnoissance 8 

Oil-burning locomotives 408 

houses 360 

Old-rail culverts 224 

Open cuts vs. tunnels 208 

OPERATING EXPENSES.— Chap. XX. 

detailed classification — Table XLI. pp. 540, 541 

per train mile 481 

reasons for uniformity per train mile 482 

Operation of trains, effect of curvature on ... 509 

Oscillatory and concussive velocity resistances, train i 430 

Ordinates of a spiral 78 

Paper location in location surveys 18 

preparation of notes for field-work 19 

Physical tests of steel splice bars 285 

steel rails .: S 274-14 

Picks, use in loosening earth ^ 138, h 



INDEX. 821 

Pae bents 161, 165 

driving 162, 167 

driving formulse 163 

points and shoes 164 

trestles, cost 168 

design , 165 

PILE TRESTLES 161-168 

Piles, timber, specifications 166 

Pilot truck of locomotive, action 399 

PIPE CULVERTS 218-221 

advantages 218 

construction 219 

iron 220 

tile 221 

Pipe compensator, 393 

Pipes, use in block signaling 393 

Pit cattle guards 228 

Platforms, station 328 

Ploughs, use in loosening earth 138, a 

Point of curve 51 

inaccessible, in curve location 62, h 

Point of tangency 51 

inaccessible, in curve location 62, 6 

Point-rails of switches, construction 300 

Point-switches 300 

Pony truck of locomotive, action 399 

Portals, tunnels, methods of excavation 207 

Posts, trestle, design of 191 

Pounds of steam per I.H.P. hour at various cut-offs 455 

per pound of coal 452 

POWER OF A LOCOMOTIVE.— Chap. XVIII. 

Preliminary financiering of railroads, Chap. XIX, and 441 

Preliminary surveys — cross-section method 11 

"first" and "second" 17 

general character 10 

value of re-surveys at critical points 17 

Preservative processes for timber, cost 256 

general principle 249 

methods 249-255 

Prismatic compass, use in reconnoissance 8 

Prismoidal correction for irregular prismoids, approximate value 115 

in earthwork computations, comparison of exact 

and approximate 

methods 116, 117 

for equivalent sections 113 
for irregular sections. 115 

for level sections 112 

for three-level sec- 
tions 114 

for triangular pris- 

moid Ill 

formula, proof » 110 



822 INDEX. 

Prismoids, in earthwork computations 97 

Profit and loss, dependence on business done 472 

small margin between them for railroad promoters. . . . 470 
Profits (and security) in the two general classes of railroad obligations. 469 

Profit, in earthwork operations 147 

PROMOTION OF RAILROAD PROJECTS.— Chap. XIX. 

Provisions, for camping 32 

Pumping, for locomotive water-tanks 325, 326 

Pusher grades 523-528 

comparative cost 528 

general principles 523 

required balance between through and pusher grades. . . 524 

required length 527 

Pusher engines, cost per mile — Table XLIV p. 583 

operation 526 

service 528 

Radial stays, in locomotive boilers 403 

Radiation from locomotives 410 

into the exhaust-steam 410 

Radii of curves — Table I pp. 612-615 

Rail braces 286 

expansion, resistance at joints and ties to free expansion 293 

RAIL FASTENINGS.— Chap. X. 

Rail gap, effect of, at joints 281 

joints 279-285 

effect of rail gap 281 

efl&ciency of angle-bar 280 

failures 283 

later designs 284 

specifications 285 

"supported" 280. 282 

"suspended" 280, 282 

theoretical requirements for perfect 279 

sections 266, 267 

A.S.C.E 267 

"bridge" 266 

"bull-headed" . 266, 267 

compound . 281 

"pear" section. 266 

radius of upper corner, effect 267 

reversible 267 

"Stevens" 266 

" Vignoles" 266 

wear, experimental determination 277 

RAILS.— Chap. IX. 

angles and dimensions of standard designs, Table XXIII ... p. 299 

branding 274-11 

cast-iron '. 266 

cost 278, 447c 

cost of renewals of 278, 485 

chemical composition 273, 274 

effect of stiffness on traction 269 



INDEX. 823 

RailSi expansion 271 

stresses caused by prevention of expansion 271 

rules for allowing for. 272 

inspection 274-17 

length 270 

allowable variation 274-8 

45- and 60-foot rails 270 

No. 2 274-16 

relation of weight, strength, and stiffness 269 

temperature when exposed to sun 272 

testing 274 

tons per mile — Table XXXI 447, c 

wear on curves 276, 277 

tangents , . . 275, 277 

weight, allowable variation ' . . 274-7 

for various kinds of traffic 268, 447 

Rates based on distance, reasons 497 

through, method of division of 499 

Rating of locomotives 467 

Receipts (railroad) , effect of distance on 498-505 

Reconnoissance over a cross-country route 4 

surveying, leveling methods 7 

surveys 1-9 

character of 1 

cross-country route 4 

distance measurements 8 

mountain route " 5 

selection of general route 2 

value of high grade work 9 

through a river valley 3 

Reduction of barometer reading to 32° F. — Table XI . ' p. 783 

Reheaters, in locomotives 406 

Reinforced-concrete culverts 225 

ties 265 

Renewal of rails, cost of 485 

of ties, cost of 246, et seq. 485 

regulations governing it 246 

Repairs and renewals of locomotives, cost 488 

Repairs of roadway, cost of 486 

Repairs, wear, depreciation, and interest on cost of plant: cost for earth- 
work operations 145 

Replacement of a compound curve by a curve with spirals 83 

simple curve by a curve with spirals 81 

Requirements, nut-locks 295 

perfect rail-joint 279 

spikes 289 

track-bolts 293 

Resistances internal to the locomotive 429 

(see Train Resistance.) 

Retardation-speed curves 463 

Revenue, gross, distribution of 480 

Roadbed, form of subgrade 93 



824 INDEX. 

Roadbed, width for single and double track. 92 

Roadway, cost of repairs of 486 

as affected by pusher engines 528 

Roadways, earthwork operations, cost of keeping in order 143 

Rock ballast 232-236 

Rock cuts, compound sections 91 

Rolling friction of wheels 431, a 

ROLLING STOCK.— Chap. XIV. 

Rotative kinetic energy of wheels of train 435, 514 

Route, selection as affected by locomotive power 466 

Rules for switch-laying 313 

Ruling grades 519-522 

choice of 520 

definition 3, 519 

proportion of traffic affected by 522 

Run-oflf for elevated outer rail 73 

Sand houses 362 

used for ballast 232 

Scales, track 383 

Scrapers, use in earthwork 140, d 

Screw-spikes, as rail-fastenings 291 

Section houses, value, construction 340 

tool houses , 361 

Selection of a general route for a railroad 2 

Semaphore boards, in block signaling 392 

Setting tie-plates, methods 288 

Shafts, tunnel, design 201 

surveying 195 

Shifting centers for locomotive pilot trucks, action 399 

Shoveling (hand) of earthwork, cost 139, a 

(steam) of earthwork, cost 139, h 

Shrinkage of earthwork 127 

allowance 128 

Side-hill work, in earthwork computations 119 

correction for curvature 122 

Signaling, block, " absolute" blocking 388 

automatic 389 

manual systems 386-388 

permissive 388 

Signals, mechanical details 392 

Signs 372-374 

division posts 374 

highway signs 372 

marker posts 374 

mile posts 374 

trespass signs 373 

whistle signs 374 

Sills for trestles, design 192 

Simple curves 46-66 

Skidding of wheels on rails , 421-422 

Slag, used for ballast 232 

Slide-rule, in earthwork computations IQO 



INDEX. 825 

Slipping of wheels on rails, lateral 396 

longitudinal 395 

Slips, for switchwork 314 

Slopes in earthwork, for cut and fill 90, 92 

effect and value of sodding 95 

Slope-stake rod, automatic 100 

Slope-stakes, determination of position 99 

Smith's formula for train resistance • 433 

Snake bites, treatment 44 

Snow fences 364 

sheds 365 

Sodding slopes, effect and value . 95 

Spacing of ties 244 

Span of trestles 172 

Specifications for earthwork 157 

steam shoveling, earthwork 139, b 

steel rails ^ 274 

steel splice-bars '. 285 

timber piles 166 

wooden ties 245 

Speed of trains, reduction due to curvature 508, a 

relation to superelevation of outer rail. . 71, 72 

relation to tractive adhesion 422, e 

Spikes 289, 291 

cost 447. d 

driving 290 

number per mile of track — Table XXXIII 447, d 

screw 291 

requirements in design 289 

"wooden" for plugging spike-holes 292 

Spirals, bridge and tunnel 5 

(see Transition Curves.) 
Splice-bars — see Angle-bars. 

Split stringers, caps, and sills 161, 177 

Sprague*s formula for train resistance 438 

Spreading earthwork, cost 142 

Stadia method for preliminary surveys 13 

form of notes 14 

for reconnoissance 8 

methods of work 13 

organization of party 13 

reduction of observations 15 

vs. cross-section method 16 

Stand pipes, for locomotive water supply 327 

Starting grade at stations, reduction of 537 

Station buildings, cost 329 

platforms 328 

Staybolts for locomotive fire-boxes 403 

Stays, in locomotive fire-box 403 

Steam pile-drivers 162 

Steam-shoveling of earthwork 139, 6 

weight per foot of stroke.— Table XXXVII and. . . , 454 



826 INDEX. 

Stiffness of rails, effect on traction 269 

Stocks of railroads, security and profits 369 

Stone ballast 232v233 

box culverts 223 

foundations for framed trestles 173, c 

Straight connecting curve between two parallel curved tracks 312 

from a curved main track 310 

Strength of timber 187 

factors of safety 189 

required elements for trestles 186 

Stringer bridges, standard, steel '. . . . 230 

Stringers, design 190 

for trestle floors ' 177 

Stub-switches 299 

Subchord, length 48 

Subgrade, of roadbed, form , 93 

Superelevation of the outer rail on curves, L. V. R. R. run-off 73 

on trestles 181 

practical rules 72 

standard on N. Y. N. H. 

& H. R. R 72 

Table XIX p. 83 

theory ! 71 

Super-heaters, in locomotives 405 

Superintendence, cost in earth operations 146 

Supported rail-joints 282 

Surface cattle guards 228, b 

surveys for tunneling : . 194 

Surveying parties, maintenance 23-35 

number of men required 22 

Surveys and engineering expenses for railroads, cost •. 442 

accuracy ^. 197 

for tunneling '. 194-197 

with compass 11 

Suspended rail-joints 282 

Swinging pilot truck on locomotive 399 

Switchbacks 5 

Switch construction 296-303 

essential elements 296 

frogs 297, 298 

guard rails 303 

point 300 

stands 301 

stub ., 299 

tie rods 302 

SWITCHES AND CROSSINGS.— Chap. XI. 

Switches, curved lead rail, rectangular coordinates — Table XXV. . . p. 358 

mathematical design 304-307 

using circular lead rails 304 

using straight frog rails and straight 

point rails 305 

Switching engines, wheel-bases 400 



iNpEX. 827 

Switching engines, used in pusher-engine service 526 

Switch leads and distances — Table III pp. 619, 620 

laying, practical rules 313 

slips ^ 314 

stands 301 

Tables, dining, camp 28 

drawing, camp 30 

Talbot's formula for culvert area 214 

Tamping for blasting 153 

Tangents for a 1° curve — Table II pp. 616-618 

Tangent distance, simple curve 51 

Tanks, water, for locomotives 324 

capacity of cylindrical tanks — Table XXVII and 324 

track 326 

Temperature allowances, while laying rails 272 

Ten-wheel locomotives, wheel-base 400 

Telegraph lines for railroads, cost 450 

Tent floors 26 

stoves 27 

Tents 25 

TERMINALS.— Chap. XIII. 

inconvenient, resulting loss 476 

justification for great expenditures 476 

Terminal pyramids and wedges, in earthwork 89 

Tests for splice bars 285 

for rails 274 

to measure the efficiency of brakes 426 

Three-level sections in earthwork, determination of area 105 

numerical example 105 

Throw of a switch 304 

Through traflSic, definition 498 

division of receipts between roads 499 

effect of changes in distances on receipts 500 

Tie-plates 286-288 

advantages 286 

elements of design 287 

method of setting 288 

Tie rods, for switches 302 

TIES.— Chap. VIII. 

cost of renewal of 246 et seq. 

metal, 258-263 

cost 262 

durability 259 

extent of use 258 

fastenings 261 

form and dimensions 260 

number per mile of track — Table XXX p. 494 

number and value, used in U. S. in 1912 — Table XXII. ... p. 217 

on trestles 180 

reinforced concrete 265 

wooden, preservative processes 249-256 

regulations for relaying 246 



828 INDEX. 

Ties, wooden 241-257 

choice of wood 241 

construction 245 

cost 249, 447, b 

dimensions 243 

durability 242 

economics 240 

quality of timber 245 

spacing 244 

specifications 245 

Tile drains, to drain roadbed 94 

pipe culverts 221 

Timber, choice for trestles 183 

piles 161 

ties 241 

moduli of rupture — Table XX p. 220 

strength of 187 

working unit stresses — Table XXI p. 221 

Topographical maps, use of, in reconnoissance 6 

Track bolts, average number in a keg of 200 pounds 447, d 

cost 447, d 

design 294 

essential requirements 293 

for various weights of rail — Table XXXII p. 496 

number required per mile — Table XXXV 447, d 

circuit for automatic signaling 394 

laying on railroads, cost 447, e 

scales 383 

Tractive power of locomotives — Tables XXXIX and XLIII and 411 

effect of grade 461 

relation to boiler and cylinder power. . . 414 

variation with velocity 457, 460 

Traffic, classification of 498 

estimation of probable volume 473 

TRAIN-BRAKES . 421-427 

automatic 425 

brake*shoes 427 

general principles 421, 422 

hand-brakes 423 

straight air-brakes 424 

tests for efficiency • • • • 426 

Train length limited by curvature 509, a 

maximum on any grade 521 

loads, methods of increasing 532, 6, 535 et seq. 

TRAIN RESISTANCE —Chap. XVI. 

formulae for 438-439 

Train service, cost of, 492, and Table XLI. 

supplies and expenses, cost of, in conducting transportation — 

493 and Table XLI 
wages — see Train service. 

Transfer cranes in freight yards 382 

Transit, adjustment of — Appendix pp. 597-601 



INDEX. 829 

Transition-curves 71-83 

Table IV pp. 621-623 

application to compound curves 82 

field-work 80 

fundamental principle 74 

replacing a compound curve by curves with spirals . 83 

simple curve by a curve with spirals. ... 81 

required length 76 

symbols 77 

their relation to tangents and simple curves 79 

to find the deflections from any point , 78 

ordinates 78 

use of Table IV 78 

varieties 75 

from level to inclined track 73 

Transportation, surveying parties 34 

TRESTLES.— Chap. IV. 

cost 184 

extent of use 158 

framed 169-184 

pile 161-168 

posts, design ' 191 

required elenients of strength 186 

sills, design 192 

stringers, design 190 

timber 183, 187 

vs. embankments 159 

Trimming cuts to proper cross-section 144 

Trucks, car 418 

four-wheeled, action on curves 396 

locomotive pilot 399 

with shifting center 399 

TUNNELS.— Chap. V. 

cost 209 

vs. open cuts 208 

Tunnel cross-sections 198 

design 198-202 

drains 202 

enlargement 204 

grade 199 

headings 203 

lining 200 

portals 207 

shafts 195, 201 

spirals 5 

Turnout, connecting curve from a straight track 308 

from a curved tract to the outside 309 

to the inside 310 

dimensions, development of approximate rule 306 

from inner side of curved track 307 

from outer side of curved track 306 



830 INDEX. 

Turnouts with straight point rails and straight frog rails, dimensions 

of— Table III „ pp. 619, 620 

Turntables for locomotives 292 

Underground surveys in tunnels 196 

Undulatory grades, advantages, disadvantages, and safe limits 518 

Unit chord, simple curves 46 

Useful formulae and constants — Table XV p. 787 

trigonometrical formulae — Table XIV pp. 785, 786 

Valley route — reconnoissance 3 

Velocity head applied to theory of motion of trains 514 

as applied to determination of train resistance 437 

of trains — Table XLII p. 570 

Velocity of trains, method of obtaining 535 

resistances, train 430 

Ventilation of a tunnel during construction , 206 

Vertex inaccessible, curve location 62, a 

of a curve 51 

Vertical curves, mathematical form 86 

necessity for use 84 

numerical example 87 

required length 85 

Virtual grade, reduction of 535-537 

profile, construction of 515 

use, value, and possible misuse 517 

Von Borrie's formula for train resistance.. 438 

Wages of engine-men 490 

trackmen 486 

trainmen 492 

Wagons, use in hauling earthwork 140, h 

Water for locomotives, chemical qualities 319 

consumption and cost 318, 325 

methods of purification 319-321 

reagents for removing corrosive or incrusting matter — Table 

XXVI p. 370 

stations and water supply 318-327 

location % 318 

pumping 325 

required qualities of water. 319 

stand-pipes 327 

tanks 324 

track tanks 326 

table in locomotive fire-box 403 

tanks for locomotives — 324 

capacity of cyUndrical tanks 324 

protection from freezing 324 

way for culverts 212-217 

Watering stock 469 

Wear of rails on curves 276 

on tangents 275 

Weight of rails, 267, 268, and Table XVI pp. 788-804 

Wellington's formula for train resistance 438 

Westinghouse air-brakes 425 



f 



INDEX. 831 

Wheelbarrows, use in hauling earthwork 140, c 

Wheel-bases of locomotives, types 400 

Wheel resistances, train 431 

Wheels and rails, mutual action and reaction 395-399 

effect of rigidly attaching them to axles 395 

White oak, use for trestles 161, 183, 187 

ties 242 

Wire-drawn steam 410 

Wires and pipes, used in block signaling 393 

Wolff's formula for train resistance 438 

Wooden box culverts 222 

spikes, for filling spike holes 246, 292 

Wounds, treatment 45 

Yard-engine expenses 489 

YARDS AND TERMINALS.— Chap. XIII. 

Yards, engine 384 

freight 378-383 

general principles 378 

hump 379 

minor 381 

relation to main tracks 380 

transfer cranes 382 

track scales 383 

value of proper design 376 

Zinc-creosote, emulsion process 254 

two-injection process 255 

-tannin process for preserving timber 253 



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